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El artículo presenta los resultados de una investigación en la asignatura de matemática, realizada con estudiantes de tercer año de secundaria en liceos municipalizados de la comuna de Talca. A partir de contenidos matemáticos específicos, se analiza el perfil inicial de los estudiantes, las capacidades que desarrollan y el cambio en las concepciones matemáticas cuando se enfrentan a procesos de modelización. Siguiendo una metodología de corte cualitativa y cuantitativa, se diseñó un plan de análisis, que permitió un estudio pormenorizado de las producciones del grupo objeto de experimentación. A nivel de conclusiones se destacan dificultades y obstáculos detectados en el trabajo con problemas en el pretest. En contraste, el postest muestra que estas dificultades pueden ser reguladas cuando se relaciona la matemática con las distintas áreas del saber y con la vida cotidiana.
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The article presents the findings in the subject of study of mathematics, accomplished with students of third year of municipal secondary schools of the Talca commune. In relation to specific mathematical contents, the initial profile of the students, the skills that they develop and the change in their math conceptions are analyzed, when they are faced to processes of modelling. Following a methodology of quantitative and qualitative modalities, a plan of analysis that enabled a detailed study of the productions of the target group of experimentation was designed. To the level of conclusions, difficulties and obstacles are noticed in the pretest in the work with problems. It contrast with the posttest, that shows that these obstacles can be regulated when Mathematics is related with the different areas of knowledge and with the everyday life. | 677.169 | 1 |
About
the Post CALC Project
The Post CALC Project presents
interactive, mathematically-based modules designed for high school students
who have finished a year-long course in calculus.
These
modules, each appropriate for several weeks work by individuals or small groups
of students, illustrate the use of calculus in a wide range of applications.
These materials have a structure
similar to the modules in the Duke Connected
Curriculum Project (CCP). The HTML pages contain discussion, instructions,
and simple Java-based interactions together with a downloadable computer algebra
system worksheet for student exploration and reporting. However, the CCP modules
are appropriate for approximately one class period of in-class work with some
guidance from an in-class instructor. While the Post CALC materials can be used
in a similar mode, they are designed for independent use. There is much more
explanatory detail and the expectation is that students will work on each module
for one
to many weeks.
Duke University has provided technical
and mathematical support to schools participating in the PostCALC Project.
The NSF grant supporting this project has expired. However, we are still
maintaining the site and will support users to the best of our ability. If
you would like more information about using these materials, send an email
message to Jim Tomberg at tomberg@math.duke.edu.
Click here
to see pictures of some Chapel Hill students combining their interests in music
and mathematics as they work through the Post CALC module on Fourier Approximations. | 677.169 | 1 |
At its heart, algebra is about symmetries, of geometric objects or of roots of equations.
The history of algebra starts with the quadratic formula,
firstIn the second semester, weWe will also study elliptic curves, which are curves having a cubic equation.
What is special about cubic curves is that there is a group law on the points on its graph.
Because of this group law, some points on the graph have special properties.
This is an inspiring topic, since elliptic curves played a crucial role in the proof of Fermat's Last Theorem and
are used ubiquitously in cryptography.
Grading:
The course grades will be computed as follows:
20% homework; 20% first midterm; 20% second midterm; 15% Project and presentation; 25% Final.
Borderline grades will be decided on the basis of class participation.
Homework: Due every Wednesday Monday 10/4 and Wednesday 11/17.
The project is due Monday 12/3. The presentations will be in class between 12/6 and 12/10.
The final exam is Monday 12/13, 1:30-3:30 TBA in Weber 118. | 677.169 | 1 |
First Course in Differential Equations
ISBN-10: 0387259643
ISBN-13: 9780387259642 standard sophomore course on elementary differential equations is typically one semester in length, most of the texts currently being used for these courses have evolved into calculus-like presentations that include a large collection of methods and applications, packaged with state-of-the-art color graphics, student solution manuals, the latest fonts, marginal notes, and web-based supplements. All of this adds up to several hundred pages of text and can be very expensive. Many students do not have the time or desire to read voluminous texts and explore internet supplements. Thats what makes the format of this differential equations book unique. It is a one-semester, brief treatment of the basic ideas, models, and solution methods. Its limited coverage places it somewhere between an outline and a detailed textbook. The author writes concisely, to the point, and in plain language. Many worked examples and exercises are included. A student who works through this primer will have the tools to go to the next level in applying ODEs to problems in engineering, science, and applied mathematics. It will also give instructors, who want more concise coverage, an alternative to existing texts.This text also encourages students to use a computer algebra system to solve problems numerically. It can be stated with certainty that the numerical solution of differential equations is a central activity in science and engineering, and it is absolutely necessary to teach students scientific computation as early as possible. Templates of MATLAB programs that solve differential equations are given in an appendix. Maple and Mathematica commands are given as well. The author taught this material on several ocassions to students who have had a standard three-semester calculus sequence. It has been well received by many students who appreciated having a small, definitive parcel of material to learn. Moreover, this text gives students the opportunity to start reading mathematics at a slightly higher level than experienced in pre-calculus and calculus; not every small detail is included. Therefore the book can be a bridge in their progress to study more advanced material at the junior-senior level, where books leave a lot to the reader and are not packaged with elementary formats.J. David Logan is Professor of Mathematics at the University of Nebraska, Lincoln. He is the author of another recent undergraduate textbook, Applied Partial Differential Equations, 2nd Edition (Springer 2004 | 677.169 | 1 |
You Will Not Learn Math Without Practicing. Watching The Lesson Videos Are Not Enough!
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Taking my free GED math course is a big step to help you prepare for the GED. The course will help you learn core middle and high school math skills, but you also need to do your part and practice! As such, I created a workbook for this course that is designed to give students practice problems and solutions to strengthen their skills- it includes access to an additional 234 great math videos perfect for GED Math skills! In order to truly master math concepts you must practice a wide variety of problems. This workbook is an extremely valuable supplement that all students using GED Math Lessons should have.
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This Package Deal Also Comes With The eBook You Need To Learn The Most Effective Way To Study For The GED – So You Can Pass ASAP! | 677.169 | 1 |
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This is a practice worksheet that contains the following topics:
Using graphs to relate to quantities
Patterns and linear functions
Patterns and NON-linear functions
Writing a function rule
Formalizing Relationships and Functions
Sequences and Functions
This is a great activity for classroom practice, homework, or test prep. Enjoy! | 677.169 | 1 |
Watering Down the Common Core Standards (I)
There is hope – at least in some quarters – that the Common Core State Standards (CCSS) will bring an improvement in math education. According to the CCSS website, the new standards "[i]nclude rigorous content and application of knowledge through high-order skills" and "[a]re informed by other top performing countries, so that all students are prepared to succeed in our global economy and society". It sounds exciting and indeed, when one reads the standards, one does see an attempt to connect concepts and to ask students to think rather than memorize a set of unrelated facts.
At our last staff meeting, I was understandably excited to see for my first time a book that claims to be aligned to the common core state standards (Algebra 2 – Glencoe/McGraw Hill). I reviewed the book for the last three days and …what a huge disappointment!
A serious concern is that, in this book, some of the CCSS standards are missing either in body or in spirit. Here is an example. F-LE1 wants students to "[d]istinguish between situations that can be modeled with linear functions and with exponential functions". The key in this standard is to distinguish between quantities that grow by equal differences over equal intervals and those that change at a constant rate per unit interval. In my opinion this is an important concept – it goes to the fundamental idea that functions model real phenomena in nature and that these phenomena represent various rates of change – linear, exponential (both growth and decay), zero and so on.
Not only is F-LE1 not in the book's index of standards, but the concept of constant rate growth is stressed in the section on series, not in that on exponential functions. There is no connection established in the book between exponential functions and geometric series and the two are separated by 3 chapters (1 – 2 months worth of school days) so the idea of exponential growth is likely forgotten by the time students get to geometric series.
Here is another example. Standard A-CED.4 wants students to "[r]earrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R." Again this is an important concept, especially for those who later go into engineering, math or the sciences. In my experience students have difficulty with this – they are not trained in middle school to manipulate abstract variables.
Standard A-CED4 appears twice in the book's index of standards. The first time it appears in the section introducing circles, but the rearranging of formulas here is not in the same spirit as Ohm's law example – it has more to do with completing the square. The second time the standard appears is in the discussion of geometric sequences and series and again I fail to see how this standard is applied in the spirit it was written. Therefore, for all practical purposes, the standard is missing.
It looks like the standards were fitted to the book, rather than the book written around the standards. Perhaps this is not surprising, given that 7 authors of the previous book's edition (non-CCSS) are also the authors of the new one. From a business point of view it's good to be early in the market in response to a market need – unfortunately this product leaves a lot to be desired.
In my next post I will address the rigor of the book and how it compares to other texts (domestic and international). For now, I am concerned that standards which aim to be rigorous and indeed world-class will be watered down by textbook publishers and possibly by those that write the state exams. | 677.169 | 1 |
Helping Students Connect Functions and Their Representations
Deborah Moore-Russo, John Golzy
A teaching method to help promote deeper understanding of both the graphical and algebraic representations of linear and quadratic functions. The authors ask students to find the graphical representation of the sum and product of given functions Various representations lead to a deeper understanding of the connections between the equation and its graph. Graphing calculators are utilized to enhance student understanding.
This is available to members of NCTM. If you are interested in a NCTM membership, join now. You may also purchase this article now for online access. | 677.169 | 1 |
COURSE DESCRIPTION: Business mathematical exercises and problems. Units of study will include decimals, fractions, banking, payroll records, taxes, percents in business, commission, discounts and markup. Other areas of study will include credit interest, installment buying, and depreciation.
PREREQUISITES:
NOTES:
STUDENT LEARNING OUTCOMES: Upon completion of this course the student will: | 677.169 | 1 |
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Introduction to Differentiable Manifolds
Finally, here are a couple of books recommendations from introductory ones to ones which describe applications of differential geometry. Text at the level of Riemannian Geometry of do Carmo's or Gallot-Hulin-Lafontaine. Differential geometry begins by examining curves and surfaces, and the extend to which they are curved. It was used by Jessica Kwasnica to create an Anamorphic Giraffe and by Joey Rollo to create an Anamorphic Elephant.
Pages: 126
Publisher: John Wiley & Sons, Inc; 1st edition (1962)
ISBN: B001NS16R4
Differential Geometry of Manifolds
Symmetries (Springer Undergraduate Mathematics Series)
An Introduction to Teichmüller Spaces
Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations (Memoirs of the American Mathematical Society)
Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport online. JDG was founded by the late Professor C.-C. Hsiung in 1967, and is owned by Lehigh University, Bethlehem, PA, U. The Journal of Differential Geometry is published at Lehigh University. Call 610-758-3726 to speak to the managing editor Professor Huai-Dong Cao Spinor Structures in Geometry and Physics. It is that part of geometry which is treated with the help of continuously and it is achieved by the use of differential calculus. There are two branches Another definition of space curve: A space curve can also be defined as the intersection of two surfaces viz., When a straight line intersects a surface in k points, we say that the surface is of degree k Lectures on Fibre Bundles and Differential Geometry (Tata Institute Lectures on Mathematics and Physics).
The Index Theorem and the Heat Equation Method (Nankai Tracts in Mathematics)
Offbeat Integral Geometry on Symmetric Spaces
Differential geometry and topology (Notes on mathematics and its applications)
Compact Lie Groups: An Introduction to Their Representation Theory and Their Differential Geometry
Coordinates in Geodesy
Click on any part of the photo of Bill Gates, hold the left mouse button down, then drag it to "warp" the photo into a topologically equivalent distortion Lectures on Fibre Bundles and Differential Geometry (Tata Institute Lectures on Mathematics and Physics). Informal Notes for the The elegant solution of this problem uses the dual ring of differential operators,. tensors geometry and applications 44 (1987), 265-282. 7 epub. You absolutely need such a book to really understand general relativity, string theory etc Introduction to Differentiable Manifolds online. Our course descriptions can be found at: My research interests are in computational algebra and geometry, with special focus on algorithmic real algebraic geometry and topology Geometric Methods in PDE's (Springer INdAM Series). It is part of the trimester programme on Topology at the Hausdorff Institute for Mathematics running from September-December, 2016. This workshop will explore topological properties of random and quasi-random phenomena in physical systems, stochastic simulations/processes, as well as optimization algorithms. Practitioners in these fields have written a great deal of simulation code to help understand the configurations and scaling limits of both the physically observed and computational phenomena Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218). It uses curvature to distinguish straight lines from circles, and measures symmetries of spaces in terms of Lie groups, named after the famous Norwegian mathematician Sophus Lie. Topology, in contrast, is the study of qualitative properties of spaces that are preserved under continuous deformations. The spaces in question can be tame like a smooth manifold, or wild and hard as rock Seminar On Minimal Submanifolds - Annals Of Mathematics Studies, Number 103. The module Lie groups is based on the analysis of manifolds and therefore should be completed (if possible immediately) after it pdf. The conference is supported by the Journal of Differential Geometry and Lehigh University, and NSF. Limited travel support is available, and the priority will be given to recent PhD's, current graduate students and members of underrepresented groups. Participants interested in being considered for this support should complete the request for travel support form no later than May 13, 2016. More information will be available at this site as it becomes available Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations (Reviews in Mathematics and Mathematical Physics). Differential topology - Congresses, Discrete Geometry - Congresses, Geometry - Data Processing - Congresses, Geometry, Differential The aim of this volume is to give an introduction and overview to differential topology, differential geometry and computational geometry with an emphasis on some interconnections between these three domains of mathematics download Introduction to Differentiable Manifolds pdf. In such a case you must rotate them to be parallel, because no matter what the metric is or how it weights various directions, if the vectors are parallel then the weighting will be the same for both of them, there's no unfair bias Encyclopedia of Distances. Our aim is to provide an opportunity for both experts and young researchers to discuss their results and to start new collaboration download. | 677.169 | 1 |
Introduction To Differentiable Manifolds 1ST Edition
It was one of the two fields of pre-modern mathematics, the other being the study of numbers. The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;[3] Euclid himself wrote eight more advanced books on geometry. A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid ), as well as twa divergin ultraparallel lines.
Pages: 0
Publisher: Interscience Publishers; 1St Edition edition (1966)
ISBN: B0063BIG3S
Projective Differential Geometry of Submanifolds
Curvature and Betti Numbers. (AM-32) (Annals of Mathematics Studies)
Nonlinear Waves and Solitons on Contours and Closed Surfaces (Springer Series in Synergetics)
Complex Surfaces theory gives you where you will be working and Complex Geometry techniques that are more or less Algebraic Geometry gives you the tools to understand 'geometrical' understanding of your topological operations. (blowing up the points, Bezout's theorem for finding intersections, Riemann-Hurwitz formula for finding degree of ramification divisors and so on) Geometries in Interaction: GAFA special issue in honor of Mikhail Gromov ferienwohnung-roseneck-baabe.de. The many related strands of research in DDG have been demonstrated at the series of Oberwolfach workshops in the area (2006, 2009, 2012) and in two new books on the subject pdf. The wide variety of topics covered make this volume suitable for graduate students and researchers interested in differential geometry. The subjects covered include minimal and constant-mean-curvature submanifolds, Lagrangian geometry, and more. This book provides full details of a complete proof of the Poincare Conjecture following Grigory Perelman's preprints NON-RIEMANNIAN GEOMETRY. marcustorresdesign.com. By the way, the only thing the reader learns about what an 'open set' is, is that it contains none of its boundary points. All the topology books I have read define open sets to be those in the topology. This is another point of confusion for the reader. In fact, points of confusion abound in that portion of the book. 2) On page, 17, trying somewhat haphazardly to explain the concept of a neighborhood, the author defines N as "N := {N(x) Great care has been taken to make it accessible to beginners, but even the most seasoned reader will find stimulating reading here ... The appeal of the book is due first of all to its choice of material, which is guided by the liveliest geometric curiosity pdf. When) Riemannian Geometry (Philosophie Und Wissenschaft) Riemannian Geometry (Philosophie Und.
Singular learning theory draws from techniques in algebraic geometry to generalize the Bayesian Information Criterion (BIC) to a much wider set of models. (BIC is a model selection criterion used in machine-learning and statistics.) (1) This leads to longer battery life. (2) The proofs are complex but geometric intuition can be used to explain some of the key ideas, as explained here by Tao D-Modules and Microlocal Calculus (Translations of Mathematical Monographs, Vol. 217) download epub. The theorem of Gauss–Bonnet now tells us that we can determine the total curvature by counting vertices, edges and triangles. In the last sections of this book we want to study global properties of surfaces Lectures on Seiberg-Witten download epub This is well-known for gauge theory, but it also applies to quaternionic geometry and exotic holonomy, which are of increasing interest in string theory via D-branes. Kotschick: On products of harmonic forms Duke Math. Kotschick: Cycles, submanifolds, and structures on normal bundles, Manuscripta math. 108 (2002), 483--494. Terzic: On formality of generalised symmetric spaces, Math Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics) download epub.
The primary target audience is sophomore level undergraduates enrolled in a course in vector calculus. Later chapters will be of interest to advanced undergraduate and beginning graduate students , e.g. Regularity of Minimal Surfaces read online Regularity of Minimal Surfaces Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310) The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BCE , e.g. Clifford Algebras and Lie Theory (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 3. Folge a) tiny-themovie.com. Like Renaissance artists, Desargues freely admitted the point at infinity into his demonstrations and showed that every set of parallel lines in a scene (apart from those parallel to the sides of the canvas) should project as converging bundles at some point on the "line at infinity" (the horizon). With the addition of points at infinity to the Euclidean plane, Desargues could frame all his propositions about straight lines without excepting parallel ones—which, like the others, now met one another, although not before "infinity." A farther-reaching matter arising from artistic perspective was the relation between projections of the same object from different points of view and different positions of the canvas Theorems on Regularity and Singularity of Energy Minimizing Maps (Lectures in Mathematics. ETH Zürich) tiny-themovie.com. Some of this material has also appeared at SGP Graduate schools and a course at SIGGRAPH 2013. Peter Schröder, Max Wardetzky, and Clarisse Weischedel provided invaluable feedback for the first draft of many of these notes; Mathieu Desbrun, Fernando de Goes, Peter Schröder, and Corentin Wallez provided extensive feedback on the SIGGRAPH 2013 revision , e.g. Introduction to Differential read here The de Rham cohomology of a manifold is the subject of Chapter 6 The Real Fatou Conjecture read online The Real Fatou Conjecture.
Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7-14, 1997 (Progress in Mathematics)
The topology of fibre bundles (Princeton mathematical series)
Complex Geometry (Lecture Notes in Pure and Applied Mathematics)
Riemannian Geometry (Graduate Texts in Mathematics, Vol. 171)
Mathematical Adventures in Performance Analysis: From Storage Systems, Through Airplane Boarding, to Express Line Queues (Modeling and Simulation in Science, Engineering and Technology)
The elementary differential geometry of plane curves (Volume 2)
Geometry and Differential Geometry: Proceedings of a Conference Held at the University of Haifa, Israel, March 18-23, 1979
Topological Quantum Field Theory and Four Manifolds (Mathematical Physics Studies)
Geometry Part 2 (Quickstudy: Academic)
Probably I will end up with my own notes extracted from different sources. Nevertheless, I have found the following books, and some of them seem to be useful for learning (from easiest to hardest): C A Differential Approach to download online tiny-themovie.com. In addition to the combination of all your previous mathematical learning, differential geometry moves everything into three-dimensional world and brings it to life through equations. If your differential geometry assignment has you stressed out – fading into the two dimensional planes of your textbooks, you can get differential geometry help to assist you in completing all your assignments successfully Introductory Differential read pdf Introductory Differential Geometry For. Click on Secret for the solution and the link to a Print & Play version of the postcard for practice. This ancient puzzle is easy to make and uses inexpensive materials. Available commercially under a variety of names, such as Two Bead Puzzle and Yoke Puzzle ref.: Advances in Geometry download here Archimedes also came West in the 12th century, in Latin translations from Greek and Arabic sources. Apollonius arrived only by bits and pieces. Ptolemy's Almagest appeared in Latin manuscript in 1175. Not until the humanists of the Renaissance turned their classical learning to mathematics, however, did the Greeks come out in standard printed editions in both Latin and Greek , cited: Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods (Applied and Numerical Harmonic Analysis) read online. I will try to post there as often as possible The Mathematical Works Of J. download pdf A comprehensive textbook on all basic structures from the theory of jets. It begins with an introduction to differential geometry. After reduction each problem to a finite order setting, the remaining discussion is based on properties of jet spaces. This book provides a route for graduate students and researchers to contemplate the frontiers of contemporary research in projective geometry Analytic Geometry For example, the graphs of maps from a circle to itself lie on the surface of a torus (which is topologically the product space the same number of times; then they have the same degree ref.: Elementary Topics in Differential Geometry Guillermo Peñafort Sanchis is a Spanish PhD student. He obtained his master's degree from Universitat de València and has recently submmited his PhD Thesis, supervised by Juan José Nuño Ballesteros (Valencia) and Washington Luiz Marar (Universidade de São Paulo) Differential Manifolds (Dover read epub read epub. See the chapter on We also note that if the curve is a helix, which the helix is drawn, and rectifying developable is the cylinder itself Symposium on the Differential read for free tiny-themovie.com. American mathematician Edward Kasner found it easier to teach topology to kids than to grownups because "kids haven't been brain-washed by geometry". Two figures are said to be topologically equivalent if one can be transformed into the same shape as the other without connecting or disconnecting any points. Distorted as viewed in a fun-house mirror, Jill Britton's face is topologically equivalent to its rippling counterpart: a single point and its neighbourhood on one correspond to a single point and its neighbourhood on the other Differential and Riemannian Geometry read pdf. I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now Pfaffian Systems, k-Symplectic Systems | 677.169 | 1 |
Description
For physicists, mechanics is quite obviously geometric, yet the classical approach typically emphasizes abstract, mathematical formalism. Setting out to make mechanics both accessible and interesting for non mathematicians, Richard Talman uses geometric methods to reveal qualitative aspects of the theory. He introduces concepts from differential geometry, differential forms, and tensor analysis, then applies them to areas of classical mechanics as well as other areas of physics, including optics, crystal diffraction, electromagnetism, relativity, and quantum mechanics.For easy reference, the author treats Lagrangian, Hamiltonian, and Newtonian mechanics separately exploring their geometric structure through vector fields, symplectic geometry, and gauge invariance respectively. Practical perturbative methods of approximation are also developed. This second, fully revised edition has been expanded to include new chapters on electromagnetic theory, general relativity, and string theory. "Geometric Mechanics" features illustrative examples and assumes only basic knowledge of Lagrangian mechanics.show more
Author information
Richard M. Talman is Professor of Physics at Cornell University, Ithaca, New York. He studied physics at the University of Western Ontario and received his Ph.D. at the California Institute of Technology in 1963. After accepting a full professorship for Physics at Cornell in 1971, he spent time as Visiting Scientist in Stanford, CERN, Berkeley, and the S.S.C. in Dallas and Saskatchewan. In addition, he has delivered lecture series at several institutions including Rice and Yale Universities. Professor Talman has been engaged in the design and construction of a series of accelerators, with special emphasis on x-rays.show more | 677.169 | 1 |
Waner and Costenoble's FINITE MATHEMATICS AND APPLIED CALCULUS, Seventh Edition, helps your students see the relevance of mathematics in their lives. A large number of the applications are based on real, referenced data from business, economics, and the life and social sciences. Spreadsheet and TI Graphing Calculator instruction appears throughout the text, supplemented by the WebAssign course, and an acclaimed author website. The end-of-chapter Technology Notes and Technology Guides are optional, allowing you to include in your course precisely the amount of technology instruction you choose. Praised for its accuracy and readability, FINITE MATHEMATICS AND APPLIED CALCULUS is perfect for all types of teaching and learning styles and supportWhat's New
AN UNSURPASSED COLLECTION OF EXERCISES at all difficulty levels, and exercises based on real, referenced data on topics that students relate to -- including social media, the 2008 economic crisis and the 2009-2016 economic recovery, the 2014 Ebola epidemic, the SARS outbreak of 2003, the 2010 stock market "flash crash" and many others. The inside back cover lists over 90 corporations referenced in the applications.
MANY NEW CONCEPTUAL COMMUNICATION AND REASONING EXERCISES, including many dealing with common student errors and misconceptions, have been added.
LOGARITHMS ARE NOW DISCUSSED IN THE PRECALCULUS REVIEW CHAPTER, up through solving for unknowns in the exponent. Students who need additional preparation in the basis of logarithms can now be assigned this material before studying the section on logarithmic functions and models in Chapter 2. This also makes it easier for instructors who wish to use logarithms in discussions of exponential functions and the mathematics of finance.
The chapter on nonlinear functions and models has been moved to appear earlier in the book: It is now Chapter 2 rather than Chapter 9. Although this material is not required for the finite mathematics chapters, it fits logically with Chapter 1, which discusses functions in general and linear models, and many instructors prefer to cover this material earlier rather than later.
Chapter 3, on the mathematics of finance, has been substantially revised, specifically the sections on simple and compound interest and annuities and the exercise sets.
CASE STUDIESFeatures
COMMUNICATION AND REASONING EXERCISES FOR WRITING AND DISCUSSIONAUTHOR WEBSITE: The authors' website gives instructors powerful online tools that can be used in the classroom to do everything from graphing and evaluating functions to displaying and computing integrals and Riemann sums and solving linear programming problems both graphically and with the simplex method. Instructors often use the online interactive tutorials as teaching tools that invite class participation, and the randomized "game tutorials" may be used for in-class quizzes that teach as they test. An interactive e-book provides exercises and topics not in the printed book.QUICK EXAMPLES: Most definition boxes include quick, straightforward examples that students can use to solidify their understanding of each new conceptMARGINAL TECHNOLOGY NOTES AND END-OF-CHAPTER TECHNOLOGY GUIDES required appear throughout the exercise sets. | 677.169 | 1 |
My Assignment Help
Calculus Homework Help
Facing problem with Calculus assignment?
Calculus is the main portion of mathematics that can be defined as the study of change in the form of derivatives and integration. This subject have basically two main divisions, the differential calculus and the integral portion.
Understanding the basic of calculus is most to have a good grasp of this topic. But before going for calculus students need to be sure that they knows algebra and geometry pretty well. It is because the combined knowledge of both this subject is important to understand the basics of calculus.
Though it seems that use of calculus is tightened basically with mathematics but this is not the case; rather calculus is used in different fields of physics, chemistry, mathematics and engineering.
Major topics under calculus
You may find derivations and list of formulas in calculus uninteresting and useless to be applied in any career oriented field. You might think that these bunch of equations are only concise within your textbook. But mathematics is also a part of science and when science encloses real facts, mathematics can have no topic that is senseless for students. Calculus denotes change and hence, this change is applied in existing world including your career field. There are mainly two section enclosed in calculus:
Derivatives: It is the rate of change of function at precise value of x. This can be better understood with the example below:
Question: g(x) = 4-16x
Solution: From the definition of derivative and using the basic formula of derivative:
Integral: It is the other important section of calculus that can be inferred as an area. Example below illustrate the use of calculus:
Therefore, these equations seems easy but they can be complicated at times. Getting a good grasp in calculus is Important and this can be achieved by constant practice.
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Mathematical Literacy and College Mathematics Education
Abstract:
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Mathematics, being an important part of human culture, plays a significant role in both spiritual life and material life of people. Mathematical literacy is one of the most basic scientific literacies, which is of far-reaching importance to cultivate mathematical literacy of students in college mathematics education. This paper elaborates the contents of mathematical literacy, discusses the necessity of cultivating mathematical literacy of college students, and proposes effective approaches of improving the literacy of college students. | 677.169 | 1 |
25, 2012
The mini-lessons presented in this book are based on a general algebra curriculum. Many of the mini-lessons in Sections 1 and 2 focus on prerequisite skills that students must master if they are to succeed in algebra. Each mini-lesson, consisting of teaching notes and a reproducible worksheet, concentrates on a specific algebraic concept or skill students often have trouble mastering. Each mini-lesson requires only a few minutes to deliver and can be used with individual students, groups, or the whole class. | 677.169 | 1 |
Higher Level Mathematical Reasoning
Mathematics presents students with a mode of thinking/reasoning. It includes observation, attention to detail, analysis, synthesis, relevant question asking, and problem solving. It involves some valuable traits like the ability to handle sweat, frustration, dead ends, perseverance, and the discovery that there is wonder, joy, and even some exhilaration at the end. We invite students deeper into or higher up this mode of reasoning year-by-year, subject-by-subject. So what is higher level mathematical reasoning?
A look at some of the approved and adopted texts suggests that a typical answer is, "algebra." Algebra is generally considered to be higher level math thinking for today's school students. Constants, variables, coefficients, expressions, equations, quadratic equations, real, rational, and irrational numbers, and combining like terms… If we can just get upper and even lower elementary students to start thinking about some of this, we believe that there is more of an opportunity for higher level math reasoning.
But what about geometry students who have already passed Algebra I, but still have not mastered basic number sense concepts involving fractions? For example, I tutored a high school geometry student recently who did not realize that if amount A is half as much as amount B, then amount B must be twice as much as amount A. This student had memorized the formula for determining the measure of an inscribed angle (it is 1/2 the measure of its intercepted arc), and had solved many problems correctly. But when asked to find the measure of the arc when given the measure of the angle, the student was stumped. It seems that for this student, thinking about basic fractional relationships was actually higher level mathematical reasoning—higher than the current level of understanding.
Higher level math reasoning for students is simply whatever the next step is from where they are now. The relationship between 1/2 and twice, or that a group can be both one and many, or that a "1" sitting in the tens column has a different value than a "1" in the ones column are all higher level math thinking for students who do not yet understand those concepts. People generally consider algebra more abstract than arithmetic, because it appears to be less concrete—and therefore it must be the flagship of "higher level mathematical reasoning." But any concept is "abstract" to the student who does not understand it yet!
The critical element is not the level of difficulty of the work, but whether or not the work is being addressed through reasoning. Students who can factor quadratic equations because they have memorized a bunch of rules cannot be said to be applying higher level mathematical reasoning, unless they actually understand why they are doing what they are doing. There is a big difference between "higher level activities" and "higher level mathematical reasoning." When higher level activities are taught through mere memorization or repetitive activities devoid of real understanding, they do not involve any reasoning at all. When lower level activities are taught in ways that make students really think, then those students are involved with higher level mathematical reasoning. And math teaching need not hang its head and feel inferior to other academic disciplines while focusing on these lower level activities.
Algebra is not the problem in itself. Thinking that it accomplishes the need for higher level reasoning and application is.
Another unfortunate answer to what is higher level mathematical reasoning can be seen in the rush to complicate problem sets in textbooks. The geometry book that the student I tutor is using in school, published by a major publisher and state adopted, has outstanding higher level math reasoning problems to solve. I'm having as much fun with some of them as I'm sure that authors and state committee members had. But my student and many in her class are not. There are precious few problems in any section of this book that allow students to develop a confident understanding of the basic concepts and procedures before "higher level math reasoning" is introduced in the form of clever and complicated levels of application.
Rather than leaping to higher level activities that require fluent reasoning that has not yet been developed, the interests of students would be better served if this book (and others like it) presented step-by-step contexts of problems of graduated difficulty—each problem based on the reasoning developed in the previous problem, and preparing students for the next step of reasoning represented in the following problem. The proper function of a math book is to develop mathematical reasoning, not merely to create problems that require its use. By rushing to over-complicate the problems, textbooks unwittingly exclude many students from success, actually thwarting the development of their reasoning and forcing them to rely on mere memorization to cope with their work.
Yes we need to keep earlier concepts and procedures alive by integrating them into problems in subsequent chapters, and yes students need to explore multiple uses and applications, and yes they need to use all of this to solve mathematical problems and not merely perform arithmetic calculations. I am not arguing against any of this. But enrichment is enriching and higher level mathematical reasoning is only reasoning when students have access to it. We should take as much pride in opening up and developing that next level of higher mathematical reasoning, whatever it may be, as we do in the creative, clever, complicated, and fun problems our mathematical minds conceive. We should remember what it's like for those who are new to all of this. What is higher level mathematical reasoning for them? | 677.169 | 1 |
Fast Track
Fast Track Math Assessment Prep Workshop
This 8-day workshop is designed to provide students with an intensive review of elementary
and intermediate algebra. Instruction on important algebra topics, one-on-one help,
and workbooks with extra practice are all provided. This workshop prepares qualified
students to potentially place into a higher-level math course, getting them on track
to complete their coursework sooner. Session: June 5 – 15 Monday – Thursday, 5:30 pm – 8 pm MA-105 – Instructor:Salvador Rico | 677.169 | 1 |
Discrete mathematics forms the mathematical foundation of computer and information science. It is also a fascinating subject in itself. Learners will become familiar with a broad range of mathematical objects like sets, functions, relations, graphs, that are omnipresent in computer science. Perhaps more importantly, they will reach a certain level of mathematical maturity - being able to understand formal statements and their proofs; coming up with rigorous proofs themselves; and coming up with interesting results.
Count | 677.169 | 1 |
How to Use a TI 84 Calculator
by Nicholas Smith
The TI 84 is an advanced graphic calculator. You can view equations and their graphs, along with coordinates on the device. The device contains a process which helps to quickly present results to you. The device also has the ability to connect to your computer for expanded capability. Available capabilities include advanced statistics, regression analysis, graphical and data analysis, among others.
Basic Functions
Power on the TI84 by pressing "ON," and power off the device by pressing "2nd," and "Off."
Adjust the screen's contrast by pressing "2nd" and "Up" or "Down."
Press the "Up" or "Down" buttons to display your previous instructions to display previously entered instructions or expressions evaluated by the device.
Configure the clock by pressing, "Mode," "Down" and browsing to "Set Clock." Press "Enter" and set the clock to desired settings.
Entering Expressions on the TI-84
Enter a single desired expression on the TI 84 keypad and press "Enter."
Enter two or more expressions on one line, separated by a colon. Access the colon by pressing "Alpha" and ":".
Enter a number in scientific notation. First, enter the portion of the number that precedes the exponent. Then, press "2nd," and "EE." Enter the exponent or one or two digits.
Set the TI 84 Mode (Controls Display of Numbers and Graphs)
Press "Mode" on the device's keypad.
Review the list of available modes. For example, the "Normal" setting displays results in "Numeric notation," while "Float" displays the number of decimal places in your answers.
References
About the Author
Nicholas Smith has written political articles for SmithonPolitics.com, "The Daily Californian" and other publications since 2004. He is a former commissioner with the city of Berkeley, Calif. He holds a Bachelor of Arts in political science from the University of California-Berkeley and a Juris Doctor from St. John's University School of Law. | 677.169 | 1 |
Copyrighted materials reproduced herein are used under the provisions of the Copyright Act 1968 as amended, or
as a result of application to the copyright owner.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any
means electronic, mechanical, photocopying, recording or otherwise without prior permission.
Produced by theDistance and e-Learning Centre using FrameMaker7.1 on a Pentium workstation.
TABLE OF
CONTENTS
PAGE
Introduction
1
A word about starting out
2
1.
Addition and subtraction
3
2.
Multiplication and division
7
3.
Brackets
9
4.
Powers
10
5.
Fractions
16
6.
Using the x-1 key
18
7.
Scientific notation
19
8.
Factorial x!
219.
Using memory
22
10.
Statistics
24
11.
Linear regression
30
12.
Trigonometric functions
33
13.
Exponential and logarithmic functions
35
14.
Degrees, minutes, seconds
37
Review calculator exercises
39
Calculator solutions
40
M astering the Calculator using the Casio f x -82TL
Introduction
This is one in a series ofbooklets prepared to assist students who are learning to use a
calculator. They have been prepared by staff in The Learning Centre from Learning and
Teaching Support Unit (LTSU) at USQ. The series comprises:
Mastering the calculator
•
•
•
•
•
•
•
•
Using the Casio fx-100s (also suitable for Casio fx-570)
Using the Casio fx-100AU
Using the Casio fx-82LB
Using the Casio fx-82TL
Using theCasio fx-82MS
Using the Sharp EL-531LH
Using the Sharp EL-556L
Using the Sharp EL-531RH
The instructions in this booklet only explain some of the keys available on your calculator
necessary for basic work in data manipulation. If you require more assistance please contact
The Learning Centre. If you would like information about other support services available
from The Learning Centre,please contact
The Learning Centre (TLC)
Learning and Teaching Support Unit (LTSU), S-Block
The University of Southern Queensland
Telephone: 07 4631 2751
Email: tlc@usq.edu.au
Fax: 07 4631 1801
Home page:
1
2
M astering the Calculator using the Casio f x -82TL
A word about starting out
• Make sure you are in the correct mode selection and that allprevious data is cleared.
• For e.g. To perform arithmetic operations press
• To clear all values press
• To clear memory press
• If your calculator has FIX or SCI on the display press
three times
appears on the screen
press 3 then 2 so you are in Normal mode.
• If your calculator has RAD or GRAD on the display press
appears on the screen
press 1 so you are in Degree mode.
twiceM astering the Calculator using the Casio f x -82TL
1.
Addition and subtraction
1.1 To add numbers
addition key
Find the
key
(it is shown on the photograph of the calculator here).
Example
To add 7 and 3, type
The display should read 10
Example
I want to find the total amount I earned in the past four weeks. If I earned $471, $575, $471
and $528, the key strokeswould be
The display should read 2045.
3
4
M astering the Calculator using the Casio f x -82TL
1.2 Sometimes you make an error when typing in a number
If this happens use the
key to cancel the number and then type in the correct number
and continue.
Example
If you want to enter 3 + 4 but accidentally type
press | 677.169 | 1 |
This article summarizes the research of Texas A&M faculty Robert M. Capraro and Mary Capraro comparing students' interpretation of the equal sign internationally and its relationship to achievement in mathematics. The page includes a 1-minute video of Robert Capraro discussing the importance of understanding the concept of equivalence and how misunderstandings arise.
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Description
This course gives an introduction to algebraic number theory. The main objects of study are number fields, i.e. finite extensions of the field of rational numbers. To a number field $K$ we will attach its ring of integers $\mathcal{O}_K$. The ring $\mathcal{O}_K$ is a Dedekind domain and we will see that one of its invariants is the class number $h_K$, which measures "how far" $\mathcal{O}_K$ is away from being a unique factorization domain.
We will also study finite extensions $K\subset L$ of number fields, and how the prime ideals behave in the associated extension $\mathcal{O}_K\subset \mathcal{O}_L$ of Dedekind domains.
Here is a rough outline of the course (subject to change): | 677.169 | 1 |
This book includes all 6 volumes of the Light and Matter series: Newtonian Physics, Conservation Laws, Vibrations and Waves, Electricity and Magnetism, and the Modern Revolution in Physics. This is an introductory text intended for a one-year…
This textbook is designed to college sophomores and juniors the basics of linear algebra and the techniques of formal mathematics. There are no prerequisites other than ordinary algebra. The text has two goals: to teach the fundamental concepts and…
This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The first three or four chapters can stand alone as a one semester course in abstract…
Elementary Algebra is a work text 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…
From the University of Florida Department of Mathematics, this is the third volume in a three volume presentation of calculus from a concepts perspective. The emphasis is on learning the concepts behind the theories, not the rote completion of…
From the University of Florida Department of Mathematics, this is the first volume in a three volume presentation of calculus from a concepts perspective. The emphasis is on learning the concepts behind the theories, not the rote completion | 677.169 | 1 |
Hop til / Skip to:
Optimal Stopping with Applications
Course content
The theory of optimal stopping is concerned with the problem of
choosing a time to take a particular action. Some applications
are:
The valuation/pricing of financial products/contracts
where the holder has the right to exercise the contract at
any time before the date of expiration is
equivalent to solving optimal stopping problems.
Examples:
1. American options in finance
2. Surrender options in life insurance
3. Prepayment of mortage loans
In financial engineering, where the problem is to
determine an optimal time to sell an asset. Examples
1. Optimal prediction problem, to sell the asset when the price is,
or close to, the ultimate maximum.
2. Mean-variance stopping problem, to sell the asset so as to
maximise the return and to minimise the risk.
The content of the course.
Optimal stopping:
Definitions
General theory
Methods of solutions
Areas of applications:
Pricing financial products with exercise feature in
mathematical finance and life insurance | 677.169 | 1 |
Big Idea:
This lesson builds on the "concept level" understanding that the students have obtained on "e" and extends that knowledge to skills practice and the natural logarithm function. It also focuses on mathematics specific readings strategies.
Big Idea:
This lesson builds nicely on the students' knowledge of inverse functions and introduces them to the world of logarithms. Rather than trying to memorize mathematical notation, the students are guided through an in-depth study under the umbrella of exponen | 677.169 | 1 |
This math book teaches Integers, a topic that is completely neglected in school
except some mention of "there are positive and negative numbers" and then discussion of the rules in operations with integers.
Fact 1: Many kids see their grades drop when they start studying Algebra.
Fact 2: Many kids see their grades drop, or drop more for some, when they start studying Calculus.
Fact 3: Those are not bad students but many are 'A' students.
The common explanation: Algebra and Calculus are conceptually hard (nonsense), and most students are just not naturally inclined to math. Sometimes the explanation is that "the student is not working hard enough." Not true in many cases.
The real explanation: Algebra and Calculus require understanding of numbers, operations, and our place value system. Not in a superficial and operational, can only use but doesn't understand, way. They require a deeper level of understanding. They require students to really understand numbers, what they represent, and how they work.
That's where Integers come into the picture.
Integers are a very important concept. They are not just "negative and positive numbers". An integer is a new idea compared to whole (positive) number (it's an idea about direction).
Fact 4: Schools and math books DO NOT TEACH Integers! None does. Students are told (or read) that "integers are just both positive and negative numbers". They they learn the Rules (e.g. "negative times negative is positive"), which don't really make sense to the poor students who were not taught what integers really are. Naturally, they forget those rules quickly.
Then comes Algebra, with its extensive need for understanding Integers. Without understanding the idea of Integers, how they work, and how to use them, students have no chance in understanding Algebra well, and of solving Word Problems (see the book on those) that require those ideas. Later, when students study calculus, which introduces real numbers, it is even more critical to understand Integers fully and deeply.
This book teaches EVERYTHING to know about Integers. It does so simply. It explains what Integers are; the meaning of operations with integers; and how to use integers. Those "negative times negative is positive" rules become logical and easy to remember because students understand why they are so. Then, when they learn Algebra and Calculus, they are not confused, and they often do better (you do want them to use the Word Problems book before Algebra).
Get the book and make your kids' experience with math a positive one!
Testimonials
Alicia
It's the first time I don't have to tell my kids to do their math. They even asked me if they can do it over other things. | 677.169 | 1 |
Listing Detail Tabs
NOTE: Item will ship from our Australian warehouse. A handling time of 2-4 business days applies for all orders. Delivery will then take 3-7 working days.
Understanding Maths
Description: In Focus- A Studymates Series Who else wants to spend less time studying and be able to calculate correctly each time? This book is aimed at anyone - adults and children alike - who is having trouble with basic mathematics This book includes details on: * Numbers and place value, * Dealing with fractions, * Calculating with percentages, * Working with decimals, * Using a calculator, * Angles and turning, * Area and volume, * Data handling, * Probabilities and chance, * Estimating and checking | 677.169 | 1 |
Explorations of Vector Calculus using Virtual Reality
- Calcflow for Math 20E -
Welcome to the Virtual Reality (VR) Supplement for Vector Calculus! In this series of VR labs, we will explore the subject of Vector Calculus using Calcflow, a high-performance mathematical tool for VR. With Calcflow, you can actually plot and see many of the two- and three-dimensional objects that you will encounter in your study of Vector Calculus.
The goal of these assignments is to explore the conceptual side of this beautiful subject. In class, you will undoubtedly have many homework problems that ask you to perform numerous computations. This is important, especially when working in an applied-mathematics environment. However, in order to really internalize what these computations are telling you, it is important to be able to conceptually explain the finer details behind the calculations. Not only does this solidify your understanding of the material at hand, but this can help you build intuition for more advanced topics. Having a strong conceptual understanding of a subject, as well as the necessary computational skills, can make you a stronger mathematician in the future!
We hope you enjoy exploring Vector Calculus in Calcflow! Now, let's get started. Below, you will find links to the lab assignments, information about the "VR Quiz", and further resources about VR and mathematics.
IMPORTANT: As a reminder, please note that the VR component is worth a percentage of your final grade. Treat these lab assignments as you would treat your "pen-and-paper" homework.
Lab Assignments
The following lab assignments give you the opportunity to visually explore the concepts you will be learning in Math 20E. Please click the links below to access the information page for each lab. There, you will find the assignment details, as well as the questions to answer and submit.
The Labs will be turned in at the beginning of class time (1pm) by 5pm on the indicated dates above. Labs may be turned in early, but must be given to the TA no later than the aforementioned time.
VR Quiz
The VR Quiz consists of two feedback surveys. These surveys are designed assess how much you learned using the Calcflow software, as well as gather information for how to further improve this course component.
The first quiz will be released at 8pm on Tuesday, August 7th. It is due at 12pm on Wednesday, August 8th.
The second quiz will be released during Week 5. More details will come at a later time.
Tutor Hours
The VR lab hours for the 2017 Summer Academy can be found on the class website, located here.
Further Resources and References
(Under construction)
Books:
Marsden, Jerrold and Anthony Tromba. Vector Calculus. 6th ed. W. H. Freeman and Company, 2012.
This is the main textbook for Math 20E.
Websites:
Nanome, Inc.
Nanome, Inc. is the company that develops Calcflow. Product information can be found here.
UCSD Math
This is the home page for the UCSD Department of Mathematics. | 677.169 | 1 |
Exponential Functions and Equations
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Here's a fantastic way for you to introduce the next new unit. This investigation activity allows students to work in pairs to discover exponential functions and equations. You can use this lesson to drive your instruction and to thread throughout the unit. | 677.169 | 1 |
This practice guide provides three recommendations for teaching algebra to students in middle school and high school. Each recommendation includes implementation steps and solutions for common roadblocks. The recommendations also summarize and rate supporting evidence. This guide is geared toward teachers, administrators, and other educators who want to improve their students' algebra knowledge.
Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure. This guide provides eight specific recommendations intended to help teachers, principals, and school administrators use Response to Intervention (RtI) to identify students who need assistance in mathematics and to address the needs of these students through focused interventions. The guide provides suggestions on how to carry out each recommendation and explains how educators can overcome potential roadblocks to implementing the recommendations. | 677.169 | 1 |
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
Fundamentals of Statics
Get to grips with static structural analysis once and for all. In this course we focus on getting the basics right prior understanding or knowledge of static or mechanical analysis, just a pencil, paper, calculator and a few of hours of effort!
An understanding of basic high school level algebra and geometry will be helpful but not essential.
Description
This course covers fundamental concepts and methods in static structural analysis. Starting with the very basics, we consider forces, moments and how to use the principle of static equilibrium. We then move on to look at pin-jointed structures or trusses; what are they and how do we analyse them? We'll cover the joint resolution method and method of sections in detail. Worked examples are used extensively to demonstrate the practical application of theory.
Based on my experience lecturing to engineering undergraduates, the course focuses on those areas students find particularly tricky when starting out. The link between theory and practice is reinforced using my experience as a structural design engineer. The course includes video lectures which combine screencast voice over with traditional style lectures.
The emphasis is on worked examples with students encouraged to try questions before the detailed solution is presented. The teaching philosophy is 'learn by doing!'.
This course is suitable for engineering students who find their structures lectures confusing and feel a little lost when it comes to structural analysis. Students wishing to get a head start before starting their degree programme or more advanced engineering students who need a refresher would also benefit from taking this course.
Who is the target audience?
This course is great for undergraduate engineering students who feel a little lost in their structures lectures. We start from scratch, establishing the basics and build from there.
More advanced engineering students who want a refresher or who didn't quite grasp the fundamentals first time around would also benefit from this course.
If you're about to start an engineering program, this course is a great way to get a head start.
If you're a structures genius already, this course is probably not for you.
In this lecture we'll start by introducing forces and their defining characterises. By the end of the lecture you should understand what a force is and how it influences mass. You'll also understand what a 2D force system is.
2D Force Systems
07:37
Force resultants and components
07:55
When dealing with 2D force systems, determining the resultant force is only half the story, we also need to determine the combined twisting or rotational effect generated by the force system. By the end of this lecture you'll understand what the moment of a force is and how to determine the resultant moment from a 2D force system.
The Moment of a Force
09:47
Now that you can analyse 2D force systems, in this lecture we put these skills to use by determining equivalent systems of moments and forces. By the end of this lecture you'll be able to replace complex systems of moments and forces with equivalent simplified systems. We'll also cover the concept of force couples.
Couples and Equivalent Systems
09:44
Test Yourself - Worked Example #1
06:14
Test Yourself - Worked Example #2
10:54
Test Yourself - Worked Example #3
07:27
+–
Static Equilibrium and Reaction Forces
11 Lectures
46:37
Equilibrium is the foundation on which almost all mechanical analysis is built. In this lecture we'll first introduce the principle of equilibrium and show how it can be used to determine the reaction forces for simple structures. We'll also look at different types of structural support, how we can model structural supports and critically how this model behaviour compares to real life support conditions.
Equilibrium and Reaction Forces
11:17
The first step in analysing most structures is determining the support reactions. It's a critical skill that needs to be honed before we can advance. In this lecture we'll present a variety of structures and demonstrate how the support reactions are calculated. We'll also introduce different load types such as uniformly and triangularly distributed loads.
Calculating Reaction Forces
08:54
The three equilibrium equations can only get us so far. In this lecture we'll identify the limitations of using equilibrium equations to determine support reactions. By the end of this lecture you will understand the difference between statically determinate and indeterminate structures.
Limitations of the Equations of Statics
02:32
Test Yourself - Worked Example #4
03:05
Test Yourself - Worked Example #5
03:39
Test Yourself - Worked Example #6
03:56
Test Yourself - Worked Example #7
03:56
Test Yourself - Worked Example #8
02:09
Test Yourself - Worked Example #9
02:47
Test Yourself - Worked Example #10
01:51
Test Yourself - Worked Example #11
02:31
+–
Analysis of Pin-Jointed Structures (trusses)
11 Lectures
01:15:47
In this lecture we introduce one of the most common forms of structure in the world, the pin-jointed truss. We'll focus on the details of the theoretical models we use in our analyses, discussing the key features of this type of structure and what makes it such an attractive structural form for civil and structural engineers.
What is a pin-jointed structure in theory
04:42
As with all models we use in our analyses, they're an approximation of the behaviour of the real structure. As such their behaviour will be different depending on the simplifying assumptions we make. In this lecture we focus on some of the important differences between our analysis models and real trusses. By the end of this lecture you will have an appreciation for how your analysis relates to reality.
What is a pin-jointed structure in reality
05:09
In this lecture we get into analysing pin-jointed structures to determine how forces are transmitted through the structure. By the end of this lecture you'll be able to use the 'Joint Resolution' method to determine the internal member forces within trusses and will appreciate how these elegant structures really work.
Force calculation using 'Joint Resolution'
10:56
Now we introduce an alternative analysis method known as the Method of Sections. Based on the principle of equilibrium just like the joint resolution method, this analysis procedure gives us more flexibility when analysing larger structures. By the end of this lecture you will be able to employ the method of sections to analyse trusses and will understand when to use this method joint resolution.
Force calculation using the 'Method of Sections'
08:19
The concept of statical determinacy introduced in lecture 7 also extends to pin-jointed structures. In this lecture we look at what it means in the context of pin-jointed structures. We'll introduce the concept of a mechanism as well as establish a simple test to determine the if a truss is statically determinate, indeterminate or a mechanism.
Statical determinacy
04:50
Test Yourself - Worked Example #12
12:22
Test Yourself - Worked Example #13
09:47
Test Yourself - Worked Example #14
11:41
Test Yourself - Worked Example #15
03:34
Test Yourself - Worked Example #16
02:31
In this lecture I give you some closing words of advice and some ideas on where to go next.
Wrapping up
01:56
+–
BONUS CONTENT
1 Lecture
03:02
In this lecture I just want to let you know about another one of my fundamentals courses; 'Fundamentals of Internal Bending Moments'.
Use promo code: SEANSUDEMYENGINEERS to enrol in this ($100) course for FREE.
Seán is a UK based lecturer in structural engineering and a chartered engineer. He holds a degree in structural engineering, a masters degree in civil engineering and a PhD also in civil engineering. After graduating in 2006, Seán worked for several years as a structural design engineer before pursuing a PhD in structural dynamics. In addition to his civil engineering research, Seán's other passion is teaching. Whether that's live in the lecture hall or here on Udemy. Seán is also a Fellow of the Higher Education Academy in the UK, a recognition of his commitment to professionalism in teaching and learning in higher education. | 677.169 | 1 |
If mathematics are one of the most complicated subjects, they are even more so when we speak about algebra and geometry. That is why programs like GeoGebra, that help to clear ideas and perform calculations can be so useful for students.
Dynamic maths for learning and teaching.
GeoGebra is an application that will allow you to represent geometrical and algebraic functions in a very simple manner thanks to an interface that is more similar to that of a drawing program than to that of a maths application. Nevertheless, calculations of positions, angles and other data are calculated automatically.
With GeoGebra it will be possible to recreate an endless amount of exercises that will range from school level all the way up to real university problems, because the software in itself doesn't lack any calculation or drawing possibility. It includes vectors, segments, rays, regular and irregular geometric figures,... Everything necessary to learn algebra, mathematics and geometry is included in the program.
Furthermore, if you need to you can input the calculations directly so that the program's engine represents them. Thus, we'll be able to know if our calculations coincide with those of the application | 677.169 | 1 |
algebra
Algebra is a method of thinking about mathematics in a general way. It provides rules about how equations must be put together and how they can be changed. The word algebra comes from the title of a book on mathematics written in the early 800s. The book was written by an Arab astronomer and mathematician named al-Khwarizmi. The rules of algebra are older than that, however. The ancient Greeks wrote down some of the rules | 677.169 | 1 |
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Students under 18 who have conducted advanced research in mathematics might be interested in the Davidson Fellows Scholarship program.
Many high school students are interested in trying their hand at doing mathematics
research (Appendix IX). This effort is hampered by the fact that even 12th graders,
not to mention 9th graders, do not have a huge collection of mathematical tools at
their disposal. Furthermore, many high school mathematics department libraries and public
libraries do not have books that are of value to students who are interested in doing
research. High schools can help support their mathematics research programs by systematically attempting to acquire the resources listed in the appendices. Despite impediments
there are surprisingly many options by which high school students can get started.
One important source of ideas for high school students to do research on is to extend the problems that occur in Mathematics Team competitions to more general settings.
Students can attempt this with problems they have been challenged by in recent competitions.
Books which contain such problems appear in some of the lists shown in the appendices. Below are some suggestions for getting students going on research projects.
Working on mathematics research problems is exciting and rewarding.
1. Martin Gardner's books
There is no better introduction to a variety of very rich mathematical problems and
ideas explained at a level that can be understood by high schools students than the
works of Martin Gardner. (see Appendix I)
2. Ian Stewart's column in Scientific American Magazine
Stewart's monthly column (dropped by Scientific American recently) has problems of current interest that are accessible to high
school students. (Stewart has also published some excellent books. See Appendix VIII)
3. Quantum Magazine
This magazine is written specifically for high school students. It is published by
Springer-Verlag. (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010-78580.
Phone: 1-800-SPRINGER) Quantum should be part of every high school mathematics department library. (Unfortunately, Quantum has ceased publication. Some libraries have the issues that were published.)
4. The New Mathematics Library
This series of nearly 40 volumes is written specifically for high school students.
The books in the series cover a wide range of topics from graph theory and number
theory to game theory. (see Appendix II)
5. Dolciani Mathematical Expositions
This series of expertly written monographs has wonderful materials not easily available
elsewhere. The level is comparable to that of the New Mathematics Library. (See Appendix III.)
6. Mathematics Teacher
The NCTM (National Council of Teachers of Mathematics) publishes a wonderful journal
primarily for high school mathematics teachers. This journal has many articles that
can be read by students and that can suggest interesting research problems.
7. Mathematics World
This series of books, (see Appendix IV) are published by the American Mathematical
Society. Many of the volumes consist of materials developed in other countries that
have been translated into English.
8. College Mathematics and Mathematics Magazine
These two magazines, published by the Mathematical Association of America, are available
through the Mathematics Department office of many high schools. They contain many
articles that can be read by high school students.
9. HIMAP and GEOMAP Modules (High School Mathematics and its Applications Project
and Geometry and its Applications Project)
This series of modules (see Appendix V) is aimed at high school teachers and students and treats many
topics at the high school level that have been traditionally only discussed at higher
levels. The series is published by COMAP (Consortium for Mathematics and Its Applications). COMAP's quarterly newsletter
CONSORTIUM has articles that may be of value, and COMAP also has available both in
CD-ROM and print form hundreds of applications-oriented modules (aimed for college
students), many of which are accessible to high school students.
NCTM (National Council of Teachers of Mathematics) publishes yearly volumes devoted
to a specific theme. Some of these volumes contain pointers to ideas that lead quickly
to research investigations.
12. World Wide Web
Many resources to assist with high school mathematics research are available on the
web. These include being able to access the card catalogues for major libraries,
access the home pages of the mathematics professional societies, access the home
pages of particular individuals that have research problems in various areas of mathematics,
and look at materials related to the history of mathematics, etc. Appendix VI lists
some web pages that may be of use.
13. Miscellaneous Resources
Some miscellaneous books and resources can be found in Appendix VII and Appendix VIII In particular in Appendix VII you can take a look at a collection of projects that I generated for students to work on.
Appendix I (Martin Gardner's Books)
1. The Scientific American Book of Puzzles and Diversions.
2. The Second Scientific American Book of Mathematical Puzzles and Diversions.
3. New Mathematical Diversions.
4. The Numerology of Dr. Matrix.
5. The Unexpected Hanging.
6. The Sixth Book of Mathematical Games from Scientific American.
7. Mathematical Carnival.
8. Mathematical Magic Show.
9. Mathematical Circus.
10. Wheels, Life and other Mathematical Amusements.
11. Knotted Doughnuts.
12. Time Travel and other Mathematical Bewilderments.
13. Penrose Tiles to Trapdoor Ciphers.
14. Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American
Magazine
15. Entertaining Mathematical Puzzles.
16. Mathematics Magic and Mystery.
17. The Last Recreations.
18. The Colossal Book of Mathematics.
Other books by Martin Gardner:
Aha! Gotcha
Aha! Insight
The Ambidextrous Universe
Fads and Fallacies in the Name of Science
The Incredible Dr. Matrix
Logic Machines and Diagrams
The Magic Numbers of Dr. Matrix
The Relativity Explosion
Riddles of the Sphinx
Note 1: Many of these books consist of edited versions of articles Gardner wrote
for Scientific American Magazine for many years.
Note 2: Gardner's books have been published by a variety of publishers including W.
H. Freeman. Recently, the Mathematical Association of America has been republishing
new versions of some of the older books with corrections and some updating.
Note 3: I know this list is not complete.
Note 4: Carl Lee (University of Kentucky) has a web page giving where in Martin Gardner's works one can find various topics.
Several books which though not written by Martin Gardner either honor him or are very much in the spirit of what he wrote. Here are some examples:
Appendix II (New Mathematical Library)
This series was originally begun by Random House and subsequently taken over by MAA.
The books are specifically designed to be readable by secondary school students.
Of course, they are of interest to other audiences as well.
The Amercian Mathematica Society also publishes the ongoing series of monographs by Barry Cipra devoted to new developments in the Mathematical Sciences:
What's Happening in the Mathematical Sciences, Volume 1, 1993.
What's Happening in the Mathematical Sciences, Volume 2, 1994.
What's Happening in the Mathematical Sciences, Volume 3, 1995-1996.
What's Happening in the Mathematical Sciences, Volume 4, 1998-1999.
Also noteworthy from AMS:
Knuth, D., Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, American Mathematical Society, Providence, 1997.
The American Mathematical Society recently started the Student Mathematical Library, aimed at undergraduates. The books in this series differ greatly in level but many of the books are self-contained and are accessible to high school students. Here are the titles in this series. The first book in the series was Charles Radin's miles of tiles, but rather than list all of the titles here is the link to the AMS page where the books are listed in reverse order of publication.
The American Mathematical Society also publishes the DIMACS Series in Discrete Mathematics and Theoretical Computer Science. (DIMACS is housed at Rutgers University and sponsors research and education programs related to Discrete Mathematics and Theoretical Computer Science.) Although books in this series are designed for researchers, some of the articles in the books are survey papers that can be read by novices. These books also pose many research problems some of them easily understood by individuals with little technical background.
Appendix VI World Wide Web Locations
Although the web sites below do not in all cases have research problems for high school students, they have a rich variety of information about mathematics, as well as many links to resources that will help high school students interested in doing mathematical research.
History of Mathematics
A large collection of biographies of mathematicians and lots of other useful information can be found here.
An attempt is being made to list all individuals who ever received a doctoral degree in Mathematics together with each individuals doctoral thesis advisor, dissertation title, and list of doctoral students. A lot interesting history can be gleaned from this project.
The Mathematics Forum at Swarthmore College provides a broad array of resources for people interested in mathematics. The site provides extensive searchable archives and well as many links to other sites dealing with mathematics.
The Institute for Electrical and Electronic Engineers provides lots of information about mathematical support for computer science and engineering, in particular, how mathematics is being used to help create new technologies.
The Institute for Operations Research and Management Science is the major professional
organization in the areas of operations research and management science. These two
areas of knowledge aim at improving the efficiency with which business and governments deliver services internally and to the public.
Appendix VII Special sites:
David Eppstein maintains a staggering collection of pointers to exciting ideas and
results in geometry and the application of geometry. His home page also has materials related to recreational mathematics, combinatorial games and number theory, etc.
Appendix VIII Miscellaneous Resources
This book is designed for undergraduates and deals with a wide variety of basic tools
(e.g. logic, functions, graph theory, probability, etc.) and proof techniques (e.g.
mathematical induction, parity, etc.). However, it serves as a good introduction
to these ideas for high school level students.
Cofman, J., What to Solve? Problems and Suggestions for Young Mathematicians, Oxford,
New York, 1990.
A collection of interesting combinatorial problems and methods to solve and extend
them.
This lovely book contains a wealth of accessible research problems and a lot of exciting
mathematics.
Gale, D., Tracking the Automatic Ant and Other Mathematical Explorations, Springer-Verlag,
New York, 1998. This book contains a series of wonderful brief explorations in the spirit of Martin
Gardner's writings.
This book surveys methods of problem solving techniques. (This book is aimed at teachers
but it is readable by students as well.)
Stewart, I., The Problem of Mathematics, Oxford U. Press, New York, 1987. A nice survey of important ideas in mathematics.
Stewart, I., Game, Set and Math, Penguin, New York, 1989.
A lovely collection of short articles on many varied topics.
Stewart, I., Another Fine Math You've Got Me Into..., W. H. Freeman, New York, 1992. A compendium of some of the articles that Stewart wrote for Scientific American Magazine. Most of them are very well done.
Journals:
American Scientist
Published by Sigma Xi, the honor society for those interested in Science (and Mathematics), this journal has superb survey articles dealing with science and mathematics. It also features excellent articles by Brian Hayes dealing with the interface between mathematics and computer science.
College Mathematics
Published by the Mathematical Association of America (MAA) and originally aimed for students and faculty at Two Year (Community) Colleges, this journal now specializes in topics primarily treated in the first two years of college.
This journal has interesting problems and articles in the areas of geometry and combinatorics.
Soifer also publishes several collections of problem oriented books.
Math Horizons
Published by the MAA (Mathematical Association of America, 1529 Eighteenth Street,
N.W., Washington, DC, 20036-1385; Phone: 1-800-331-1622) for undergraduates this
quarterly has information about careers in mathematics, articles about mathematicians
and mathematics, and a section with problems for solution.
Mathematical Intelligencer
Published by Springer-Verlag (Springer-Verlag, 175 Fifth Avenue, New York, New York
10010-78580. Phone: 1-800-SPRINGER) this journal although it has many articles at
an advanced level, also has materials dealing with the history of mathematics, mathematical biographies, book reviews, and a lovely column entitled Mathematical Entertainments.
Mathematical Spectrum
This journal published in Australia has many interesting articles that could inspire
high school student research.
Mathematics Magazine
This journal is published by the Mathematical Association of America (MAA) and treates all of undergraduate mathematics. However, there are often historical and survey articles that can be read by a wide variety of students.
Pi Mu Epsilon Journal
This journal is published by Pi Mu Epsilon which is an honor society for undergraduates interested in mathematics. It contains research articles written by undergraduates, problems, and surveys.
Quantum
Published by Springer-Verlag this journal is aimed specifically at high school students. Many articles are translations from the Russian journal Qvant. In Qvant articles for high school students written by many of Russia's most emminent mathematicians appeared. The articles are a wonderful mix of classical and recent topics in mathematics. Unfortunately, publication of this journal has been discontinued, but old issues are still available in some libraries.
Scientific American
This journal has survey articles about mathematics and science of the highest calibre. A regular feature is a column by Ian Stewart about recreational mathematics. Recently, Stewart's column has been dropped.
SIAM News
This journal has wonderful expositiory articles about applications of mathematics and applicable mathematics. I regular feature is a wonderful column of Philip Davis and the columns of Barry Cipra are excellent.
Spectrum
Published by IEEE this wonderful magazine features articles about engineering and how science and mathematics support new developments in engineering.
UMAP Journal
Published by COMAP this journal has articles related to mathematical modeling, book reviews, and articles of general interest about mathematics. | 677.169 | 1 |
Showing 1 to 30 of 31
Running Maple Maple should be on your desktop, but if not, the usual place to find it is under All programs -> Maple 10 -> Maple 10 You should see the Maple 10 splash screen. Either Maple will start right up or you'll get a menu screen like the one b
Summing a series of real numbers
Kowalski
Convergence tests. If you can nd a formula for the terms an of a series an , it is often possible to determine
if the series (i.e. the innite sum) converges by inspecting the only the sequence of terms, rather tha
The Deadly Sins of Mathematics
Kowalski
Motivation. Unfortunately, students often abuse or misuse the rules of algebra, trigonometry, and calculus.
I have classified four common errors that so profoundly and fundamentally miss the point of a problem that,
Final Exam: Math 125, Fall 2011io
ENGL 279-06
SPRING 2008
Guidelines and Requirements for Instructional Team-Based Presentations
For the second set of oral presentations, you'll present, as part of a team, a section of a chapter from our textbook. This "instructional" speech puts y
Chapter 13
Finding Exact Words and Adjusting Tone (pg 264-282)
Use
and
Wording.
Don't use three syllables when one will do
- demonstrate =
- endeavor
=
- initiate
=
Weak
"Six rectangular grooves around the outside edge of the steel plate
are needed for th
Hum 100 300 Spartans
Heath Milton 07 Feb 2008
During the Persian Empire, Greece was made up of separate city states, not a country. Each city state fought against the others. It was the way of life, and in order to survive that kind of life you nee
Guidelines for Writing a Summary
1. Preview the article by noting (a) what kind of article it is and (b) how it is structured. Does it appear to be primarily research-oriented or primarily informative? Is it more explanatory or more persuasive? Note
Final Exam: Math 125, Spring 2012
This class is not a required course for most majors, and this is because it is a higher level math class that is not for everyone. I plan on going into Mechanical Engineering so I need to take this class and i had a great experience and I definitely would do it agian
Course highlights:
This class teaches an expanded knowledge of derivatives and anti-derivatives learned in Calc 1 as well as vectors and limits.
Hours per week:
9-11 hours
Advice for students:
You need to put in a lot of time into this course. In homework alone I had put in 70+ hours into the course. | 677.169 | 1 |
Full coverage of 12 different math topics in just one volume! Statistis, plane trigonometry, calculus, boolean algebra--most likely your mathematics needs don't fit neatly into books on single math topics. That's why we created the new Handbook of Applied Mathematics for Engineers and Scientists--the first math problemsolver to solve tough daily problems in 13 different math fields. put this new Hnadbook at your fingertips, and you'll be able to review essential math principles and definitions in seconds, and use its data as a springboard for creative problems-solving. Each math area features definitions, principles and equations, and all applications are cross-indexed for maximum convenience. Algebra; Plane and Spherical Trigonometry; Plane and Solid Analytic Geometry; Permutations and Combinations; Matrix Algebra; Boolean Algebra; Transformation of Space in Computer Graphics; Statistics; Probability; Regression Analysis; Cybernetics; Calculus | 677.169 | 1 |
Showing 1 to 13 of 13
THE IMPLICIT FUNCTION THEOREM
1. A
SIMPLE VERSION OF THE IMPLICIT FUNCTION THEOREM
1.1. Statement of the theorem. Theorem 1 (Simple Implicit Function Theorem). Suppose that is a real-valued functions dened on a domain D and continuously differentiable on
CHARACTERISTIC ROOTS AND VECTORS
1. DEFINITION OF CHARACTERISTIC ROOTS AND VECTORS 1.1. Statement of the characteristic root problem. Find values of a scalar for which there exist vectors x = 0 satisfying Ax = x (1)
where A is a given nth order matrix. Th
REVIEW OF SIMPLE UNIVARIATE CALCULUS
1. APPROXIMATING
CURVES WITH LINES
1.1. The equation for a line. A linear function of a real variable x is given by y = f(x) = ax + b, a and b are constants (1)
The graph of linear equation is a straight line. The numb
INTRODUCTION TO MATRIX ALGEBRA
1. DEFINITION
OF A MATRIX AND A VECTOR
1.1. Denition of a matrix. A matrix is a rectangular array of numbers arranged into rows and columns. It is written as a11 a12 . a1n a21 a22 . a2n . . (1) . . . . am1 am2 . amn The abov
FUNCTIONS AND EQUATIONS
1. SETS
AND SUBSETS
1.1. Denition of a set. A set is any collection of objects which are called its elements. If x is an element of the set S, we say that x belongs to S and write x S. If y does not belong to S, we write y S. The s
SINGLE VARIABLE OPTIMIZATION
1. DEFINITION OF LOCAL MAXIMA AND LOCAL MINIMA 1.1. Note on open and closed intervals. 1.1.1. Open interval. If a and b are two numbers with a < b, then the open interval from a to b is the collection of all numbers which are
SIMPLE MULTIVARIATE OPTIMIZATION
1. DEFINITION
OF LOCAL MAXIMA AND LOCAL MINIMA
2
1.1. Functions of 2 variables. Let f(x1 , x2 ) be dened on a region D in
containing the point (a, b). Then
a: f(a, b) is a local maximum value of f if f(a, b) f(x1 , x2 ) fo
SYSTEMS OF EQUATIONS
1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which we would like to solve. The variables are typical
CONVEXITY AND OPTIMIZATION
1. CONVEX
SETS
1.1. Denition of a convex set. A set S in Rn is said to be convex if for each x1 , x2 S, the line segment x1 + (1-)x2 for (0,1) belongs to S. This says that all points on a line connecting two points in the set ar | 677.169 | 1 |
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This essential book: * Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs * Reinforces the foundations of calculus and algebra * Explores how to use both a direct and indirect proof to prove a theorem * Presents the basic properties of real numbers * Discusses how to use mathematical induction to prove a theorem * Identifies the different types of theorems * Explains how to write a clear and understandable proof * Covers the basic structure of modern mathematics and the key components of modern mathematics
A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty.
Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra.
Details
ISBN-13: 9780470042953
ISBN-10: 0470042958
Publisher: Wiley-Interscience
Publish Date: July 2006
Page Count: 318
Dimensions: 9.56 x 6.14 x 0.8 inches
Shipping Weight: 1.27 pounds
Series: Pure and Applied Mathematics: A Wiley-Interscience Series of #52 | 677.169 | 1 |
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MATH 1101. Intro to Mathematical Modeling. 3-0-3 Units.
This course is not intended to supply sufficient algebraic background for students who intend to take precalculus or the calculus sequence for mathematics and science majors. This course is an introduction to mathematical modeling using graphical, numerical, symbolic, and verbal techniques to describe and explore real-world data and phenomena. Emphasis is on the use of linear, polynomial, exponential, and logarithmic functions to investigate and analyze applied problems and questions, supported by the use of appropriate technology, and on effective communication of quantitative concepts and results.(F,S,M)
Corequisites: MATH 0998 unless exempt from learning support | 677.169 | 1 |
4: Exponential and Logarithmic Functions
In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.
Contributors
The OpenStax College name, OpenStax College logo, OpenStax College book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact partners@openstaxcollege.org. "Download for free at | 677.169 | 1 |
-7-
Algebra Handbook
Table of Contents
Page
Description
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
Chapter 19: Sequences and Series
Introduction to Sequences and Series
Fibonacci Sequence
Summation Notation and Properties
Some Interesting Summation Formulas
Arithmetic Sequences
Arithmetic Series
Pythagorean Means (Arithmetic, Geometric)
Pythagorean Means (Harmonic)
Geometric Sequences
Geometric Series
A Few Special Series (π, e, cubes)
Pascal's Triangle
Binomial Expansion
Gamma Function and n !
Graphing the Gamma Function
170
Index
Useful Websites
Mathguy.us – Developed specifically for math students from Middle School to College, based on the
author's extensive experience in professional mathematics in a business setting and in math
tutoring. Contains free downloadable handbooks, PC Apps, sample tests, and more.
Wolfram Math World – Perhaps the premier site for mathematics on the Web. This site contains
definitions, explanations and examples for elementary and advanced math topics.
Purple Math – A great site for the Algebra student, it contains lessons, reviews and homework
guidelines. The site also has an analysis of your study habits. Take the Math Study Skills Self‐
Evaluation to see where you need to improve.
Math.com – Has a lot of information about Algebra, including a good search function.
Version 2.5
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8.
-8-
Algebra Handbook
Table of Contents
Schaum's Outlines
An important student resource for any high school math student is a Schaum's Outline. Each book
in this series provides explanations of the various topics in the course and a substantial number of
problems for the student to try. Many of the problems are worked out in the book, so the student
can see examples of how they should be solved.
Schaum's Outlines are available at Amazon.com, Barnes & Noble, Borders and other booksellers.
Note: This study guide was prepared to be a companion to most books on the subject of High
School Algebra. In particular, I used the following texts to determine which subjects to include
in this guide.
Algebra 1 , by James Schultz, Paul Kennedy, Wade Ellis Jr, and Kathleen Hollowelly.
Algebra 2 , by James Schultz, Wade Ellis Jr, Kathleen Hollowelly, and Paul Kennedy.
Although a significant effort was made to make the material in this study guide original, some
material from these texts was used in the preparation of the study guide.
Version 2.5
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9.
-9-
Algebra
Order of Operations
To the non‐mathematician, there may appear to be multiple ways to evaluate an algebraic
expression. For example, how would one evaluate the following?
3·4·7
6·5
You could work from left to right, or you could work from right to left, or you could do any
number of other things to evaluate this expression. As you might expect, mathematicians do
not like this ambiguity, so they developed a set of rules to make sure that any two people
evaluating an expression would get the same answer.
PEMDAS
In order to evaluate expressions like the one above, mathematicians have defined an order of
operations that must be followed to get the correct value for the expression. The acronym that
can be used to remember this order is PEMDAS. Alternatively, you could use the mnemonic
phrase "Please Excuse My Dear Aunt Sally" or make up your own way to memorize the order of
operations. The components of PEMDAS are:
P
E
M
D
A
S
Anything in Parentheses is evaluated first.
Usually when there are multiple
operations in the same category,
for example 3 multiplications,
they can be performed in any
order, but it is easiest to work
from left to right.
Items with Exponents are evaluated next.
Multiplication and …
Division are performed next.
Addition and …
Subtraction are performed last.
Parenthetical Device. A useful device is to use apply parentheses to help you remember
the order of operations when you evaluate an expression. Parentheses are placed around the
items highest in the order of operations; then solving the problem becomes more natural.
Using PEMDAS and this parenthetical device, we solve the expression above as follows:
Initial Expression:
3 · 4 · 7
6·5
Add parentheses/brackets:
3·4·7
Solve using PEMDAS:
84
6 · 25
150
84
Final Answer
234
Version 2.5
6· 5
Note: Any expression which is
ambiguous, like the one above, is
poorly written. Students should strive
to ensure that any expressions they
write are easily understood by others
and by themselves. Use of parentheses
and brackets is a good way to make
your work more understandable.
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10.
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Algebra
Graphing with Coordinates
Graphs in two dimensions are very common in algebra and are one of the most common
algebra applications in real life.
y
Coordinates
The plane of points that can be graphed in 2 dimensions is
called the Rectangular Coordinate Plane or the Cartesian
Coordinate Plane (named after the French mathematician
and philosopher René Descartes).
Quadrant 2
Quadrant 1
x
Quadrant 3
Quadrant 4
•
Two axes are defined (usually called the x‐ and y‐axes).
•
Each point on the plane has an x value and a y value, written as: (xvalue, yvalue)
•
The point (0, 0) is called the origin, and is usually denoted with the letter "O".
•
The axes break the plane into 4 quadrants, as shown above. They begin with Quadrant 1
where x and y are both positive and increase numerically in a counter‐clockwise fashion.
Plotting Points on the Plane
When plotting points,
•
the x‐value determines how far right (positive) or left (negative) of the origin the point is
plotted.
•
The y‐value determines how far up (positive) or down (negative) from the origin the point is
plotted.
Examples:
The following points are plotted in the figure to
the right:
A = (2, 3)
B = (‐3, 2)
C = (‐2, ‐2)
D = (4, ‐1)
O = (0, 0)
Version 2.5
in Quadrant 1
in Quadrant 2
in Quadrant 3
in Quadrant 4
is not in any quadrant
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Algebra
Linear Patterns
Recognizing Linear Patterns
The first step to recognizing a pattern is to arrange a set of numbers in a table. The table can
be either horizontal or vertical. Here, we consider the pattern in a horizontal format. More
advanced analysis generally uses the vertical format.
Consider this pattern:
x‐value
y‐value
0
6
1
9
2
12
3
15
4
18
5
21
To analyze the pattern, we calculate differences of successive values in the table. These are
called first differences. If the first differences are constant, we can proceed to converting the
pattern into an equation. If not, we do not have a linear pattern. In this case, we may choose
to continue by calculating differences of the first differences, which are called second
differences, and so on until we get a pattern we can work with.
In the example above, we get a constant set of first differences, which tells us that the pattern
is indeed linear.
x‐value
y‐value
First Differences
0
6
1
9
3
2
12
3
3
15
3
4
18
3
5
21
3
Converting a Linear Pattern to an Equation
Creating an equation from the pattern is easy if you have
constant differences and a y‐value for x = 0. In this case,
• The equation takes the form
, where
• "m" is the constant difference from the table, and
• "b" is the y‐value when x = 0.
In the example above, this gives us the equation:
.
Note: If the table does not have a
value for x=0, you can still obtain
the value of "b". Simply extend the
table left or right until you have an
x‐value of 0; then use the first
differences to calculate what the
corresponding y‐value would be.
This becomes your value of "b".
Finally, it is a good idea to test your equation. For example, if
4, the above equation gives
3·4
6 18, which is the value in the table. So we can be pretty sure our equation is
correct.
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12.
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ADVANCED
Algebra
Identifying Number Patterns
When looking at patterns in numbers, is is often useful to take differences of the numbers you
are provided. If the first differences are not constant, take differences again.
n
‐3
‐1
1
3
5
7
∆
n
2
5
10
17
26
37
∆
∆2
3
5
7
9
11
2
2
2
2
2
2
2
2
2
When first differences are constant, the pattern represents a
linear equation. In this case, the equation is: y = 2x ‐ 5 . The
constant difference is the coefficient of x in the equation.
When second differences are constant, the pattern represents a
quadratic equation. In this case, the equation is: y = x 2 + 1 . The
constant difference, divided by 2, gives the coefficient of x2 in the
equation.
When taking successive differences yields patterns that do not seem to level out, the pattern
may be either exponential or recursive.
n
5
7
11
19
35
67
n
2
3
5
8
13
21
Version 2.5
∆
∆2
2
4
8
16
32
2
4
8
16
∆
∆2
1
2
3
5
8
1
1
2
3
In the pattern to the left, notice that the first and second
differences are the same. You might also notice that these
differences are successive powers of 2. This is typical for an
exponential pattern. In this case, the equation is: y = 2 x + 3 .
In the pattern to the left, notice that the first and second
differences appear to be repeating the original sequence. When
this happens, the sequence may be recursive. This means that
each new term is based on the terms before it. In this case, the
equation is: y n = y n‐1 + y n‐2 , meaning that to get each new term,
you add the two terms before it.
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Algebra
Basic Number Sets
Number Set
Definition
Examples
Natural Numbers (or,
Counting Numbers)
Numbers that you would normally
count with.
1, 2, 3, 4, 5, 6, …
Whole Numbers
Add the number zero to the set of
Natural Numbers
0, 1, 2, 3, 4, 5, 6, …
Integers
Whole numbers plus the set of
negative Natural Numbers
… ‐3, ‐2, ‐1, 0, 1, 2, 3, …
Any number that can be expressed
All integers, plus fractions and
mixed numbers, such as:
in the form , where a and b are
Rational Numbers
integers and
4
2 17
,
, 3
3
6
5
0.
Any number that can be written in
decimal form, even if that form is
infinite.
Real Numbers
All rational numbers plus roots
and some others, such as:
√2 , √12 , π, e
Basic Number Set Tree
Real Numbers
Rational
Irrational
Integers
Fractions and
Mixed Numbers
Whole
Numbers
Natural
Numbers
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Negative
Integers
Zero
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Algebra
Operating with Real Numbers
Absolute Value
The absolute value of something is the distance it is from zero. The easiest way to get the
absolute value of a number is to eliminate its sign. Absolute values are always positive or 0.
| 5| 5 |3| 3
|0| 0
|1.5| 1.5
Adding and Subtracting Real Numbers
Adding Numbers with the Same Sign:
•
•
•
Add the numbers without regard
to sign.
Give the answer the same sign as
the original numbers.
Examples:
6
3
9
12 6 18
Adding Numbers with Different Signs:
•
•
•
Ignore the signs and subtract the
smaller number from the larger one.
Give the answer the sign of the number
with the greater absolute value.
Examples:
6
3
3
7
11 4
Subtracting Numbers:
•
•
•
Change the sign of the number or numbers being subtracted.
Add the resulting numbers.
Examples:
6
3
6
3
3
13 4 13
4
9
Multiplying and Dividing Real Numbers
Numbers with the Same Sign:
Version 2.5
•
•
•
Multiply or divide the numbers
without regard to sign.
Give the answer a "+" sign.
Examples:
6 · 3
18 18
12 3
4 4
Numbers with Different Signs:
•
•
•
Multiply or divide the numbers without
regard to sign.
Give the answer a "‐" sign.
Examples:
6 · 3
18
12
3
4
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Algebra
Solving Multi‐Step Equations
Reverse PEMDAS
One systematic way to approach multi‐step equations is Reverse PEMDAS. PEMDAS describes
the order of operations used to evaluate an expression. Solving an equation is the opposite of
evaluating it, so reversing the PEMDAS order of operations seems appropriate.
The guiding principles in the process are:
•
•
Each step works toward isolating the variable for which you are trying to solve.
Each step "un‐does" an operation in Reverse PEMDAS order:
Subtraction
Inverses
Division
Inverses
Multiplication
Exponents
Inverses
Logarithms
Parentheses
Inverses
Remove Parentheses (and repeat process)
Addition
Note: Logarithms are the
inverse operator to exponents.
This topic is typically covered in
the second year of Algebra.
The list above shows inverse operation relationships. In order to undo an operation, you
perform its inverse operation. For example, to undo addition, you subtract; to undo division,
you multiply. Here are a couple of examples:
Example 1
Example 2
Solve:
Step 1: Add 4
3
4
4
Result:
3
Step 2: Divide by 3 3
Result:
14
4
Solve:
Step 1: Add 3
18
3
Result:
2· 2
Step 2: Divide by 2 2
5
2
2
6
Result:
2
Step 3: Remove parentheses
5
1
2
5
5
1
5
Notice that we add and subtract before we
multiply and divide. Reverse PEMDAS.
Result:
Step 4: Subtract 5
2· 2
5
3
3
5
3
Result:
2
6
With this approach, you will be able to
Step 5: Divide by 2
2
2
solve almost any multi‐step equation. As
Result:
3
you get better at it, you will be able to use
some shortcuts to solve the problem faster.
Since speed is important in mathematics, learning a few tips and tricks with regard to solving
equations is likely to be worth your time.
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Algebra
Tips and Tricks in Solving Multi‐Step Equations
Fractional Coefficients
Fractions present a stumbling block to many students in solving multi‐step equations. When
stumbling blocks occur, it is a good time to develop a trick to help with the process. The trick
shown below involves using the reciprocal of a fractional coefficient as a multiplier in the
solution process. (Remember that a coefficient is a number that is multiplied by a variable.)
Example 1
Multiply by :
·
·
Explanation: Since is the reciprocal of ,
8
·8
Solve:
Result:
when we multiply them, we get 1, and
1·
. Using this approach, we can avoid
dividing by a fraction, which is more difficult.
12
Example 2
Solve:
Explanation: 4 is the reciprocal of
Multiply by 4:
Result:
2
·
4
·
4
2 ·
4
8
, so
when we multiply them, we get 1. Notice
the use of parentheses around the negative
number to make it clear we are multiplying
and not subtracting.
Another Approach to Parentheses
In the Reverse PEMDAS method, parentheses
are handled after all other operations.
Sometimes, it is easier to operate on the
parentheses first. In this way, you may be able
to re‐state the problem in an easier form before
solving it.
Example 3, at right, is another look at the
problem in Example 2 on the previous page.
Use whichever approach you find most to your
liking. They are both correct.
Version 2.5
Example 3
Solve:
2· 2
Step 1: Eliminate parentheses
5
3
5
Result:
4
Step 2: Combine constants
10
3
5
4
7
7
Result:
Step 3: Subtract 7
Result:
Step 4: Divide by 4
Result:
4
4
5
7
12
4
3
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Algebra
Probability and Odds
Probability
Probability is a measure of the likelihood that an event will occur. It depends on the number of
outcomes that represent the event and the total number of possible outcomes. In equation terms,
Example 1: The probability of a flipped coin landing as a head is 1/2. There are two equally likely events
when a coin is flipped – it will show a head or it will show a tail. So, there is one chance out of two that
the coin will show a head when it lands.
1
1
2
2
Example 2: In a jar, there are 15 blue marbles, 10 red marbles and 7 green marbles. What is the
probability of selecting a red marble from the jar? In this example, there are 32 total marbles, 10 of
which are red, so there is a 10/32 (or, when reduced, 5/16) probability of selecting a red marble.
10
32
10
32
5
16
Odds
Odds are similar to probability, except that we measure the number of chances that an event will occur
relative to the number of chances that the event will not occur.
In the above examples,
1
10
10
5
1
1
22
22 11
1
• Note that the numerator and the denominator in an odds calculation add to the total number of
possible outcomes in the denominator of the corresponding probability calculation.
•
To the beginning student, the concept of odds is not as intuitive as the concept of probabilities;
however, they are used extensively in some environments.
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Algebra
Probability with Dice
Single Die
Probability with a single die is based on the number of chances of an event out of 6 possible
outcomes on the die. For example:
2
5
Two Dice
Probability with two dice is based on the number of chances of an event out of 36 possible
outcomes on the dice. The following table of results when rolling 2 dice is helpful in this regard:
1st Die
2nd Die
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
The probability of rolling a number with two dice is the number of times that number occurs in
the table, divided by 36. Here are the probabilities for all numbers 2 to 12.
2
5
3
6
4
7
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9
8
10
6
4
12
3
11
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Algebra
Combinations
Single Category Combinations
The number of combinations of items selected from a set, several at a time, can be calculated
relatively easily using the following technique:
Technique: Create a ratio of two products. In the numerator, start with the number of
total items in the set, and count down so the total number of items being multiplied is
equal to the number of items being selected. In the denominator, start with the
number of items being selected and count down to 1.
Example: How many
combinations of 3 items can
be selected from a set of 8
items? Answer:
8·7·6
56
3·2·1
Example: How many
combinations of 4 items can
be selected from a set of 13
items? Answer:
13 · 12 · 11 · 10
715
4·3·2·1
Example: How many
combinations of 2 items can
be selected from a set of 30
items? Answer:
30 · 29
435
2·1
Multiple Category Combinations
When calculating the number of combinations that can be created by selecting items from
several categories, the technique is simpler:
Technique: Multiply the numbers of items in each category to get the total number of
possible combinations.
Example: How many different
pizzas could be created if you
have 3 kinds of dough, 4 kinds
of cheese and 8 kinds of
toppings?
Answer:
3 · 4 · 8 96
Example: How many different
outfits can be created if you
have 5 pairs of pants, 8 shirts
and 4 jackets?
Answer:
5 · 8 · 4 160
Example: How many designs
for a car can be created if you
can choose from 12 exterior
colors, 3 interior colors, 2
interior fabrics and 5 types of
wheels? Answer:
12 · 3 · 2 · 5 360
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Algebra
Statistical Measures
Statistical measures help describe a set of data. A definition of a number of these is provided in the table below:
Concept
Description
Calculation
Example 1
Example 2
Data Set
Numbers
35, 35, 37, 38, 45
15, 20, 20, 22, 25, 54
Average
Add the values and
divide the total by the
number of values
Median
Middle
Arrange the values from
low to high and take the
middle value(1)
37
21(1)
Mode
Most
The value that appears
most often in the data
set
35
20
Size
The difference between
the highest and lowest
values in the data set
45 – 35 = 10
54 – 15 = 39
Oddballs
Values that look very
different from the other
values in the data set
none
54
Mean
(1)
Range
(2)
Outliers
35
35
37
5
38
45
38
15
18
22
22
25
6
54
26
Notes:
(1) If there are an even number of values, the median is the average of the two middle values. In Example 2, the median is 21,
which is the average of 20 and 22.
(2) The question of what constitutes an outlier is not always clear. Although statisticians seek to minimize subjectivity in the
definition of outliers, different analysts may choose different criteria for the same data set.
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Algebra
Introduction to Functions
Definitions
•
•
A Relation is a relationship between variables, usually expressed as an equation.
In a typical xy equation, the Domain of a relation is the set of x‐values for which y‐
•
values can be calculated. For example, in the relation
0
√ the domain is
because these are the values of x for which a square root can be taken.
In a typical xy equation, the Range of a relation is the set of y‐values that result for all
•
•
values of the domain. For example, in the relation
0 because
√ the range is
these are the values of y that result from all the values of x.
A Function is a relation in which each element in the domain has only one
corresponding element in the range.
A One‐to‐One Function is a function in which each element in the range is produced by
only one element in the domain.
Function Tests in 2‐Dimensions
Vertical Line Test – If a vertical line passes through the graph of a relation in any two locations,
it is not a function. If it is not possible to construct a vertical line that passes through the graph
of a relation in two locations, it is a function.
Horizontal Line Test – If a horizontal line passes through the graph of a function in any two
locations, it is not a one‐to‐one function. If it is not possible to construct a horizontal line that
passes through the graph of a function in two locations, it is a one‐to‐one function.
Examples:
Figure 1:
Figure 2:
Not a function.
Figure 3:
Fails vertical line test.
Is a function, but not a one‐
to‐one function.
Passes vertical line test.
Passes vertical line test.
Passes horizontal line test.
Version 2.5
Is a one‐to‐one function.
Fails horizontal line test.
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Algebra
Special Integer Functions
Greatest Integer Function
Also called the Floor Function, this function gives the
greatest integer less than or equal to a number. There
are two common notations for this, as shown in the
examples below.
Notation and examples:
3.5
3
2.7
3
6
6
2.4
2
7.1
8
0
0
In the graph to the right, notice the solid dots on the left of the segments (indicating the points are
included) and the open lines on the right of the segments (indicating the points are not included).
Least Integer Function
Also called the Ceiling Function, this function gives the
least integer greater than or equal to a number. The
common notation for this is shown in the examples
below.
Notation and examples:
3.5
4
2.7
2
6
6
In the graph to the right, notice the open dots on the
left of the segments (indicating the points are not included) and the closed dots on the right of the
segments (indicating the points are included).
Nearest Integer Function
Also called the Rounding Function, this function gives
the nearest integer to a number (rounding to the even
number when a value ends in .5). There is no clean
notation for this, as shown in the examples below.
Notation and examples:
3.5
4
2.7
3
6
6
In the graph to the right, notice the open dots on the
left of the segments (indicating the points are not
included) and the closed dots on the right of the segments (indicating the points are included).
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Algebra
Operations with Functions
Function Notation
Function notation replaces the variable y with a function name. The x in parentheses indicates
that x is the domain variable of the function. By convention, functions tend to use the letters f,
g, and h as names of the function.
Operations with Functions
The domain of the combination
of functions is the intersection
of the domains of the two
individual functions. That is,
the combined function has a
value in its domain if and only if
the value is in the domain of
each individual function.
Adding Functions
Subtracting Functions
·
Multiplying Functions
·
,
Dividing Functions
0
Examples:
Let:
1 Then:
2
1
1
·
1,
Note that in
there is the requirement
1
1. This is because
1
0 in the
denominator would require dividing by 0, producing an undefined result.
Other Operations
Other operations of equality also hold for functions, for example:
·
·
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·
·
·
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Algebra
Composition of Functions
In a Composition of Functions, first one function is performed, and then the other. The
. In both of these notations,
notation for composition is, for example:
or
the function g is performed first, and then the function f is performed on the result of g.
Always perform the function closest to the variable first.
Double Mapping
A composition can be thought of as a double mapping. First g maps from its domain to its
range. Then, f maps from the range of g to the range of f:
Range of g
Domain of g
Range of f
Domain of f
g
f
The Words Method
Example: Let
1
and
Then:
And:
In the example,
• The function says square the argument.
• The function says add 1 to the argument.
Sometimes it is easier to think of the functions in
words rather than in terms of an argument like x.
says "add 1 first, then square the result."
says "square first, then add 1 to the result."
Using the words method,
Calculate:
o
g: add 1 to it 12
f: square it
Version 2.5
12
Calculate:
1
f: square it
g: add 1 to it 4
2
o
2
4
1
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Algebra
Inverses of Functions
In order for a function to have an inverse, it must be a one‐to‐one function. The requirement
for a function to be an inverse is:
The notation
is used for the Inverse Function of
Another way of saying this is that if
.
, then
for all in the domain of .
Deriving an Inverse Function
The following steps can be used to derive an inverse function. This process assumes that the
original function is expressed in terms of
.
•
•
•
•
•
•
•
Make sure the function is one‐to‐one. Otherwise it has no inverse. You can accomplish
this by graphing the function and applying the vertical and horizontal line tests.
Substitute the variable y for
.
Exchange variables. That is, change all the x's to y's and all the y's to x's.
Solve for the new y in terms of the new x.
(Optional) Switch the expressions on each side of the equation if you like.
Replace the variable y with the function notation
.
Check your work.
Examples:
Substitute for
Subtract 2:
Multiply by 3:
Switch sides:
2
3
6
3
o
Version 2.5
3
1
2
Switch sides:
:
6
Change Notation:
6
1
Divide by 2:
To check the result, note that:
To check the result, note that:
2
Add 1:
Change Notation:
1
3
3
1
Exchange variables:
2
Exchange variables:
1
2
Substitute for
2
:
2
Derive the inverse of:
2
Derive the inverse of:
2
6
o
1
2
2
1
2
1
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Algebra
Transformation – Translation
A Translation is a movement of the graph of a relation to a different location in the plane. It
preserves the shape and orientation of the graph on the page. Alternatively, a translation can
be thought of as leaving the graph where it is and moving the axes around on the plane.
In Algebra, the translations of primary interest are the vertical and horizontal translations of a
graph.
Vertical Translation
Starting form:
Vertical Translation:
At each point, the graph of the translation is units higher or
lower depending on whether is positive or negative. The
letter is used as a convention when moving up or down. In
algebra, usually represents a y‐value of some importance.
Note:
• A positive shifts the graph up.
• A negative shifts the graph down.
Horizontal Translation
Starting form:
Horizontal Translation:
At each point, the graph of the translation is units to
the left or right depending on whether is positive or
negative. The letter is used as a convention when
moving left or right. In algebra, usually represents an
x‐value of some importance.
Note:
• A positive shifts the graph to the left.
• A negative shifts the graph to the right.
For horizontal translation, the direction of movement of the graph is counter‐intuitive; be
careful with these.
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Algebra
Transformation – Vertical Stretch and Compression
A Vertical Stretch or Compression is a stretch or compression in the vertical direction, relative
to the x‐axis. It does not slide the graph around on the plane like a translation. An alternative
view of a vertical stretch or compression would be a change in the scale of the y‐axis.
Vertical Stretch
Starting form:
Vertical Stretch:
·
,
1
At each point, the graph is stretched vertically by a factor of
. The result is an elongated curve, one that exaggerates all
of the features of the original.
Vertical Compression
Starting form:
·
Vertical Compression:
,
1
At each point, the graph is compressed vertically by a
factor of . The result is a flattened‐out curve, one that
mutes all of the features of the original.
Note: The forms of the equations
for vertical stretch and vertical
compression are the same. The
only difference is the value of " ".
Value of " " in
·
Resulting Curve
0
reflection
x‐axis
1
compression
Version 2.5
original curve
1
stretch
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Algebra
Transformation – Horizontal Stretch and Compression
A Horizontal Stretch or Compression is a stretch or compression in the horizontal direction,
relative to the y‐axis. It does not slide the graph around on the plane like a translation. An
alternative view of a horizontal stretch or compression would be a change in the scale of the x‐
axis.
Horizontal Stretch
Starting form:
,
Horizontal Stretch:
At each point, the graph is stretched horizontally
by a factor of . The result is a widened curve, one
that exaggerates all of the features of the original.
Horizontal Compression
Starting form:
,
Horizontal Compression:
At each point, the graph is compressed horizontally by a
factor of . The result is a skinnier curve, one that mutes
Note: The forms of the equations
for the horizontal stretch and the
horizontal compression are the
same. The only difference is the
value of " ".
all of the features of the original.
Value of " " in
0
reflection
horizontal line
1
Resulting Curve
stretch
original curve
1
compression
Note: For horizontal stretch and compression, the change in the graph caused by the value
of "b" is counter‐intuitive; be careful with these.
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Algebra
Transformation – Reflection
A Reflection is a "flip" of the graph across a mirror in the plane. It preserves the shape the
graph but can make it look "backwards."
In Algebra, the reflections of primary interest are the reflections across an axis in the plane.
X‐Axis Reflection
Starting form:
x‐axis Reflection:
Note the following:
•
•
•
Version 2.5
Y‐Axis Reflection
Starting form:
At each point, the graph is
reflected across the x‐axis.
The form of the transformation is
the same as a vertical stretch or
compression with
.
The flip of the graph over the x‐
axis is, in effect, a vertical
transformation.
y‐axis Reflection:
Note the following:
•
•
•
At each point, the graph is
reflected across the y‐axis.
The form of the transformation is
the same as a horizontal stretch
or compression with
.
The flip of the graph over the y‐
axis is, in effect, a horizontal
transformation.
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Algebra
Transformations – Summary
Starting form:
For purposes of the following table, the variables h and k are positive to make the forms more
like what the student will encounter when solving problems involving transformations.
Transformation Summary
Form of Transformation
Result of Transformation
Vertical translation up k units.
Vertical translation down k units.
Horizontal translation left h units.
Horizontal translation right h units.
·
·
,
,
,
,
1
Vertical stretch by a factor of .
Vertical compression by a factor of .
1
1
Horizontal compression by a factor of .
1
Horizontal stretch by a factor of .
Reflection across the x‐axis (vertical).
Reflection across the y‐axis (horizontal).
Transformations based on the values
of "a" and "b" (stretches,
compressions, reflections) can be
represented by these graphics.
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Algebra
Building a Graph with Transformations
The graph of an equation can be built with blocks made up of transformations. As an example,
we will build the graph of
2
3
4.
Step 1: Start with the basic
quadratic equation:
Step 2: Translate 3 units to
the right to get equation:
Step 3: Stretch vertically by
a factor of 2 to get equation:
Step 5: Translate up 4
units to get equation:
Final Result: Show the graph
of the final equation:
Step 4: Reflect over the
x‐axis to get equation:
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Algebra
Slope of a Line
The slope of a line tells how fast it rises or falls as it moves from left to right. If the slope is
rising, the slope is positive; if it is falling, the slope is negative. The letter "m" is often used as
the symbol for slope.
The two most useful ways to calculate the slope of a line are discussed below.
Mathematical Definition of Slope
The definition is based on two points with
coordinates ,
and ,
. The definition,
then, is:
Comments:
•
You can select any 2 points on the line.
•
A table such as the one at right can be helpful for doing
your calculations.
Note that
Point 2
Point 1
•
implies that
.
So, it does not matter which point you assign as Point 1
and which you assign as Point 2. Therefore, neither does
it matter which point is first in the table.
•
x‐value
Difference
y‐value
It is important that once you assign a point as Point 1 and another as Point 2, that you use
their coordinates in the proper places in the formula.
Examples:
For the two lines in the figure above, we get the following:
Green Line
x‐value
y‐value
Point A
1
4
Point C
‐3
‐4
Difference
4
Red Line
x‐value
y‐value
Point D
4
‐2
Point B
‐4
2
Difference
8
‐4
8
Green Line:
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Red Line:
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Algebra
Slope of a Line (cont'd)
Rise over Run
An equivalent method of calculating slope that is more
visual is the "Rise over Run" method. Under this
method, it helps to draw vertical and horizontal lines
that indicate the horizontal and vertical distances
between points on the line.
The slope can then be calculated as follows:
=
The rise of a line is how much it increases (positive) or decreases (negative) between two
points. The run is how far the line moves to the right (positive) or the left (negative) between
the same two points.
Comments:
•
You can select any 2 points on the line.
•
It is important to start at the same point in measuring both the rise and the run.
•
A good convention is to always start with the point on the left and work your way to the
right; that way, the run (i.e., the denominator in the formula) is always positive. The only
exception to this is when the run is zero, in which case the slope is undefined.
•
If the two points are clearly marked as integers on a graph, the rise and run may actually be
counted on the graph. This makes the process much simpler than using the formula for the
definition of slope. However, when counting, make sure you get the right sign for the slope
of the line, e.g., moving down as the line moves to the right is a negative slope.
Examples:
For the two lines in the figure above, we get the following:
Green Line:
Red Line:
Version 2.5
Notice how similar the
calculations in the examples
are under the two methods
of calculating slopes.
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Algebra
Slopes of Various Lines
4
5
line is steep and going down
2
line is vertical
When you look at a line, you
should notice the following
about its slope:
•
Whether it is 0, positive,
negative or undefined.
•
If positive or negative,
whether it is less than 1,
about 1, or greater than 1.
1
line goes down at a 45⁰ angle
1
2
line is steep and going up
3
The purpose of the graphs on
this page is to help you get a feel
for these things.
This can help you check:
•
Given a slope, whether you
drew the line correctly, or
•
1
line goes up at a 45⁰ angle
Given a line, whether you
calculated the slope
correctly.
3
17
line is shallow and going down
2
11
line is shallow and going up
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0
line is horizontal
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Algebra
Various Forms of a Line
There are three forms of a linear equation which are most useful to the Algebra student, each
of which can be converted into the other two through algebraic manipulation. The ability to
move between forms is a very useful skill in Algebra, and should be practiced by the student.
Standard Form
The Standard Form of a linear equation is:
Standard Form Examples
3
where A, B, and C are real numbers and A and B are not both zero.
Usually in this form, the convention is for A to be positive.
2
6
2
7
14
Why, you might ask, is this "Standard Form?" One reason is that this form is easily extended to
additional variables, whereas other forms are not. For example, in four variables, the Standard
Form would be:
. Another reason is that this form easily lends itself
to analysis with matrices, which can be very useful in solving systems of equations.
Slope‐Intercept Form
The Slope‐Intercept Form of a linear equation is the one most
familiar to many students. It is:
Slope‐Intercept Examples
3
3
4
6
14
where m is the slope and b is the y‐intercept of the line (i.e., the
value at which the line crosses the y‐axis in a graph). m and b must also be real numbers.
Point‐Slope Form
The Point‐Slope Form of a linear equation is the one used least by
the student, but it can be very useful in certain circumstances. In
particular, as you might expect, it is useful if the student is asked for
the equation of a line and is given the line's slope and the
coordinates of a point on the line. The form of the equation is:
Point‐Slope Examples
3
2
4
7
5
2
3
is any point on the line. One strength of this form is that
where m is the slope and ,
equations formed using different points on the same line will be equivalent.
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Algebra
Slopes of Parallel and Perpendicular Lines
Parallel Lines
Two lines are parallel if their slopes are equal.
•
In
the same.
form, if the values of are
Example:
•
3
1
and
In Standard Form, if the coefficients of and
are proportional between the equations.
Example: 3
6
•
2
2
2
4
5
and
7
Also, if the lines are both vertical (i.e., their
slopes are undefined).
Example:
3 and
2
Perpendicular Lines
Two lines are perpendicular if the product of their
slopes is . That is, if the slopes have different
signs and are multiplicative inverses.
•
In
form, the values of
multiply to get 1..
Example:
•
5
and
3
In Standard Form, if you add the product of
the x‐coefficients to the product of the y‐
coefficients and get zero.
Example: 4
3
•
6
6
2
4 and
5 because 4 · 3
6·
2
0
Also, if one line is vertical (i.e., is undefined) and one line is horizontal (i.e.,
Example:
6 and
3
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0).
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Algebra
Parallel, Perpendicular or Neither
The following flow chart can be used to determine whether a pair of lines are parallel,
perpendicular, or neither.
First, put both lines in:
form.
Are the
slopes of the
two lines the
same?
yes
Result: The
lines are
parallel.
yes
Result: The lines
are
perpendicular.
no
Is the
product of
the two
slopes = ‐1?
no
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Result: The
lines are
neither.
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Algebra
Parallel, Coincident or Intersecting
The following flow chart can be used to determine whether a pair of lines are parallel,
coincident, or intersecting. Coincident lines are lines that are the same, even though they may
be expressed differently. Technically, coincident lines are not parallel because parallel lines
never intersect and coincident lines intersect at all points on the line.
First, put both lines in:
form.
Are the
slopes of the
two lines the
same?
yes
Are the y‐
intercepts of
the two lines
the same?
yes
Result: The
lines are
coincident.
no
no
Result: The
lines are
intersecting.
Result: The
lines are
parallel.
The intersection of the two lines is:
•
For intersecting lines, the point of intersection.
•
For parallel lines, the empty set, .
•
For coincident lines, all points on the line.
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Algebra
Properties of Inequality
For any real numbers a, b, and c:
Property
Definition
Addition
Property
,
,
Subtraction
Property
,
,
Multiplication For
Property
0,
For
0,
For
·
·
,
·
·
,
Division
Property
,
·
·
,
·
·
0,
For
0,
,
,
,
,
Note: all properties which hold for "<" also hold for "≤", and all properties which hold for ">"
also hold for "≥".
There is nothing too surprising in these properties. The most important thing to be obtained
from them can be described as follows: When you multiply or divide an inequality by a
negative number, you must "flip" the sign. That is, "<" becomes ">", ">" becomes "<", etc.
In addition, it is useful to note that you can flip around an entire inequality as long as you keep
the "pointy" part of the sign directed at the same item. Examples:
4
is the same as
4
2
is the same as
3
3
Version 2.5
2
One way to remember this
is that when you flip around
an inequality, you must also
flip around the sign.
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Algebra
Graphs of Inequalities in One Dimension
Inequalities in one dimension are generally graphed on the number line. Alternatively, if it is
clear that the graph is one‐dimensional, the graphs can be shown in relation to a number line
but not specifically on it (examples of this are on the next page).
One‐Dimensional Graph Components
•
The endpoint(s) – The endpoints for the ray or segment in the graph are shown as either
open or closed circles.
o If the point is included in the solution to the inequality (i.e., if the sign is ≤ or ≥), the
circle is closed.
o If the point is not included in the solution to the inequality (i.e., if the sign is < or >),
the circle is open.
•
The arrow – If all numbers in one direction of the number line are solutions to the
inequality, an arrow points in that direction.
o For < or ≤ signs, the arrow points to the left ( ).
o For > or ≥ signs, the arrow points to the right ( ).
•
The line – in a simple inequality, a line is drawn from the endpoint to the arrow. If there are
two endpoints, a line is drawn from one to the other.
Examples:
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Algebra
Compound Inequalities in One Dimension
Compound inequalities are a set of inequalities that must all be true at the same time. Usually,
there are two inequalities, but more than two can also form a compound set. The principles
described below easily extend to cases where there are more than two inequalities.
Compound Inequalities with the Word "AND"
An example of compound inequalities with the word "AND" would be:
12
2
(Simple Form)
or
1
(Compound Form)
These are the same conditions,
expressed in two different forms.
Graphically, "AND" inequalities exist at points where the graphs of the individual inequalities
overlap. This is the "intersection" of the graphs of the individual inequalities. Below are two
examples of graphs of compound inequalities using the word "AND."
A typical "AND" example: The result is a
segment that contains the points that overlap
the graphs of the individual inequalities.
"AND" compound inequalities sometimes result
in the empty set. This happens when no
numbers meet both conditions at the same time.
Compound Inequalities with the Word "OR"
Graphically, "OR" inequalities exist at points where any of the original graphs have points. This
is the "union" of the graphs of the individual inequalities. Below are two examples of graphs of
compound inequalities using the word "OR."
A typical "OR" example: The result is a pair of
rays extending in opposite directions, with a
gap in between.
Version 2.5
"OR" compound inequalities sometimes result in
the set of all numbers. This happens when every
number meets at least one of the conditions.
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Algebra
Inequalities in Two Dimensions
Graphing an inequality in two dimensions involves the following steps:
•
Graph the underlying equation.
•
Make the line solid or dotted based on whether the inequality contains an "=" sign.
o For inequalities with "<" or ">" the line is dotted.
o For inequalities with "≤" or "≥" the line is solid.
•
Determine whether the region containing the solution set is above the line or below the
line.
o For inequalities with ">" or "≥" the shaded region is above the line.
o For inequalities with "<" or "≤" the shaded region is below the line.
•
Shade in the appropriate region.
Example:
Graph the solution set of the following system of inequality:
1
Step 1: Graph the underlying
equation.
Step 2: Determine whether the line
should be solid or dotted:
1 the > sign does not
contain "=", so the line is dotted
Step 3: Determine the region to be
shaded based on the sign in the
equation:
1 the > sign indicates
shading above the line
The solution set is the shaded area.
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Algebra
Graphs of Inequalities in Two Dimensions
Dashed Line
Below the Line
Dashed Line
Above the Line
Solid Line
Below the Line
Solid Line
Above the Line
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Algebra
Absolute Value Functions (cont'd)
Inequalities
Since a positive number and a negative number can have the same absolute value, inequalities
involving absolute values must be broken into two separate equations. For example:
3
3|
|
The first new equation is simply the original
equation without the absolute value sign.
4
4
3
Sign that determines
use of "AND" or "OR"
4
In the second new equation, two things
change: (1) the sign flips, and (2) the value on
the right side of the inequality changes its sign.
At this point the absolute value problem has converted into a pair of compound inequalities.
Equation 1
Equation 2
Solve:
Step 1: Add 3
Result:
3
3
4
3
7
Solve:
Step 1: Add 3
Result:
3
3
4
3
1
Next, we need to know whether to use "AND" or "OR" with the results. To decide which word
to use, look at the sign in the inequality; then …
•
•
Use the word "AND" with "less thand" signs.
Use the word "OR" with "greator" signs.
Note: the English is poor, but the math
is easier to remember with this trick!
The solution to the above absolute value problem, then, is the same as the solution to the
following set of compound inequalities:
7
1
The solution set is all x in the range (‐1, 7)
Note: the solution set to this example is given in "range" notation. When using this notation,
• use parentheses ( ) whenever an endpoint is not included in the solution set, and
• use square brackets [ ] whenever an endpoint is included in the solution set.
• Always use parentheses ( ) with infinity signs ( ∞ ∞).
Examples:
Version 2.5
The range:
6
Notation: 2, 6
2
The range:
Notation:
2
∞, 2
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Algebra
Systems of Equations
A system of equations is a set of 2 or more equations for which we wish to determine all
solutions which satisfy each equation. Generally, there will be the same number of equations
as variables and a single solution to each variable will be sought. However, sometimes there is
either no solution or there is an infinite number of solutions.
There are many methods available to solve a system of equations. We will show three of them
below.
Graphing a Solution
In the simplest cases, a set of 2 equations in 2 unknowns can be solved using a graph. A single
equation in two unknowns is a line, so two equations give us 2 lines. The following situations
are possible with 2 lines:
•
•
•
They will intersect. In this case, the point of intersection is the only solution.
They will be the same line. In this case, all points on the line are solutions (note: this is
an infinite set).
They will be parallel but not the same line. In this case, there are no solutions.
Examples
Solution Set:
The point of intersection
can be read off the graph;
the point (2,0).
Version 2.5
Solution Set:
The empty set;
these parallel lines
will never cross.
Solution Set:
All points on the line.
Although the equations look
different, they actually
describe the same line.
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Algebra
Systems of Equations (cont'd)
Substitution Method
In the Substitution Method, we eliminate one of the variables by substituting into one of the
equations its equivalent in terms of the other variable. Then we solve for each variable in turn
and check the result. The steps in this process are illustrated in the example below.
Example: Solve for x and y if:
and: 2
.
Step 1: Review the two equations. Look for a variable that can be substituted from one
equation into the other. In this example, we see a single "y" in the first equation; this is a prime
candidate for substitution.
We will substitute
from the first equation for in the second equation.
Step 2: Perform the substitution.
becomes:
Step 3: Solve the resulting equation for the single variable that is left.
Step 4: Substitute the known variable into one of the original equations to solve for the
remaining variable.
After this step, the solution is tentatively identified as:
,
, meaning the point (3, 1).
Step 5: Check the result by substituting the solution into the equation not used in Step 4. If the
solution is correct, the result should be a true statement. If it is not, you have made a mistake
and should check your work carefully.
Version 2.5
Since this is a true mathematical
statement, the solution (3, 1) can
be accepted as correct.
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Algebra
Systems of Equations (cont'd)
Elimination Method
In the Substitution Method, we manipulate one or both of the equations so that we can add
them and eliminate one of the variables. Then we solve for each variable in turn and check the
result. This is an outstanding method for systems of equations with "ugly" coefficients. The
steps in this process are illustrated in the example below. Note the flow of the solution on the
page.
Example: Solve for x and y if:
and: 2
Step 1: Re‐write the equations in
standard form.
2
.
Step 2: Multiply each equation by a value
selected so that, when the equations are added,
a variable will be eliminated.
(Multiply by 2)
(Multiply by ‐1)
2
Step 3: Add the resulting equations.
Step 5: Substitute the result into
one of the original equations and
solve for the other variable.
2
U
Step 4: Solve for the variable.
Step 6: Check the result by substituting
the solution into the equation not used in
Step 5. If the solution is correct, the
result should be a true statement. If it is
not, you have made a mistake and should
check your work.
Version 2.5
Since this is a true mathematical statement, the
solution (3, 1) can be accepted as correct.
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Algebra
Systems of Equations (cont'd)
Classification of Systems
There are two main classifications of systems of equations: Consistent vs. Inconsistent, and
Dependent vs. Independent.
Consistent vs. Inconsistent
•
Consistent Systems have one or more solutions.
•
Inconsistent Systems have no solutions. When you try to solve an inconsistent set of
equations, you often get to a point where you have an impossible statement, such as
"1 2." This indicates that there is no solution to the system.
Dependent vs. Independent
•
Linearly Dependent Systems have an infinite number of solutions. In Linear Algebra, a
system is linearly dependent if there is a set of real numbers (not all zero) that, when
they are multiplied by the equations in the system and the results are added, the final
result is zero.
•
Linearly Independent Systems have at most one solution. In Linear Algebra, a system is
linearly independent if it is not linearly dependent. Note: some textbooks indicate that
an independent system must have a solution. This is not correct; they can have no
solutions (see the middle example below). For more on this, see the next page.
Examples
One Solution
Consistent
Independent
No Solution
Inconsistent
Independent
Infinite Solutions
Consistent
Dependent
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ADVANCED
Algebra
Linear Dependence
Linear dependence is a concept from Linear Algebra, and is very useful in determining if
solutions to complex systems of equations exist. Essentially, a system of functions is defined
to be linearly dependent if there is a set of real numbers (not all zero), such that:
…
0 or, in summation notation,
0
If there is no set of real numbers , such that the above equations are true, the system is said
to be linearly independent.
is called a linear combination of the functions . The
The expression
importance of the concept of linear dependence lies in the recognition that a dependent
system is redundant, i.e., the system can be defined with fewer equations. It is useful to note
that a linearly dependent system of equations has a determinant of coefficients equal to 0.
Example:
Consider the following system of equations:
Notice that:
.
Therefore, the system is linearly
dependent.
Checking the determinant of the coefficient matrix:
3
1
1
2
1
0
1
2
5
1
2
1
1
2
0
3
1
1
2
5
3
1
2
1
1
5
0 7
5 1
0.
It should be noted that the fact that D 0 is sufficient to prove linear dependence only if there
are no constant terms in the functions (e.g., if the problem involves vectors). If there are
constant terms, it is also necessary that these terms combine "properly." There are additional
techniques to test this, such as the use of augmented matrices and Gauss‐Jordan Elimination.
Much of Linear Algebra concerns itself with sets of equations that are linearly independent. If
the determinant of the coefficient matrix is non‐zero, then the set of equations is linearly
independent.
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Algebra
Systems of Inequalities in Two Dimensions
Systems of inequalities are sets of more than one inequality. To graph a system of inequalities,
graph each inequality separately (including shading in the appropriate region). The solution set,
then, is either the overlap of the regions of the separate inequalities ("AND" Systems) or the
union of the regions of the separate inequalities ("OR" Systems).
Examples:
Graph the solution set of the following system of inequalities:
2
(a)
3 AND
1
(b)
2
3 OR
1
Step 1: Graph the underlying equations.
Step 2: Determine whether each line should be
solid or dotted:
2
3 the ≤ sign contains "=", so the
line is solid
1 the > sign does not contain "=",
so the line is dotted
Step 3: Determine the regions to be shaded based on the signs in the equations:
•
2
•
3 the ≤ sign indicates shading below the line
1 the > sign indicates shading above the line
Step 4: Determine the final solution set.
(a) If the problem has an "AND" between
the inequalities, the solution set is the
overlap of the shaded areas (i.e., the
green part in the graph below).
Version 2.5
(b) If the problem has an "OR" between
the inequalities, the solution set is the
union of all of the shaded areas (i.e.,
the blue part in the graph below).
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Algebra
Parametric Equations
Parametric Equations in 2 dimensions are functions that express each of the two key variables
in terms of a one or more others. For example,
in terms of a one or more others. For example,
Parametric equations are sometimes the most useful way to solve a problem.
Pythagorean Triples
As an example, the following parametric equations can be used to find Pythagorean Triples:
Let , be relatively prime integers and let
. Then, the following equations produce a set
of integer values that satisfy the Pythagorean Theorem:
Examples:
s
t
a
b
c
3
2
5
12
13
5
12
13
4
3
7
24
25
7
24
25
5
2
21
20
29
21
20
29
5
3
16
30
34
16
30
34
Pythagorean Relationship
Creating a Standard Equation from Parametric Equations
To create a standard equation from a set of
parametric equations in two dimensions,
•
•
•
Solve one parametric equation for t.
Substitute this value of t into the other
equation.
Clean up the remaining expression as
necessary.
Note: any other method of solving
simultaneous equations can also be used for
this purpose.
Example: Create a standard equation for the
parametric equations:
Solving for t in the first equation, we get:
Substituting into the second equation gives:
Cleaning this up, we get the solution we seek:
seek:
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Algebra
Exponent Formulas
Word Description
Math Description
Limitations
of Property
of Property
on variables
·
Product of Powers
Examples
·
·
Quotient of Powers
·
Power of a Power
Anything to the zero power is 1
, if , ,
Negative powers generate the
reciprocal of what a positive
power generates
Power of a product
·
·
Power of a quotient
Converting a root to a power
√
√
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Algebra
Scientific Notation
Format
A number in scientific notation has two parts:
•
A number which is at least 1 and is less than 10 (i.e., it must have only one digit before
the decimal point). This number is called the coefficient.
A power of 10 which is multiplied by the first number.
•
Here are a few examples of regular numbers expressed in scientific notation.
32
3.2
1,420,000
0.00034
10
1.42
10
3.4
1000
1
10
10
1
450
1
10
4.5
10
How many digits? How many zeroes?
There are a couple of simple rules for converting from scientific notation to a regular number or
for converting from a regular number to scientific notation:
•
If a regular number is less than 1, the exponent of 10 in scientific notation is negative.
The number of leading zeroes in the regular number is equal to the absolute value of
this exponent. In applying this rule, you must count the zero before the decimal point in
the regular number. Examples:
Original Number
Action
Conversion
0.00034
Count 4 zeroes
3.4 x 10‐4
6.234 x 10‐8
Add 8 zeroes before the digits
0.000 000 062 34
•
If the number is greater than 1, the number of digits after the first one in the regular
number is equal to the exponent of 10 in the scientific notation.
Action
Conversion
4,800,000
Count 6 digits after the "4"
4.8 x 106
•
Original Number
9.6 x 103
Add 3 digits after the "9"
9,600
As a general rule, multiplying by powers of 10 moves the decimal point one place for
each power of 10.
o Multiplying by positive powers of 10 moves the decimal to the right.
o Multiplying by negative powers of 10 moves the decimal to the left.
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Algebra
Adding and Subtracting with Scientific Notation
When adding or subtracting numbers in scientific notation:
•
•
•
Adjust the numbers so they have the same power of 10. This works best if you adjust
the representation of the smaller number so that it has the same power of 10 as the
larger number. To do this:
o Call the difference between the exponents of 10 in the two numbers "n".
o Raise the power of 10 of the smaller number by "n", and
o Move the decimal point of the coefficient of the smaller number "n" places to
the left.
Add the coefficients, keeping the power of 10 unchanged.3.2
10
0.32 10
9.9
10
9.90
10
10. 22
10
1.022
10
Explanation: A conversion of the smaller
number is required prior to adding because the
exponents of the two numbers are different.
After adding, the result is no longer in scientific
notation, so an extra step is needed to convert it
into the appropriate format.
6.1
10
6.1
10
2.3
10
2.3
10
8. 4
10
1.2
10
1.20
10
4.5
10
0.45
10
0.75
10
Explanation: No conversion is necessary
because the exponents of the two numbers are
the same. After adding, the result is in scientific
notation, so no additional steps are required.
Version 2.5
7.5
10
Explanation: A conversion of the smaller
number is required prior to subtracting because
the exponents of the two numbers are different.
After subtracting, the result is no longer in
scientific notation, so an extra step is needed to
convert it into the appropriate format.
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Algebra
Multiplying and Dividing with Scientific Notation
When multiplying or dividing numbers in scientific notation:
•
Multiply or divide the coefficients.
•
Multiply or divide the powers of 10. Remember that this means adding or subtracting
the exponents while keeping the base of 10 unchanged.
o If you are multiplying, add the exponents of 10.
o If you are dividing, subtract the exponents of 10.
•4
10
· 5
10
20
10
10
2.0
Explanation: The coefficients are multiplied and
the exponents are added. After multiplying, the
result is no longer in scientific notation, so an
extra step is needed to convert it into the
appropriate format.
10
10
2. 4
10
3.3
1.2
· 2.0
10
Explanation: The coefficients are multiplied and
the exponents are added. After multiplying, the
result is in scientific notation, so no additional
steps are required.
5.5
10
0.6
10
Version 2.5
6.0
10
Explanation: The coefficients are divided and
the exponents are subtracted. After dividing,
the result is no longer in scientific notation, so
an extra step is needed to convert it into the
appropriate format.
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Algebra
Introduction to Polynomials
What is a Polynomial?
A polynomial is an expression that can be written as a term or a sum of terms, each of which is
the product of a scalar (the coefficient) and a series of variables. Each of the terms is also called
a monomial.
Examples (all of these are polynomials):
3
Monomial
2
Binomial
4
8
15
6
Trinomial
4
Other
9
6
4
12
7
1
2
6
3
3
8
2
Definitions:
Scalar: A real number.
Monomial: Polynomial with one term.
Binomial: Polynomial with two terms.
Trinomial: Polynomial with three terms.
Degree of a Polynomial
The degree of a monomial is the sum of the exponents on its variables.
The degree of a polynomial is the highest degree of any of its monomial terms.
Examples:
Polynomial
Degree
6
0
3
3
1
15
3
3
Polynomial
Degree
6
6
9
12
7
3
5
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Algebra
Adding and Subtracting Polynomials
Problems asking the student to add or subtract polynomials are often written in linear form:
2
Add: 3
4
2
4
6
The problem is much more easily solved if the problem is written in column form, with each
polynomial written in standard form.
Definitions
Standard Form: A polynomial in standard form has its terms written from highest degree to
lowest degree from left to right.
3
Example: The standard form of
4 is 3
4
Like Terms: Terms with the same variables raised to the same powers. Only the numerical
coefficients are different.
, 6
Example: 2
, and
are like terms.
Addition and Subtraction Steps
Step 1: Write each polynomial in standard form. Leave blank spaces for missing terms. For
example, if adding 3
2
4 , leave space for the missing ‐term.
Step 2: If you are subtracting, change the sign of each term of the polynomial to be subtracted
and add instead. Adding is much easier than subtracting.
Step 3: Place the polynomials in column form, being careful to line up like terms.
Step 4: Add the polynomials.
Examples:
: 3
2
4
2
4
Solution:
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3
6
: 3
2
4
2
4
6
Solution:
2
4
3
2
4
6
3
2
6
2
2
4
2
4
6
3
2
10
2
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Algebra
Multiplying Binomials
The three methods shown below are equivalent. Use whichever one you like best.
FOIL Method
FOIL stands for First, Outside, Inside, Last. To multiply using the FOIL method, you make four
separate multiplications and add the results.
Example: Multiply 2
First:
2 ·3
Outside:
2 ·
Inside:
3· 3
Last:
3·
3 · 3
4
8
6
The result is obtained by adding the results of
the 4 separate multiplications.
4
9
4
2
3 · 3
F O I L
4
12
6
8
9
12
12
6
Box Method
The Box Method is pretty much the same as the FOIL method, except for the presentation. In
the box method, a 2x2 array of multiplications is created, the 4 multiplications are performed,
and the results are added.
Example: Multiply 2
Multiply
3x
2x
6
+3
3 · 3
4
The result is obtained by adding the results of
the 4 separate multiplications.
9
2
8
3 · 3
4
8
6
6
12
9
12
12
Stacked Polynomial Method
A third method is to multiply the binomials
like you would multiply 2‐digit numbers.
The name comes from how the two
polynomials are placed in a "stack" in
preparation for multiplication.
Example: Multiply 2
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3 · 3
2
3
·
6
8
9
6
3
4
12
12
4
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Algebra
Multiplying Polynomials
If the polynomials to be multiplied contain more than two terms (i.e., they are larger than
binomials), the FOIL Method will not work. Instead, either the Box Method or the Stacked
Polynomial Method should be used. Notice that each of these methods is essentially a way to
apply the distributive property of multiplication over addition.
The methods shown below are equivalent. Use whichever one you like best.
Box Method
The Box Method is the same for larger polynomials as it is for binomials, except the box is
bigger. An array of multiplications is created; the multiplications are performed; and like terms
are added.
2
Example: Multiply
Multiply
3 · 2
2
2
4
Results:
3
3
4
2
4
6
8
4
6
9
2
8
6
2
3 · 2
3
4
3
4
6
4
6
12
4
6
8
9
8
Stacked Polynomial Method
In the Stacked Polynomial Method, the
polynomials are multiplied using the same
technique to multiply multi‐digit numbers
One helpful tip is to place the smaller
polynomial below the larger one in the
stack.
Results:
2
3
· 2
3
4
4
8
8
12
3
6
6
9
2
4
4
6
2
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7
6
8
17
12
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Algebra
Dividing Polynomials
Dividing polynomials is performed much like dividing large numbers long‐hand.
Long Division Method
This process is best described by example:
This process is best described by example:
Example: 2
5
2
2
Step 1: Set up the division like a typical long hand
division problem.
2
2
Step 3: Multiply the new term on top by the divisor
and subtract from the dividend.
and subtract from the dividend.
2
2
4
2
2
2
2 2
5
2 2
2
2
2
5
2
5
4
2
2
This process results in the final answer appearing
above the dividend, so that:
5
Step 4: Repeat steps 2 and 3 on the remainder of
the division until the problem is completed.
2
2 2
Step 2: Divide the leading term of the dividend by
the leading term of the divisor. Place the result
above the term of like degree of the dividend.
2
2
2
1
5
4
2
2 2
2
2
1
2
2
2
Remainders
0
If there were a remainder, it would be appended to
the result of the problem in the form of a fraction, just like when dividing integers. For
the result of the problem in the form of a fraction, just like when dividing integers. For
example, in the problem above, if the remainder were 3, the fraction
would be added to
the result of the division. 2
1
5
1
2
2
Alternatives
This process can be tedious. Fortunately, there are better methods for dividing polynomials
than long division. These include Factoring, which is discussed next and elsewhere in this
Guide, and Synthetic Division, which is discussed in the chapter on Polynomials – Intermediate.
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In the late 50s, CP Snow, in his famous lecture, "The Two Cultures," lamented the fragmentation of the learning in academe and the deepening wedge between the humanities and social sciences on one hand, and math and the sciences on the other. There are humanists who feel that math is merely a tool for technology, a collection of formulas and symbols with no connection to the great themes of our culture. On the other hand, there are teachers of math who teach math as nothing but an IQ test, forcing students through the drudgery of drills in algebra and calculus.
The goal of GE mathematics is to dispel these notions and convey the message that math has been and is linked in fundamental ways to the development of culture and our ways of thought. Our GE program, guided by the spirit of the liberal arts, is an opportunity to teach math and science not as an academic obstacle course, but as an adventure in ideas that is exciting and relevant to understanding the world.
Math is commonly reflected in math books and research articles (or even in the classroom) as a formal axiomatic system, a collection of formulas students memorize. But math is about ideas not formulas. It is the systematic study of patterns; a way of looking and making sense of the world. It is the language of science and its practical applications pervade almost every aspect of our lives. In this paper, we will discuss the goals and format of our GE math offerings (Math 1 and 2). These courses are designed to provide the student a deeper understanding of and appreciation for the nature of mathematics through an exploration of its applications, as well as its intellectual, aesthetic, and humanistic aspects, which are just as important as its utility.
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Dr. Fidel Nemenzo
Professor, Institute | 677.169 | 1 |
Following the format of the GCSE: Revision and Practice series, this title is targeted at the 3-5 tier of entry. It is aimed at specific tier of entry to ensure that students are focused on the mathematical concepts they need to know. It includes worked examples, showing the key techniques of how to tackle problems and approach questions. | 677.169 | 1 |
"Philosophy is written in this grand book – I mean the universe – which stands continuous open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics." — Galileo
"I think it's not so much a question of what students study, as how we equip them to lead a life that involves regular use of certain skills [of quantitative reasoning]." — Keith Devlin, quoted in Forman (1997).
Introduction
When it comes to mathematics, we know what we must teach students majoring in science, engineering, or mathematics ("SEM" for short). These students need particular calculus-based skills that they will use regularly in their careers. But what about the rest of the collegiate population, the "non-SEM" students that easily comprise half of all students we teach in mathematics courses in this country? These students outnumber their SEM peers among all college students, and they represent a large majority among students at two-year colleges (National Science Board, 1998). Nevertheless, many institutions have given little serious thought to the development of appropriate mathematics curricula for non-SEM students. One result is that approaches to non-SEM mathematics courses are all over the board. Another, of which we are constantly reminded in conversations with mathematics teachers across the United States, is widespread dissatisfaction with these courses among both faculty and students.
We have spent much of the past twelve years working on the problem of mathematics for non-SEM students. In this paper, we will present some observations regarding the nature of the problem, along with our opinions regarding its solution.
Initial Goals: Ours and Theirs
The first step in developing any kind of curriculum must be to identify the goals to be achieved. As mathematicians, we resonate with the above quotation from Galileo and would like our students to feel the same sense of awe when they think of mathematics. Thus we might state an initial goal for mathematics courses for non-SEM students as follows: Show our students some of the beauty and utility of mathematics.
Unfortunately, we almost immediately encounter a problem in trying to achieve this goal: the vast majority of non-SEM students enter mathematics courses with very negative attitudes toward mathematics. In our courses at the University of Colorado, we have found that almost all of our students will openly declare themselves – with no hint of embarrassment – to be either "math-phobics" or "math-loathers." The math-phobics are afraid of mathematics and have done poorly in past (high school) mathematics classes. The math-loathers have done well in past mathematics courses and therefore are not afraid of mathematics; they just believe that it is worthless and irrelevant to their lives. Thus there is a conflict between "our goal" of demonstrating the beauty and utility of mathematics and "their goal" of getting away from mathematics as far and as fast as they can.
An Overarching Responsibility
This conflict between our initial goal and their initial goal requires that we find a way to bring the two goals into line. The key involves thinking about our obligations as teachers. As mathematicians, we possess skills and knowledge that are of critical importance to our students. It is our responsibility to transmit this information to our students.
If we succeed, we will find that, in the process, we have also shown students some of the beauty and utility of mathematics. Moreover, the students will find that what they fear or loathe is not real mathematics, but rather a caricature of mathematics (such as the common student belief that mathematics is nothing more than a set of skills) that has been set up for them by bad experiences in past courses. Once they see the relevance of mathematics to their own lives, their goals and ours will be one and the same.
Refined Goals: Mathematics for College, Careers, and Life
We are now ready to transform our initial goal of showing students some of the beauty and utility of mathematics into a set of goals that also meets our overarching responsibility of transmitting to students the mathematics that they need. The only question is, what type of mathematics do students need to know? We believe the answer is three-fold:
Students need mathematics for college. Most students take other collegiate courses, such as core courses in natural and social sciences, in which they will use mathematical skills.
Students need mathematics for their careers. Nearly all careers today (and the average student will have several careers in a lifetime) require an ability to reason with quantitative information and to discuss quantitative issues clearly and cogently.
Students need mathematics to understand the issues that confront them in daily life. In part, this means being able to deal with issues of personal finance. But it also means dealing with citizenship issues – such as the economy, public health, and the environment – on which they will be called upon to vote intelligently.
Note that these goals – mathematics for college, careers, and life – are goals that we and our students can share. Moreover, we have found that enunciating these goals to students almost immediately begins to break down the negative attitudes that they bring to mathematics. For the first time in most of their lives, students see teachers as being on "their side," working with them to help them learn crucial and relevant skills.
It is important to recognize that the latter goal, mathematics for life, is more pervasive and more subtle than is commonly realized. In our courses, we frequently use current events to illustrate how thoroughly mathematics permeates everyday life. Virtually every major issue of our day is at least partly quantitative in nature; consider, for example, the issues surrounding the 2000 census, the federal budget, "saving" social security, finding an appropriate government settlement with tobacco companies, the statistics of crime rates, the science and economics of global warming, and even the recent impeachment of the President. The mathematics in these subjects may be fairly explicit or quite subtle, but in all cases it is necessary for a full understanding of the issues.
What Course Meets These Goals?
Now that we have a clear set of goals, it is possible to begin designing a course for non-SEM students. A brief description of how we arrived at our own course is instructive. In 1987, one of us (Bennett) was a member of an interdisciplinary faculty committee charged with developing a new mathematics requirement for non-SEM students at the University of Colorado at Boulder. The committee consisted of a dozen members; about half of whom were drawn from the Mathematics Department, while the rest represented disciplines such as astronomy, biology, psychology, geography, and economics.
Early in the discussions, several committee members suggested that all students should be expected to complete a course in calculus. While this may be a noble objective, it was quickly discarded as impractical: most non-SEM students are unprepared for calculus upon entry to college and therefore could not meet this objective within the constraints of a one- or two-semester general education mathematics requirement.
With calculus ruled out, the committee next considered the only other existing mathematics courses taught at the University of Colorado at the time: pre-calculus courses such as College Algebra or Trigonometry. But after only brief discussion, the faculty committee also rejected these courses for the new requirement, largely for the following three reasons:
While the importance of algebra and other pre-calculus courses for SEM students is clear, it is much more difficult to make the case for non-SEM students. Aside from a bit of algebra, most non-SEM students will not use the skills learned in these courses in their other college courses, their careers, or their daily lives.
At the University of Colorado, all entering students have already taken at least one year's worth of high school algebra. The committee members therefore felt that it would be largely redundant to require students to take College Algebra – although this is commonly done. To paraphrase a colleague (Darrell Abney), such an approach means teaching the students essentially the same mathematics they were taught in high school, only this time "teaching it to them louder."
The committee members recognized that whatever mathematics course was required, it would be the last mathematics course that most of these students would ever take. The members therefore saw this requirement as a tremendous opportunity to teach students something new and important about mathematics.
Thus the committee decided that a new course would need to be created, and set about the task of identifying specific content.
Content Areas
Designing a new course means deciding upon content, so the University of Colorado committee focused most of its discussion on identifying appropriate content. In the end, the committee identified four major content areas. The original statement from the committee was a bit more terse, but we now identify the four areas as follows:
Logic, Critical Thinking, and Problem Solving: Students should learn skills that will enable them to construct a logical argument based on rules of inference and to develop strategies for solving quantitative problems.
Number Sense and Estimation: Students should become "numerate," or able to make sense of the numbers that confront them in the modern world. For example, students should be able to give meaning to a billion dollars, and distinguish it from a million dollars or a trillion dollars. Part of developing such number sense involves making simple calculations or estimates to put numbers in perspective. As a simple example, a student should be able to quickly figure out that a star athlete earning $10 million per year earns about 400 times more than the average American.
Statistical Interpretation and Basic Probability: Reports about statistical research (for example, concerning diet or disease) are ubiquitous in the news. Students must have the stools needed to interpret this research. Note the emphasis on interpretation. While it is certainly useful to show students how to calculate a mean, a standard deviation, or a margin of error, our non-SEM students will rarely perform such calculations once they leave our course. But they will encounter such statistics in the news, and we must equip them to interpret these statistics critically. Because statistical interpretation involves inference from samples to populations, it also requires a basic understanding of probability. This study of probability can then be easily extended to relevant topics including lotteries, casino gambling, risk assessment, and disease and drug testing.
Interpreting Graphs and Models: Graphical displays of numbers abound in modern media, so learning how to create and interpret graphs is clearly important. Though it is a bit less obvious, an understanding of modeling is equally important, because many major issues today (such as economic or environmental issues) are studied through mathematical models. While we do not expect non-SEM students to create sophisticated mathematical models, we must teach them how to interpret what they read or hear about models. For example, they should know enough to question the assumptions of a model before accepting its predictions, and they should understand the difference between linear and exponential growth. We group graphing and modeling together because we have found that one of the easiest ways to teach students about modeling is by presenting graphs as simple mathematical models.
These areas and their associated skills form the core of what is often called quantitative reasoning. Interestingly, we have found that almost every group that identifies appropriate content areas for non-SEM students reaches similar conclusions. For example, the AMATYC standards (AMATYC, 1995) list the following standards (identifiers in parentheses are from the AMATYC report): (I-1) problem solving; (I-2) modeling; (I-3) reasoning; (I-4) connecting with other disciplines; (I-5) communicating; (I-7) developing mathematical power; and (C-1) number sense. Although this list is organized differently from our list above, we maintain that both say essentially the same thing. A similar set of goals was enunciated in the MAA report on quantitative reasoning (MAA, 1995; p. 10).
It's Not Remedial
Before we continue, it's important to point out that our four quantitative reasoning content areas are certainly not remedial, even though they do not necessarily include much in the way of formal algebra. As a first example, consider the following quotation taken from a front-page article in the New York Times (4/20/97):
Teen-age smoking rates are still lower than in the 1970's. But the percentage of 12th graders who smoked daily last year jumped 20 percent since 1991, to 22 percent. The rate among 10th graders jumped 45 percent, to 18.3 percent, and the rate for 8th graders is up 44 percent, to 10.4 percent.
Most non-SEM students (and many SEM students as well) have a very difficult time interpreting the use of percentages in this statement. But simply teaching percentages in a remedial sense – that is, divide two numbers and multiply by 100% – will not solve the problem. Instead, students need well-developed critical thinking skills (Content Area 1) and number sense (Content Area 2) to interpret this statement, and these aptitudes are often not emphasized in standard high school mathematics curricula. Thus learning to interpret this statement is not remedial, even though it deals only with percentages.
As a second example, consider the federal budget (as it stands in March 1999). The politicians are very proud that, after decades of deficits, the government ran a $69 billion surplus in 1998. But looking deeper, you'll find that some people claim there was no surplus, but rather a $30 billion deficit. Moreover, while you might expect a surplus to reduce the national debt, the debt actually rose in 1998 by $113 billion!
From the standpoint of mathematical manipulation, reconciling these budget numbers requires nothing more than addition and subtraction. But that does not make it easy or remedial. In fact, understanding federal budget numbers requires skills from all four quantitative reasoning content areas: logic to follow the convoluted path by which the numbers are derived; number sense to understand the meaning of the numbers; statistical interpretation to understand how economic data are measured; and modeling to understand how the government forecasts future surpluses or deficits.
The Key to Success: A Context-Driven Approach
There are many possible ways to integrate the four content areas into a course syllabus, but we believe most of them can be categorized either as "content-driven" or "context-driven." A couple of examples should clarify the difference and illustrate why we believe the latter approach is superior.
Consider the topic of logic and critical thinking. A "content-driven" approach looks at logic as a mathematical content area that students should study for its own sake. This approach therefore begins by establishing the important mathematical ideas behind logic, such as sets, truth tables, and Venn diagrams, and then shows students how these ideas are useful. This approach is common in textbooks for non-SEM students. However, this content-driven approach immediately sets up the conflict discussed earlier between instructors' initial goals and students' initial goals: the instructor is trying to teach some mathematics, which the students would rather avoid.
In contrast, a "context-driven" approach begins by establishing a context that helps students understand why they should care about logic. In our course, for example, we begin by discussing common logical fallacies and critical thinking problems that appear in everyday situations (such as how to choose between the fare alternatives when buying an airplane ticket), thereby showing students the immediate relevance of logic to their lives. Only then do we introduce the essential mathematical ideas. In the end, both approaches to logic cover the same mathematical content. But the context-driven approach is far more successful because it sets up a shared goal between instructor and students – discussing a topic that all can agree is important to college, careers, and life.
As a second example, consider payments on loans, such as student loans, credit cards, or mortgages. In a content-driven approach, loan payments are taught as an application of exponential growth; after all, the formula for loan payments can be derived from compound interest considerations. Thus a content-driven approach starts with the mathematics of exponential growth, and eventually shows students that this has relevance to loan payments. Again, because students are shown the mathematics before being shown its relevance, this approach fails to bridge the gap between the initial goals of students and instructors.
A context-driven approach recognizes that most students have loans of some type (usually credit cards or student loans, and sometimes mortgages) and the topic of loan payments therefore can engage students in the more general topic of exponential growth. Thus, in our course, we teach students about loan payments before we cover exponential growth more generally. Moreover, whereas the content-driven approach is usually finished once it reaches the loan payment formula (which students can rightly argue is something they will not use, since banks or real estate agents usually do the calculations for them), the formula is only the beginning in the context-driven approach. We continue discussions of loan payments to show students how they can avoid getting into credit card trouble, how they can make decisions between adjustable-rate and fixed-rate mortgages, and how closing costs and fees can affect the cost of a loan. These are mathematical topics that involve logic, problem solving, number sense, and modeling. (Loan payments are only one of many contexts in which exponential growth can be introduced.)
In fact, when it comes to actual course construction, the four content areas identified earlier can be covered through an almost endless variety of specific applications. For example, if you want to cover the mathematics of voting, do it in the context of actual elections rather than starting from mathematical theory. If you want to cover geometry, do it in the context of art or architecture that students will appreciate, rather than as a set of abstract ideas. If you want to cover exponential modeling, begin by observing a population with a fixed doubling time. If you want to solve algebraic equations, do it in the context of linear models or inverse percentage problems. Whatever the topic, a context-driven approach will allow you to convey the beauty and utility of mathematics because your students will perceive that they are working with you toward common goals.
Practical Teaching Considerations
We now address several of the most frequently asked questions about teaching courses of the type described in this paper.
What prerequisite mathematical background is required? Ideally, students have taken a year or more of high school algebra, but this is not absolutely necessary. The context-driven approach works for all types of students, and as the examples in this paper describe, you can make a non-remedial course without integrating much algebra. Basically, an algebra prerequisite will only affect whether you are able to cover certain specific applications.
Our school requires a course that is equivalent in level to college algebra. Does this course fit the bill? Our course is so different from college algebra that "equivalence" is difficult to interpret. If it means that students learn just as much new (to them) mathematics, the answer is an emphatic yes. In fact, we believe that students learn far more mathematics in our course than they would in a college algebra course – regardless of the extent to which we cover topics (such as linear and exponential modeling) with a formal algebra content.
Is this necessarily a terminal mathematics course? Nearly all of our students enter our course with the intention of it being their last mathematics course. However, typically 5-10% of our students decide to take more mathematics after completing our course. In addition, many of our students go on to take discipline-based courses that involve mathematics; for example, social science students usually take some type of statistics course within their major, and business students often take accounting or other finance courses.
Should the course format be lecture or discussion? We prefer a discussion format. Cooperative and group learning strategies are also successful. However, at many schools large lectures are unavoidable. In such cases, it is very helpful to have a recitation in addition to the lectures.
How do you evaluate students? Ideally, students do a lot of homework that forms the bulk of their grade, along with exams that do not involve memorization or heavy time pressure. In practice, most instructors do not have adequate resources for all the grading entailed in this ideal strategy. One approach we like in this case is to grade only selected homework problems (letting students check the rest themselves from a solution set). You can overcome students objections to turning-in problems that you do not grade by basing exams almost directly on the homework (including problems you did not grade); students who work hard on the homework thereby see a tangible reward on the exam.
What technology is required? The only required technology is a scientific calculator. You can easily incorporate additional technology, such as using spreadsheets for statistics or financial calculations, but any time you spend teaching a technological tool means less time for covering application areas. Given the enormous number of applications that we'd like to cover with our students, we've tended to stay away from additional technology with one exception: we now expect all our students to make use of the Web.
Is it true that this type of course is a challenge for instructors? Yes. Although the mathematical level of this course is only general education, it is a challenging course to teach, at least for first-time instructors. A high premium must be placed on making each class stimulating and motivating. Examples and applications must be current, relevant, and carefully selected. However, we have also found that the rewards of teaching this course – especially in seeing the changing attitudes of your students toward mathematics – more than make up for the extra effort.
Summary
In conclusion, we believe that all non-SEM students – including liberal arts students, business students, and pre-service elementary teachers – should take a mathematics course (one or two semesters) with the content and approach described in this paper. All of these students need to be competent in the four content areas described, and a course using a context-driven approach is the best way to build that competence. Such a course is rich and rigorous, and can serve as the cornerstone in the collegiate mathematical training of our students. By touching on topics of interdisciplinary interest, the course provides lasting benefits for students in their future courses, careers, and lifetimes. | 677.169 | 1 |
Many people suffer from an inferiority complex where mathematics is concerned, regarding figures and equations with a fear based on bewilderment and inexperience. This book dispels some of the subject's alarming aspects, starting at the very beginning and assuming no mathematical education.Written in a witty and engaging style, the text contains an illustrative example for every point, as well as absorbing glimpses into mathematical history and philosophy. Topics include the system of tens and other number systems; symbols and commands; first steps in algebra and algebraic notation; common fractions and equations; irrational numbers; algebraic functions; analytical geometry; differentials and integrals; the binomial theorem; maxima and minima; logarithms; and much more. Upon reaching the conclusion, readers will possess the fundamentals of mathematical operations, and will undoubtedly appreciate the compelling magic behind a subject they once dreaded | 677.169 | 1 |
Showing 1 to 19 of 19Review : Functions
In this section were going to make sure that youre familiar
with functions and function notation. Both will appear in almost
every section in a Calculus class and so you will need to be able
to deal with them.
First, what exactly is a f
Review : Exponential Functions
In this section were going to review one of the more common
functions in both calculus and the sciences. However, before
getting to this function lets take a much more general approach
to things.
Lets start with
,
. An exponCalculus II Advice
Showing 1 to 2 of 2
I love Shtelen. He is so kind and intelligent and I highly recommend him. I had him for Calc 2 and Linear Algebra, and I did really well in both classes. Two things you should know: 1) 85+ is an A, but tests are hard.
Course highlights:
Key here is that this is a college math class. You must practice PROBLEMS to do well. 2) He gives points back if you justify it!
Hours per week:
6-8 hours
Advice for students:
The only negatives is that he can be somewhat quiet and he can throw a curveball on exams with stuff you learned earlie | 677.169 | 1 |
At this time of year, with GCSE exams now over, it will soon be time to make your final A Level subject choices.
Here are 10 reasons why you should consider Maths when making your A Level subject choices.
Enjoyment
One of the best reasons to choose an A Level subject is the fact that you enjoy the subject.
Desirability
Employers are keen on candidates who have A Level Maths.
Problem solving skills
A Level Maths develops your ability to solve problems in the wider world as well as the mathematical world.
Logical thought
Studying A Level Maths encourages your ability to think logically.
Language of the universe
Maths is the language of the universe.
Everything Else Seems Easy
If you can master Maths at A Level, everything else will seem easier.
Business Studies
If you are thinking of studying Business Studies at university, you will find A Level Maths extremely useful, even if the course you are interested in does not list it as an essential.
Computer Sciences
If you want to study computers or computer sciences at university, you won't find it easy without A Level Maths.
Beauty of Maths
You will often hear mathematicians talk about the beauty of maths. It's not until you study maths at A Level or higher that you begin to really appreciate the beautiful nature of maths.
Last but not least,
Increase your earnings by up to 10%
This might surprise you but research has shown that people with A Level Maths tend to earn between 7% and 10% more than their peers who do not have a Maths A level. The increased earnings does not depend on having an A grade either, simply having an E grade Maths A Level or higher will significantly increase your earnings potential. | 677.169 | 1 |
The Egyptians and Greeks wanted to find a
way to describe the general relationships and connections between objects and
events in the mathematical world, just as we often want to describe connections
between people, objects and events in the real world (for example,
"Jeannie gave the shovel to Peter"). To do this the Greeks and
Egyptians developed algebra.
It must be said up front that thinking
in terms of the general relations and the abstract connections between objects
and events, takes practice.
It's very similar to developing a
physical skill, like learning to run long distances. If you aren't used to
running, or haven't been running for a long time, at first even running short
distances seems very hard. But over time, if you keep practicing, your running
skills and your endurance improve. The improvement is gradual, so you might not
notice a big difference from day to day. Eventually, however, running seems
much easier, and you can easily cover the shorter distances that once were very
difficult.
Having said that, there are also some strategies that can
help make the situation easier (just as there are ways to make your running
easier). Mathematicians constantly rely on these strategies to help them solve
difficult problems.
Here are some problem solving strategies that you
may want to make use of when working on a math or economics problem:
Problem
Solving Strategies
-Make a diagram
-Make a list of facts or known
information
-Make a table or a chart of know information
-Make a list
of questions that you want to answer
-Make a list of unknowns- facts that
you don't yet know. | 677.169 | 1 |
Reviews of Concepts and Skills Mathematics Curriculum
Basic Information and Introduction
Concepts and Skills is published my McDougal Littell. The principal
books are Algebra 1 (for Eight or Ninth Grade) and Geometry (early
High School). There is also a Middle School series designed for the
California market. The product web page may be found through
The Concepts and Skills series contains these texts.
California Middle School Mathematics Concepts and Skills,
Course 1
California Middle School Mathematics Concepts and Skills,
Course 2
Algebra 1 Concepts and Skills
Geometry Concepts and Skills
Critical Reviews and Commentaries
Concerning Algebra
1, Concepts and Skills. This is a letter by the members of the
California Content Review Panel for the 2001 adoptions concerning the
Eight Grade (Algebra 1) submission of the Concepts and Skills
curriculum. This curriculum was adopted by the State Board over
objections detailed in this letter. | 677.169 | 1 |
Special offers and product promotions
Editorial Reviews
The Algebra 2 Tutor teaches students the core topics of Algebra 2 and bridges the gap between Algebra 1 and Trigonometry. It contains essential material to help students do well in advanced mathematics. Many of the topics in this series are used in other Math courses, such as writing equations of lines, graphing equations and solving systems of equations. These skills are used time and time again in more advanced courses such as Physics and Calculus. The Algebra 2 Tutor is a complete 15 lesson series covering all of the core topics in detail. What sets this series apart from other teaching tools is that the concepts are taught entirely through step-by-step example problems of increasing difficulty. It works by introducing each new concept in an easy to understand way and using example problems that are worked out step-by-step and line-by-line to completion. If a student has a problem with coursework or homework, simply find a similar problem fully worked on in the series and review for the steps needed to solve the problem. Students will be able to work problems with ease, improve their problem-solving skills and understand the underlying concepts of Algebra 2. This lesson teaches students how to graph equations on the coordinate plane. The 'x' and 'y' coordinates are presented along with the concept of an ordered pair. This information is used to teach students how to set up a table of values to plot graphs of functions. Numerous examples are presented to illustrate the points.
When sold by Amazon.com, this product is manufactured on demand using DVD-R recordable media. Amazon.com's standard return policy will apply. | 677.169 | 1 |
Econometrics and Mathematical Economics
Course Introduction
The preliminary year is designed for students with high academic ability but lacking a sufficient background in economics, econometrics, statistics or mathematics. Its purpose is to enable students to develop their skills to the point where they are eligible for progression to the Master's in econometrics and mathematical economics | 677.169 | 1 |
This book is available in following stores
Mathematics for Future Elementary Teachers, 5th Edition connects the foundations of teaching elementary math and the "e;why"e; behind procedures, formulas and reasoning so students gain a deeper understanding to bring into their own classrooms. Through her text, Beckmann teaches mathematical principles while addressing the realities of being a teacher. With in-class collaboration and activities, she challenges students to be actively engaged. An inquiry-based approach to this course allows future teachers to learn through exploration and group work, leading to a deeper understanding of mathematics. Known for her contributions in math education, Sybilla Beckmann writes the leading text for the inquiry approach-in Mathematics for Elementary Teachers with Activities, students engage, explore, discuss, and ultimately reach a true understanding of mathematics. Beckmann's text covers the Common Core State Standards for Mathematics (CCSSM) now implemented in most states. However, states not following Common Core will not find the information intrusive in the text.
Also available with MyLab Math.
MyLab(tm) Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. The Skills Review MyLab Math provides review and skill development that complements the text, helping students brush-up on skills needed to be successful in class. The MyLab Math course doesn't mirror the problems from the text, but instead covers basic skills needed prior to class, eliminating the need to spend valuable class time re-teaching basics that students should already know. This enables students to have a richer experience in the classroom while working through the book activities and problems. In addition to basic skills review, the MyLab Math course includes a wealth of resources to help students visualize the concepts and understand how they come into play in an elementary classroom. These includes IMAP videos, Responding to Students Videos, eManipulatives, and brand new Common Core videos, Demonstration videos, and GeoGebra animations.
NOTE: You are purchasing a standalone product; MyLab(tm) Math does not come packaged with this content. If you would like to purchase both the physical text and MyLab Math, search for to | 677.169 | 1 |
Algebra Worksheets
Here at Algebra Help Resource, we are dedicated to helping students find challenging algebra worksheets so that students may practice. The algebra worksheets are arranged according to websites and are ranked according to their quality. It is important that students try these worksheets out after reading through our algebra tutorials.Don' tmisunderstand! The real secret behind math is practice so students must practice in order to succeed in it. We hope you learn the basic concepts of algebra through these worksheets.
Tied at number one positions are math.com and algebrahelp.com.
At math.com , you can generate your own customised algebra worksheets by selecting options (type of algebra equations & systems and whether they can be factorised). After you finished deciding which algebra worksheet to do, you can print it out and do them in your spare time.
At algebrahelp.com , you are not able to customise your own algebra worksheets but a wide range of algebra worksheets are availiable. These are placed in different categories and blanks besides the algebra questions are set in place for you to fill in. Then you can verify your answers and compare your results with other people who have done the same algebra worksheets.
Edhelper.com deserves a mention even though it charges a recurring charge for people to use its algebra help worksheets. Students can choose the type of algebra questions they wish to do (linear equations ,functions, exponentials etc) and include them into the algebra worksheets. These test the different skills that you would require when taking the actual algebra exam.
Aplusmath.com is good for primary students who are beginning to grasp the concept of algeba. Its algebra worksheets are mainly linear algebra equations and students can choose the number of algebra questions they wish to have in it | 677.169 | 1 |
for the Math for Liberal Arts course, TOPICS IN CONTEMPORARY MATHEMATICS helps students see math at work in the world by presenting problem solving in purposeful and meaningful contexts. Many of the problems in the text demonstrate how math relates to subjects--such as sociology, psychology, business, and technology--that generally interest students.
Available with InfoTrac® Student Collections
Meet the Authors
Ignacio Bello,
University of South Florida
Ignacio Bello attended the University of South Florida (USF), where he earned a B.A. and M.A. in Mathematics. He began teaching at USF in 1967, and in 1971 he became a member of the faculty and Coordinator of the Math and Sciences Department ay Hillsborough Community College (HCC). Professor Bello instituted the USF/HCC remedial program, which started with 17 students taking Intermediate Algebra and grew to more than 800 students with courses covering Developmental English, Reading, and Mathematics.
In addition to Topics in Contemporary Mathematics, Professor Bello has written many other books that span the mathematics curriculum, many of which have been translated to Spanish. Professor Bello is featured in three television programs on the award-winning Education Channel. He helped create and develop the USF Mathematics department website, which serves as support for the Finite Math, College Algebra, Intermediate Algebra, Introductory Algebra, and CLAST classes at USF.
Professor Bello is a member of the Mathematical Association of America (MAA) and the American Mathematical Association of Two-Year Colleges (AMATYC). He has given many presentations regarding the teaching of mathematics at the local, state, and national levels.
Anton Kaul,
California Polytechnic State University
Jack R. Britton,
Late of University of South Florida
What's New
A Getting Started outline of objectives that corresponds to the objective heads in the exposition and in the end-of-section exercise sets has been added to the beginning of each section to provide students with a map to navigate each chapter.
In response to feedback from reviewers, Chapter 5 on Number Theory and the Real Numbers has been extensively revised and condensed creating a more comprehensive chapterThroughout the text, applications have been replaced and revised for currency and student interest. Exercises now include diverse topics such as the garbage pizza, gadgets owned by Americans, hacked passwords, the correlation of healthcare to life expectancy, the total cost of college, and many more.
The following topics are now more easily accessible in the printed book rather than being available online only: the section on linear programming, right triangle trigonometry, chaos and fractals, as well as the chapters on voting and apportionment, and graph theory.
Exclusively from Cengage Learning®, Enhanced WebAssign combines the exceptional mathematics content that you know and love with the most powerful online homework solution, WebAssign. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and interactive e-books that help students to develop a deeper conceptual understanding of their subject matter. Online assignments can be built by selecting from thousands of text-specific problems or supplemented with problems from any Cengage Learning textbook.
An Instructor's Edition is available for the tenth edition providing instructors with an appendix in the book that contains answers to all exercises and an appendix that outlines the exercises that are available through Enhanced WebAssign.
Features
A strong technology focus motivates students and shows them different ways in which mathematics can be applied. Web references and Web It exercises in the text offer students ways to utilize the Internet as an educational and creative tool to study mathematical concepts. Graph It, a feature found in the book margins, provides step-by-step directions for solving specific examples using the TI-83 graphing calculator.
Motivational chapter and section opening vignettes contain applications drawn from a broad range of fields and introduce students to the techniques and ideas covered. Applications are further integrated throughout the text, examples and in the exercise sets to help students develop the skills to apply problem-solving techniques in the real-world.
The text incorporates suggestions of AMATYC's Standards for Introductory College Mathematics. For instance, the authors de-emphasize the more abstract and theoretical aspects of the subject matter, placing emphasis on promoting the understanding and use of concepts introduced.
A unique problem-solving approach emphasized throughout the text helps students learn the mathematical skills that will benefit them in their lives and careers. Using the RSTUV method (Read, Select, Think, Use, and Verify), this approach guides students through problems and includes references to similar problems in the exercise set.
FOR INSTRUCTORS
Instructor's Solutions Manual
Student Solutions Manual
ISBN: 9781285420745
FOR STUDENTS
Student Solutions Manual
ISBN: 9781285420745
Prepare for exams and succeed in your mathematics course with this comprehensive solutions manual! Featuring worked out-solutions to the problems in TOPICS IN CONTEMPORARY MATHEMATICS, 10th Edition, this manual shows you how to approach and solve problems using the same step-by-step explanations found in your textbook examples. | 677.169 | 1 |
American Mathematical Society, 2004-11-22. Paperback. New.., American Mathematical Society, 2004-11-22
[EAN: 9780821833292], Neubuch, [PU: American Mathematical Society],. | 677.169 | 1 |
Welcome to MAC1140. Precalculus Algebra
is
a college level course designed to further prepare you in
important areas such as graphing techniques, algebraic
functions-
Handouts &
Homework : Ch
2, 4 & 5 ,
Ch 6, 7 & 11
You need to print these notes. We will
use them in class and you will turn in some of them for
Homework. | 677.169 | 1 |
Parabolas
Be sure that you have an application to open
this file type before downloading and/or purchasing.
33 KB|1 page
Share
Product Description
This is to be used as notes for describing and graphing parabolas in standard form in an algebra 2 or precalculus course. Vocabulary is described, a table is given with descriptions of different parabolas in standard form, then an example given. | 677.169 | 1 |
A COMPREHENSIVE OVERVIEW OF BASIC MATHEMATICAL CONCEPTS
-
SET THEORY
PROPERTIES OF REAL
NUMBERS
NOTATION
cfw_ Brul"e. indicate the beginning and end ofa set notation; when
listed e lements or members mllst be separated by commas;
EXAMPLE: In A= cfw_4 ,
PROPERTIES OF INEQUALITY
For any real numbers a , b , and c:
For any real numbers a, b, and c:
A,Closure
a + b is a real number
I. For addition:
2. For 1l1ultiplication: a b is a real number
B, Commutative Property
I. For addition :
a + b = b +a
2. For
A. Polygons are plane shapes that are formed by line
segments that intersect only at the endpoints. These
intersecting line segments create one and only one
enclosed interior region.
Geometry means Earth measurement; early peoples
used their knowledge of
MATH 241 - Spring 2014
Instructor: Patrick Collins
Homework #10 Solutions
6.1 #12
Find the volume of the solid of revolution generated by revolving the region between the x-axis and the line x
in between y 0 and y 2 about the y-axis.
3y 2
Solution
The dis
Dr. Yuen is an AMAZING professor! Seriously. Go to class and pay attention to lectures, he puts everything on the board. He's always available at his office hours and he's very good at explaining things.
Course highlights:
We covered all of the major topics in calculus I. It is recommended you have a good foundation in trig and an excellent foundation in algebra.
Hours per week:
6-8 hours
Advice for students:
Go to class. Pay attention. Form a study group. The learning emporium is good, but don't use it as a crutch. Those TA's will do your homework for you if you let them, and you will fail the quizzes and the exams because you don't know what is going on. Trust me, I have seen it happen. | 677.169 | 1 |
Through many examples and real-world applications, Practical Linear Algebra: A Geometry Toolbox, Third Edition
teaches undergraduate-level linear algebra in a comprehensive, geometric, and algorithmic way.
Designed for a one-semester linear algebra course at the undergraduate level, the book gives instructors the option of tailoring the course for the primary interests:
math, engineering, science, computer graphics, and geometric modeling.
New to the Third Edition
More exercises and applications
Coverage of singular value decomposition (SVD)
Application of SVD to the pseudoinverse, principal components analysis, and image compression
More attention to eigen-analysis, including eigenfunctions and the Google matrix
Greater emphasis on orthogonal projections and matrix decompositions
Fundamental concepts tied to repeated themes such as the concept of least squares
To help students better visualize and understand the material, the text introduces the fundamental concepts of linear algebra first in a two-dimensional setting and then revisits these concepts and others in a three-dimensional setting. The text also discusses higher dimensions in various real-life applications. Triangles, polygons, conics, and curves are introduced as central applications of linear algebra.
Instead of using the standard theorem-proof approach, the text presents many examples and instructional illustrations to help students develop a robust, intuitive understanding of the underlying concepts.
Features
120 numerical examples
More than 280 illustrations
WYSK (What You Should Know) summary of main points at the end of each chapter
Over 200 exercises at the end of each chapter with selected solutions in an appendix
The figures are not included as window dressing, in fact they play an important role in bringing the reader to a robust understanding of the mathematics. However they are not only instructional, they are also fun! | 677.169 | 1 |
As you may be noticing, Math Tutor DVDs encompass a wide range of mathematical levels and topics. You can see a list of topics and even view free sample videos at their website.
Our family received two videos this year, Pre-Algebra (Volume 1), and a video tutorial for Texas Intruments TI-84 Calculator, a very complicated-looking graphing calculator. I have to mention that even though we don't have one of these calculators, Middlest was fascinated and watched the video with me. I got the impression that she'd enjoy exploring the calculator, but that's going to have to come after we've gone a bit further in our math studies.
Pre-Algebra Volume 1 is the first of two sets of videos. Volume 1 contains two DVDs, about five hours in all. Topics include:
Real Numbers
The Number Line
Greater Than, Less Than, Equal To
Absolute Value and Adding Integers
Subtracting Integers
Multiplying Integers
Dividing Integers
Powers and Exponents
Order of Operations
These are not the usual razzmatazz flashy animated videos so commonly seen in the ranks of educational DVDs. Actually, they're pretty simple and low-tech: a man, a plan, and a white board. (I almost said, "A man, a plan, a canal, Panama" which is a palindrome — it reads the same forwards and backwards.)
The lecturer is pleasant, encouraging, and matter of fact in his presentation. It's kind of like having your Uncle Doug explain math to you. He starts by defining terms and basic concepts, and then works through examples. Of course it's not an interactive tutorial — there's no give and take — but it has an interactive feeling to it.
The Texas Instruments TI-84 Calculator Tutor shows a graphic of the calculator's keypad and screen while a man lectures in the background. A cursor moves about and points to various buttons as the lecture proceeds, showing what buttons to push.
There were a few times when I had to stop and run through a sequence again because I had looked away from the screen for a second and missed what key had been pressed.
Numbers, functions, and graphs appear on the calculator screen as the video works through various features. I have to admit that most of the video here is way over our heads at present. Eight hours of instruction are included on three DVDs, from a basic introduction of the keys, to solving equations and graphing functions. The list of topics is too long to include here, but click on the link above and you'll see not only the list but also Lesson 1 on the video, so you can get an idea of the teaching style.
Each of these video sets is $26.99 at the Math Tutor DVD website.
The Math Tutor DVDs we've seen are all good, basic instruction. They may be a little boring if your kids are used to a lot of bells, whistles, and fancy animations, but they work just fine for our media-deprived kids. (That was only partially a joke — our girls play too many computer games, true, but they watch very little television and so their brains have not been trained to need lots of stimulus in order to sustain attention.) | 677.169 | 1 |
Math problems to solve
With millions of users and billions of problems solved, Mathway is the world's #1 math problem solver. From basic algebra to complex calculus, Mathway. Web Algebra, math homework solvers, lessons and free tutors online.Pre-algebra, Algebra I, Algebra II , help you solve your homework problems. A few examples of what you can ask Wolfram. solve an ordinary differential equation Famous Math Problems.
Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work. Get the Cymath math solving app on your smartphone. Math for Everyone - powered by WebMath. Visit Cosmeo for explanations and help with your homework problems. Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work. Get the Cymath math solving app on your smartphone. QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices. How to Solve Math Problems. Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems.
Math problems to solve
Step-by-Step Calculator Solve algebra, trigonometry and calculus problems step-by-step. Pre Algebra;. Math notebooks have been around for hundreds of years. Visa and MasterCard security codes are located on the back of card and are typically a separate group of 3 digits to the right of the signature strip. With millions of users and billions of problems solved, Mathway is the world's #1 math problem solver. From basic algebra to complex calculus, Mathway. Solve your math problems online. The free version gives you just answers. If you would like to see complete solutions you have to sign up for a free trial account. Math for Everyone - powered by WebMath. Visit Cosmeo for explanations and help with your homework problems.
A few examples of what you can ask Wolfram. solve an ordinary differential equation Famous Math Problems. Visa and MasterCard security codes are located on the back of card and are typically a separate group of 3 digits to the right of the signature strip. Welcome to the algebra calculator, an incredible tool that will help double-check your work or provide additional practice to prepare for tests or quizzes.
FREE math problem solver with step by step description and graph analysis. It solves integrals, derivatives, limits, trig, logarithms, equations, algebra. Step-by-Step Calculator Solve algebra, trigonometry and calculus problems step-by-step. Pre Algebra;. Math notebooks have been around for hundreds of years. Our basic math calculator will ensure you have the right answer – whether you're checking homework, studying for an upcoming test, or solving a real-life problem. QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices. Solve math problems online. Get free answers to math questions instantly with the help of a free online math problem solver and thus improve your math practice.
This online algebra solver can tell you the answer for your math problem, and even show you the steps (for a fee). Enter your math problems and get them solved instantly with this free math problem solver. Don't become lazy though. Do your math problems yourself and use it as a. Find practice math problems with answers in algebra & calculus from the Cymath online math solver. The Cymath equation solver makes solving math problems easy. Enter your math problems and get them solved instantly with this free math problem solver. Don't become lazy though. Do your math problems yourself and use it as a. Solve your math problems online. The free version gives you just answers. If you would like to see complete solutions you have to sign up for a free trial account.
Web Welcome to the algebra calculator, an incredible tool that will help double-check your work or provide additional practice to prepare for tests or quizzes. | 677.169 | 1 |
The place of the Mathematics Advanced Stage 6 Syllabus in the K–12 curriculum
Building on Mathematics Learning in Stage 5
The outcomes and content in the Stage 6 Mathematics Advanced syllabus are written with the assumption that students studying this course will have engaged with all substrands of Stage 5.1 and Stage 5.2 and with the following substrands of Stage 5.3 - Algebraic techniques, Surds and indices, Equations, Linear relationships, Trigonometry and Pythagoras' theorem and Single variable data analysis | 677.169 | 1 |
Applied Finite Mathematics
A very common question math faculty get is about the content of Math 124 Finite Math. Unfortunately, no simple answer can be given in twenty-five words or less. But that's why you are reading this, yes?
"Finite Math" is a catch-all title for a collection of topics that are anything but calculus. The purpose of the course is to give a survey of mathematical analysis techniques used in the working world, but you might also say that this course gives valuable experience at organizing information and then analyzing it. In a larger sense, it's also another way we use math to give people experience at analytical thinking. Business, accounting and computer majors tend to take this course, or are required to by their program. It is an excellent choice for many education majors to fill their math requirements, but these people should check with their transfer institution first.
Here is a list of the main topics covered:
-Mathematical model building. Math modelling is the act of creating functions or equations that describe a given application or situation. In this course we mainly concentrate on business-oriented ideas such as break-even analysis or depreciation.
-Matrix algebra. Matrices are collections of numbers organized in rectangular arrays. These can effectively represent certain kinds of data or systems of equations. In Finite Math you only get a brief glimpse into how they are used and manipulated, but matrix ideas can arise in both accounting and business analysis, and computer programmers use them as array variables.
-Linear programming. This topic has nothing to do with computer programming, but it is a method for optimizing situations when constraints are in place. For example, if you produce several lines of products but have budgetary constraints on labor and materials, and have production contracts in place that must be filled, then what is the most efficient, profitable way to determine how much of each line to produce, that is, how can you maximize the profit potential? Linear programming is ideally suited to problems of this nature. Linear programming can also be an application of both math modeling and matrix algebra.
-Combinatorics. This is the art of advanced counting. For example, if a room has twenty chairs and twelve people, in how many ways can these people occupy the chairs? And are you accounting for differences in who sits in particular chairs, or does it only matter if a chair has a body in it? These kinds of counting problems are the basis for . . .
-Probability. In order to calculate the chance of a particular event happening you must be able to count all the possible outcomes. Once you understand how to find probabilities then you can begin to understand . . .
-Statistics. Statistics uses probability in order to analyze data and make decisions. In Finite Math you will only get a brief introduction and overview of statistics. For information about the subject you can link to:
-Logic. Logic is the symbolic, algebraic way of representing and analyzing statements and sentences. You will only get a brief introduction to logic in this course, but the mathematics used in logic are found at the heart of computer programming and in designing electrical circuits.
More advanced courses in Finite Math topics are sometimes called Discrete Mathematics. The word discrete helps explain where Finite Math gets its name. Discrete means broken up or separated. For example, integers are discrete objects because there are non-integer numbers in between them, but real numbers are continuous numbers because there is no identifiable separation between them.
For a maddening exercise in continuity try finding the largest real (i.e., decimal) number less than one. No, it is not 0.999999 . . . (the nines repeating forever), because it can be demonstrated that 0.999999 . . . is equal to 1. Whatever this number is it is impossible to represent it in any other than the most abstract way.
Continuity is in some ways associated with infinity and infinitesimal. Since calculus is concerned with continuous numbers and continuous functions, the subject must confront the ideas of infinity and infinitesimal. Finite Math is a subject that avoids the issues of continuity encountered in calculus, so those topics are lumped into the category of "finite mathematics."
Source:
INTERESTING VIDEO
MyMathLab for Applied Calculus & Finite Mathematics
Finite Mathematics And Applied Calculus Hybrid 6th Edition ...
INTERESTING FACTS
Applied physics is a general term for physics which is intended for a particular technological or practical use. It is usually considered as a bridge or a connection between "pure" physics and engineering.
"Applied" is distinguished from "pure" by a subtle...
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a... | 677.169 | 1 |
About Us
In 2012, Study Edge joined forces with the University of Florida to dramatically improve student achievement in Algebra 1—a required course that is a key "gateway" to higher-level math courses and high-growth careers. At that time, just 50% of Florida high school students were proficient in algebra. In high needs schools, proficiency rates were often under 20%.
Study Edge and UF launched Algebra Nation in 2013 with two key goals: significantly improve algebra achievement in Florida, and reduce disparities in student performance. Since then, statewide pass rates have improved 9% and our materials have driven achievement for students in demographics that traditionally struggle in algebra.
Due to the popularity of Algebra Nation, we added resources for other key math classes and evolved to become Math Nation. Today, we provide dynamic content videos, workbooks, an online practice tool, and teacher-built lesson plans and other materials for Algebra 1, Geometry, and Algebra 2. Students in these courses can also log-on to our interactive Math Walls to receive homework help and support from our Study Experts and peers who volunteer their time answering questions and directing other students to helpful videos.
Thanks to generous funding from the Florida legislature, students and educators can access our Algebra 1, Geometry, and Algebra 2 resources at no cost. And they have: Since our launch, students, teachers, and parents have logged in more than 8.4 million times!
Math Nation is a result of ongoing collaboration among teachers, professors, administrators, parents, and students. Thanks to the input they provide, we are constantly adapting and improving our resources. Please let us know if you have any suggestions or ideas to share. We want to hear from you! | 677.169 | 1 |
Calculus I
(Lecture #5)
Introduction to the Derivative of a Function
Ziad Z. Adwan
()
Lecture 5
1 / 26
Learning Objectives
1. The Tangent Line Problem [Revisited].
2. Denition of the Derivative at a Point and in General.
3. Computing the Derivative at a P
Calculus I
Preface
Here are the solutions to the practice problems for my Calculus I notes. Some solutions will have
more or less detail than other solutions. The level of detail in each solution will depend up on
several they should be accessible to anyone wanting to learn
Calculus I or needing a refresher in some of the early topI recommend this class to those who find it hard to learn math, it is one of the most useful courses one could ever take and it is easy to learn the subject in it too.
Course highlights:
I learned more about trigonometry and integrals than I could have done in any other course of the university. I feel like learning these would help everyone in whatever math subject they would have to take
Hours per week:
3-5 hours
Advice for students:
I would tell students that even though it is easy, it is also easy to procrastinate at times, I would recommend to try to finish the assignments as the first thing they do when they get to their home or dorm.
Course Term:Fall 2016
Professor:jacobsen
Course Required?Yes
Course Tags:Great Intro to the SubjectA Few Big AssignmentsParticipation Counts
Jul 20, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
I would definitely recommend this course. She breaks down all the elements of Calculus so that it's understandable and not as intimidating.
Course highlights:
The highlights were the group project that I dreaded at first but then came to enjoy as I got to know my group. I learned that it takes a lot of patience to work in a group and that notes are very important.
Hours per week:
6-8 hours
Advice for students:
Pay attention! If you turn your head for one minute you'd have lost half a pages worthat of notes and maybe even more. Read when she says to because there are pop quizzes. | 677.169 | 1 |
TU Delft LA-system
e-learning tutor for math
Introduction
There is a growing demand for e-learning tools to improve math skills at the university. Students start nowadays less prepared with their university study due to chances at the secondary school. During their first year at the university, students are fully engaged in more active types of learning (projects, group work) and easily neglect their math classes. After failing the first examinations, there is not much opportunity to repair their deficiencies, hence the need for facilities to support self study.
Several web-based tutor systems have been introduced lately, such as Aleks , My Math Lab and WebAssign . These interactive web sites offer a large collection of exercises, in addition to web lectures and explanations. Students can check whether their answer is correct and can ask for hints or see full worked-out examples. However, what really lacks is direct feedback. If the student is stuck or has made an error then there is no support to explain what he has done wrong and how he should proceed.
The SURF-project "Intelligent feedback" has developed techniques to track a student while making an assignment. With a special developed strategy language, all possible paths to a solution can be described compactly, and the actions the student takes can be compared with the prescribed strategy. With this information, direct and meaningful feedback can be given.
E-learning with intelligent feedback
A consortium of computer scientist and mathematicians from the Open University in the Nederlands, Eindhoven University of Technology, and Delft University of Technology, has developed several tools applying the new strategy language and parser. One of these tools, the LA-system (for Linear Algebra) of Delft, is an interactive tutor system developed on top of Mathematica. The system has a graphical front-end for formula formatting. An exercise can be made by graphical selecting terms of an equation and applying operations chosen from a menu. The system carries out the operations and after a number of steps, typically in the order of five till ten, the exercise is solved. The interface of the system is comparable with MathXpert and DirectMath . See the next section for an example.
Although the system offers at each step a number of feasible operations, the student still has to choose the correct one. It is not feasible to reach the solution without knowing where to head. The tool eliminates a lot of the nitty-gritty hand work of rewriting complex formulas and matrices, and allows students to concentrate on the strategies to solve the exercises. Further, the system can now follow the student step by step and provide adequate feedback.
The operations that can be specified are based on a "rule set", a collection of rewrite rules implemented in Mathematica. The rule set is domain specific, i.e. for each topic within calculus and linear algebra, different sets of rewrite operations are needed. However, with a given rule set, a large number of exercises can be created. The level of the rewrite operations can be adapted, such that novice users can be offered basic step-by-step procedures, while more advanced users can rely on more powerful procedures and short cuts.
Example
To give an idea of how the system works we show a simple exercise. The system starts with offering two windows: the working window, that shows the exercise, and the rule window that gives the operations to manipulate the equations in the working window. (See Figure 1).
Figure 1. Working window with equations (left) and the rule window (right). There are no rules specified yet, because there is no term selected so far.
To proceed, the student first selects a term that he wants to re-write. The system now offers a number of feasible operations in the rule window. This gives the situation of figure 2.
Figure 2. Selecting the set of equations brings verschijnen in het regel-window de toepasbare regels.
Selecting the rule "Convert set of equations to augmented matrix" rewrites the set of equations to a matrix and the first step towards the solution is taken (Figure 3).
Figure 3. The applied rule is listed between the old and the new formulation.
After a number of steps we reach the situation of figure 4.
Figure 4. Completed exercise.
There may be multiple paths towards a solution, both in the order of steps, as well as in the strategy, eg. with or without use of an augmented matrix. By being able to recognize and analyse the chosen strategy the system can provide semantically-rich feedback.
Strategy language and parser
The parser interpretes the steps the student takes and compares them with the possible paths specified in the strategy. The strategy language is designed to compactly represent all possible sequences and allowable permutations. For instance, (3 + 5) * 4 can be calculated by first adding 3+5, but the student could as well have chosen for the more complex but also correct sequence of first eliminating the brackets, i.e. 3*4 + 3*5.
The strategy language may contain strategy rules, rewrite rules, and code blocks. A strategy rule is of the format:
Strategie := term, term, term....
A term can be the name of a (sub)strategy (starting with an upper case character), or the name of a rewrite rule (starting with an lower case). The strategy rule specifies in which order the rewrite rules are to be applied.
A term can also be a code fragment to execute some Mathematica code. The variables (names with a $-prefix) used within the code block can be passed as arguments to a strategy or rewrite rule, for example:
multiply[$x] - multiplies the selected term with the value of $x.
For a certain strategy we may specify multiple, alternative, rules. This is the power of the strategy language. The language has a special 'parallel' operator to execute strategies in parallel, and a 'not' operator to test whether a strategy still goes for a certain expression. With these constructs we can define complex strategies compactly.
Below we give the strategy for the example of figure 1 which solves a set of linear equations by row reducing a matrix (Gaussian elimination). First the set of equations is converted into an 'augmented matrix' en then the matrix is brought into the reduced echelon form (with only 1's at the diagonal and zero's under the diagonal and above the diagonal). Then the matrix is converted back to a trivial set of equations. If there is an inconsistent equation eg. (0=1) then the set of equations does not have a solution. If there are one or more free variables, the user has to specify the requested parameters, and the set of equations can be rewritten to a vector/parameter representation of a point, line or plane.
The code below is a simplified version of the real syntax.
SolveLinearEqns :=
// strategy to solve a set of linear equations
eqns2aug,
// convert to augmented matrix
MakeReducedEchelon,
// rewrite to reduced echelon form
aug2eqns,
// convert back to equations
If[Code[InConsistentQ[$CurrentTerm]],
GiveSolution,
// if consistent: specify solution
nosolution]
// if not consistent: reply: 'no solution'
GiveSolution :=
// strategy for specifying the solution
Code[$freevars = FreeVariables[$CurrentTerm]],
ForEach[$var,
// for each $var
$freevars,
// out $freevars
assignparam[$var]],
// assign parameter to $var
eqns2eq,
// rewrite to vector/parameter representation
MakeReducedEchelon :=
// rewrite to reduced echelon form
ForwardPass,
// bring into echelon form
BackwardsPass
// get rid of as much zero's above diagonal
ForwardPass :=
// bring into echelon form
RepeatExhaust[
FindColumn,
// find next pivot column
ExchangeNonZero,
// exchange rows when needed
ScaleToOne,
// make pivot 1
MakeZeroesFP
// make zero's under pivot
]
BackwardPass :=
// reduce zero's above the diagonal
Code[$nonzerorows = Select[Rows[$CurrentTerm],!ZeroVector]]
ForEach[$row,
// for each row
$nonzerorows
// with other values than zero
MakeZeroesBP
// make zero's above pivot
]
Feedback
The strategy can be used to generate automatic feedback. At each point the system can track the user to see whether he is still on track. Feedback can be given to the user when:
an alternative path is taken, which leads away from the shortest path but may still lead to the correct solution
a step is taken which irrevocably diverts from the correct solution.
In the first case, the system can give a hint to return to the shortest path, although the system can also wait a while until the user recognizes his error. In that case the system should be able to recognize "detour" strategies. In the second case the system could give feedback immediately and explain why the chosen option is not correct and how to proceed. In that case it would be beneficial if the strategy could also include well-known "pitfalls". These would be errors that are easily made and are known by the instructor. The "detour" and "pitfall" options are currently being implemented.
The student can also ask for a hint. Depending on the pedagogical strategy there are several options:
Hint On request, the system can give a strategic hint that does provide some insight but does not give away the complete solution. The full strategy can be decomposed into several steps and substrategies. The system can first base the hint on the highest level in the strategic hierarchy (see "Solve Linear Equation" in the above example). When the student is inside a substrategy, the system could give a hint related to the highest level within that procedure, and descend until the next elementary step.
Next step The system shows the next step. This is only of value when the student understands the strategy, but does not know which rule to apply.
Full solution The system shows the complete procedure. This is only of value when the student does not understand the strategic hints at all. | 677.169 | 1 |
Compilation of math videos from popular video server such as: youtube, teachertube, metacafe, revver,blip.tv, etc and also video lectures from respected university like MIT, etc.
The material ranging from elementary (school) math up to professional mathematics (MSRI videos). | 677.169 | 1 |
basedA top-selling teacher resource line, The 100+ Series(TM) features over 100 reproducible activities in each book! --Algebra links all the activities to the NCTM Standards and provides students with practice in the skill areas necessary to master the concepts presented in a first course in algebra. Reinforcing operations skills plus activities that focus on order of operations, solving equations, dealing with inequalities, monomials, binomials and polynomials, factoring, plotting coordinates, graphing, and exercises involving radicals are all part of this book. Examples of solution methods are presented at the top of each page and puzzles and riddles gauge the success of skills learned. The numbers of the related standards for each activity can be found in the table of contents. An answer key is also provided.
Introduction to Logic Design by Alan Marcovitz is intended for the first course in logic design, taken by computer science, computer engineering, and electrical engineering students. As with the previous editions, this edition has a clear presentation of fundamentals and an exceptional collection of examples, solved problems and exercises. The text integrates laboratory experiences, both hardware and computer simulation, while not making them mandatory for following the main flow of the chapters. Design is emphasized throughout, and switching algebra is developed as a tool for analyzing and implementing digital systems. The presentation includes excellent coverage of minimization of combinational circuits, including multiple output ones, using the Karnaugh map and iterated consensus. There are a number of examples of the design of larger systems, both combinational and sequential, using medium scale integrated circuits and programmable logic devices. The third edition features two chapters on sequential systems. The first chapter covers analysis of sequential systems and the second covers design. Complete coverage of the analysis and design of synchronous sequential systems adds to the comprehensive nature of the text. The derivation of state tables from word problems further emphasizes the practical implementation of the material being presented.Provides engineers with an in-depth look at the key concepts in the field. It incorporates new discussions on emerging areas of heat transfer, discussing technologies that are related to nanotechnology, biomedical engineering and alternative energy. | 677.169 | 1 |
10.1 Collision detection using circles and spheres
Circles and spheres
Intersecting line and circle
Intersecting circles and spheres
10.2 Collision detection using vectors
Location of a point with respect to other points
Altitude to a straight line
Altitude to a plane
Frame rate issues
Location of a point with respect to a polygon
10.3 Exercises
As this chapter offers all necessary mathematical skills for a full mastering of all further
topics explained in this book, we strongly recommend it. To serve its purpose, the successive paragraphs below refresh some required aspects of mathematical language as used
on the applied level.
1.1 Algebra
Real Numbers
We typeset the set of:
natural numbers (unsigned integers) as N including zero,
integer numbers as Z including zero,
rational numbers as Q including zero,
real numbers (floats) as R including zero.
All the above make a chain of subsets: N ⊂ Z ⊂ Q ⊂ R.
To avoid possible confusion, we outline a brief glossary of mathematical terms. We recall
that using the correct mathematical terms reflects a correct mathematical thinking. Putting
down ideas in the correct words is of major importance for a profound insight.
Sets
We recall writing all subsets in between braces, e.g. the empty set appears as {}.
We define a singleton as any subset containing only one element, e.g. {5} ⊂ N, as
a subset of natural numbers.
We define a pair as any subset containing just two elements, e.g. {115, −4} ⊂ Z,
as a subset of integers. In programming the boolean values true and false make up
a pair {true, f alse} called the boolean set which we typeset as B.
We define Z− = {. . . , −3, −2, −1} whenever we need negative integers only. We
express symbolically that −1234 is an element of Z− by typesetting −1234 ∈ Z− .
We typeset the setminus operator to delete elements from a set by using a backslash, e.g. N \ {0} reading all natural numbers except zero, Q \ Z meaning all pure
rational numbers after all integer values left out and R \ {0, 1} expressing all real
numbers apart from zero and one.
ARITHMETIC REFRESHER
19
Calculation basics
operation
example
a
b
c
to add
a+b = c
term
term
sum
to subtract
a−b = c
term
term
difference
factor
factor
product
numerator
divisor
or denominator
quotient
or fraction
=c
base
exponent
power
a=c
radicand
index
radical
to multiply
to divide
to exponentiate
to take root
a·b = c
a
b
= c, b = 0
ab
√
b
We write the opposite of a real number r as −r, defined by the sum r + (−r) = 0. We
typeset the reciprocal of a nonzero real number r as 1r or r−1 , defined by the product
r · r−1 = 1.
We define subtraction as equivalent to adding the opposite: a − b = a + (−b). We define
division as equivalent to multiplying with the reciprocal: a : b = a · b−1 .
When we mix operations we need to apply priority rules for them. There is a fixed priority
list 'PEMDAS' in performing mixed operations in R that can easily be memorized by
'Please Excuse My Dear Aunt Sally'.
First process all that is delimited in between Parentheses,
then Exponentiate,
then Multiply and Divide from left to right,
finally Add and Subtract from left to right.
A fraction is what we call any rational number written as nt given t, n ∈ Z and n = 0,
wherein t is called the numerator and n the denominator. We define the reciprocal of a
−1
. We define the opposite fraction as
nonzero fraction nt as 1t = nt or as the power nt
− nt =
−t
n
=
t
−n .
n
We summarize fractional arithmetics:
sum
t
n
+ ab =
t¡b+n¡a
n¡b ,
difference
t
n
t¡b−n¡a
n¡b ,
product
t
n
− ab =
division
t
n
a
b
exponentiation
singular fractions
¡ ab =
t¡a
n¡b ,
= nt ¡ ba ,
t m t m
= nm ,
n
1
0
0
0
= Âąâˆž infinity,
=? indeterminate.
Powers
We define a power as any real number written as gm , wherein g is called its base and m
its exponent. The opposite of gm is simply −gm . The reciprocal of gm is g1m = g−m , given
g = 0.
We insist on avoiding typesetting radicals like 7 g3 and strongly recommend their contemporary notation using radicand g and exponent 37 , consequently exponentiating g to
√
3
1
g 7 . We recall the fact that all square roots are non-negative numbers, a = a 2 ∈ R+ for
a ∈ R+ .
As well knowing the above exponent types as understanding the above rules to calculate
them are inevitable to use powers successfully. We advise memorizing the integer squares
running from 12 = 1, 22 = 4, . . ., up to 152 = 225, 162 = 256 and the integer cubes running
from 13 = 1, 23 = 8, . . ., up to 73 = 343, 83 = 512 in order to easily recognize them.
Recall that the only way out of any power is exponentiating with its reciprocal exponent.
For this purpose we need to exponentiate both left hand side and right hand side of any
given relation (see also paragraph 1.2).
√
7
Example: Find x when x3 = 5 by exponentiating this power.
3 7
7
3
3
x 7 = 5 �⇒ x 7
= (5) 3 �⇒ x ≈ 42.7494.
We emphasize the above strategy as the only successful one to free base x from its exponent, yielding its correct expression numerically approximated if we like to.
Example: Find x when x2 = 5 by exponentiating this power.
1
1
1
x2 = 5 �⇒ x2 2 = (5) 2 or − (5) 2 �⇒ x ≈ 2.23607 or − 2.23607.
We recall the above double solution whenever we free base x from an even exponent,
yielding their correct expression as accurate as we like to.
22
A N I M AT I O N M AT H S
Mathematical expressions
Composed mathematical expressions can often seem intimidating or cause confusion.
To gain transparancy in them, we firstly recall indexed variables which we define as
subscripted to count them: x1 , x2 , x3 , x4 , . . . , x99999 , x100000 , . . ., and α0 , α1 , α2 , α3 , α4 , . . . .
It is common practice in industrial research to use thousands of variables, so just picking
unindexed characters would be insufficient. Taking our own alphabet as an example, it
would only provide us with 26 characters.
We define finite expressions as composed of (mathematical) operations on objects (numbers, variables
or structures). We can for instance analyze the expression (3a + x)4 by drawing its tree form. This
example reveals a Power having exponent 4 and a
subexpression in its base. The base itself yields a
sum of the variable x Plus another subexpression.
This final subexpression shows the product 3 Times
a.
Let us also evaluate this expression (3a + x)4 . Say
a = 1, then we see our expression partly collaps
to (3 + x)4 . If we on top of this assign x = 2, our
expression then finally turns to the numerical value
(3 + 2)4 = 54 = 625.
When we expand this power to its pure sum expression 81a4 + 108a3 x + 54a2 x2 +
12ax3 + x4 , we did nothing but reshape its pure product expression (3a + x)4 .
We warn that trying to solve this expression - which is not a relation - is completely in
vain. Recall that inequalities, equations and systems of equations or inequalities are the
only objects in the universe we can (try to) solve mathematically.
Relational operators
We also refresh the use of correct terms for inequalities and equations.
We define an inequality as any variable expression comparing a left hand side to a right
hand side by applying the 'is-(strictly)-less-than' or by applying the 'is-(strictly)-greaterthan' operator. For example, we can read (3a + x)4 (b + 4)(x + 3) containing variables
a, x, b. Consequently we may solve such inequality for any of the unknown quantities a, x
or b.
We define an equation as any variable expression comparing a left hand side to a right
hand side by applying the 'is-equal-to' operator. For example (3a + x)4 = (b + 4)(x + 3)
is an equation containing variables a, x, b. Consequently we also may solve equations for | 677.169 | 1 |
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Тема: Wolfram|Alpha Examples: Mathematics | 677.169 | 1 |
Understanding Mathematics for College (ages 14-17)
ID : 82218
This math preparation course will provide students with an overview of arithmetic, algebra, linear algebra, geometry, trigonometry, and an introduction to calculus so that they are ready for college level math courses. | 677.169 | 1 |
This lesson and worksheet guide pupils on solving equations graphically. All examples and questions are embedded in the PowerPoint and a worksheet is included to allow students to practise the topic during the lesson or as homework.
Presentation covering the whole of chapter 3 (4-6hrs) from the Pearson edexcel Pure Mathematics Year 1/AS book. Includes examples and notes for;
Linear simultaneous equations.
Quadratic simultaneous equations.
Simultaneous equations on graphs.
Linear inequalities.
Quadratic inequalities.
Inequalities on graphs.
Regions.
Suitable for the new A-Level specification year 1 or AS-level. Also suitable for able GCSE students.
Presentation designed to be used as teaching aid for entire block of lessons. Opportunities for students to work through examples on whiteboards, and notes on key points. All learning objectives included and maintained throughout the presentation. Editable for you to select you own questions or add consolidation work.
OTHER CHAPTERS AVAILABLE | 677.169 | 1 |
Saturday, January 4, 2014
Function Junction, Part 1
I was talking to my coach Sarah this week when I realized that I haven't been doing a very good job of telling my backstory. There are things that I've seen, done, and learned in life that no one, not even my closest friends, really know about. These are things that make me who I am and fuel my vision.
I was the math and science curriculum specialist at the Robert Adams Middle School in Holliston, Massachusetts from 2007 to 2010. In the fall of 2008, I worked with the 6th-grade math teachers to develop and implement a new functions unit. In most middle school math programs, students learn a little about slope in pre-algebra. But their first real introduction to functions is in Algebra I, where they are expected to master a number of discrete skills, including:
Using the slope and y-intercept to graph a line
Writing the equation for a line in slope-intercept form
Finding the slope and y-intercept of a line given two points
Working with linear equations in point-slope form
Working with linear equations in standard form
All of these skills should sit on top of a single conceptual framework, but because very few students enter Algebra I knowing anything about slope, there really isn't time to develop a conceptual framework and then get students to mastery in the time allotted. Instead, we are forced to ask students to learn each skill in isolation.
In Holliston, about 60% of our students take Algebra I in eighth-grade, and only half of those students do well on the functions chapter test. I would describe functions as a significant pain point for the eighth-grade math teachers; they really wanted their students to do better on this key concept, but they didn't know how to make that happen.
I decided to target the functions pain point in 2008 for five reasons:
I didn't have a very good working relationship with the eighth-grade math teachers and alleviating this eighth-grade math pain point would be a big step toward establishing one.
Functions is something that we could do completely in-house in the middle school. We could introduce functions in sixth-grade without relying on any help from the fifth-grade math teachers based in the elementary school, and we could have everything wrapped up and tied in a bow by the end of the eighth-grade, so we wouldn't need the high school math teachers to do anything different to take advantage of what we had done.
The math teachers all recognized that procedural approaches weren't working and wanted students to have a conceptual understanding of functions.
If we could help students develop conceptual understanding and double the number of students reaching mastery, then people would notice and applaud the work we were doing. It would also go a long way toward changing some of the core beliefs held by the middle school math teachers themselves.
The functions pain point isn't something that any one teacher can successfully tackle on his or her own. This would reinforce the fact that I wasn't pointing my finger and saying that individual teachers weren't doing their jobs. It would also highlight the good things that could happen if we set aside our egos and worked together as a department.
I spent about a month working with the sixth-grade math teachers to design a four-week functions unit for sixth-grade. We did this work after school and during their prep periods. As curriculum specialist, I had no say over how they spent their prep periods, so this was a purely volunteer effort. When we finally implemented the unit in November, I spent some time modeling lessons and coaching teachers in their classrooms.
Here are some of the problems we used in the summative assessment we gave at the end of the unit:
3) At the start of an experiment, a bacteria colony has an area of 22 cm2. It grows at a constant rate of 3 cm2 per day. Fill in the table and come up with a rule for predicting the area of this bacteria colony on Day x.
Day
Area (cm2)
0
1
2
3
4
28
55
4) Bob uses the following rule to predict the area of a bacteria colony on Day x:
area = 21 + 4x. How fast is the bacteria colony growing? What was its area at the start of the experiment? Explain how you can tell by looking at the rule.
5) FindUse the graph to answer problems 7-10 and the bonus:
7) Which bacteria colony is growing at a constant rate? How can you tell just by looking at the graph?
8) When is Bacteria Colony B growing faster than Bacteria Colony A? How can you tell just by looking at the graph?
9) What was the average growth rate per day for Bacteria Colony B between Days 4 and 7?
10) Describe the growth of Bacteria Colony B over time. Explain when it is growing faster or slower, when it is shrinking, and when it is staying the same size.
11) Bonus: Write a rule for the growth of Bacteria A over time.
We spent a professional day in January analyzing the results of the summative assessment as a department. 85% of the students scored a B or higher. The seventh- and eighth-grade math teachers were floored by both how well the students did and the quality of their written responses. We estimated that 75% of the students had a solid conceptual understanding of linear functions and could fluently translate among tables, equations, graphs, and written descriptions as long as functions were presented in a real-world context. 55% of the students completed the bonus correctly even though we had never asked them to write an equation from a graph with a fractional rate of change before, and another 20% understood what they needed to do, but just made a procedural error somewhere along the way.
One of the sixth-grade special education teachers worked closely with us as we developed the sixth-grade functions unit. She used a pared down version of the unit with the students in her substantially-separate math class. (A substantially-separate math class is for students with math learning disabilities so severe that it has been determined that their needs cannot be met in a general math class.) Her students loved the unit and got 75% of the pattern, relationship, and algebra questions on the state test correct when they only got 45% of the questions in the other four strands correct. The state test was administered in May, about five months after she taught the functions unit.
In 2010, I developed the seventh-grade functions unit with the seventh-grade math teachers. By this time, the first group of sixth-graders that had taken our sixth-grade functions unit were now seventh-graders. The seventh-grade math teachers had been excited by how well these students had done on the sixth-grade summative assessment, but they were anxious about how much those students would actually retain. We decided to start the unit with a warm up activity designed to re-activate what the students had learned sixteen months ago.
Although the students said that they had forgotten everything about functions from sixth-grade, the warm up activity demonstrated that the conceptual framework for functions they had developed was still there, and everything came flooding back to them.
Here are some of the problems we used in the summative assessment we gave at the end of the end of the seventh-grade unit:
A plant is growing at a constant rate. Use the table to answer problems 1-5.
Day
Plant Height (cm)
0
20
17
28
23
36
29
44
36
1) What is the rate of growth of the plant? Explain how you found it.
2) What is the starting height on Day 0 of the plant?
3) Write a rule for the height of the plant (y) after x days.
4) Use the rule to find the day when the height of the plant will be 80 cm. Write the rule and show all steps.
5) Create a Plant Height vs. Time graph.
Use the following rule to answer problems 6-8: w = 50m/3 + 100 where w = gallons of water in a tank and m = minutes.
6) Use the rule to fill in the rest of the table. Show all steps.
Minute
Water in Tank (gal)
0
10
15
620
800
11) Graph: y = 3x/4 − 2
13) Write an equation for the given graph.
In problems 14 and 15, graph the two given points, then find the slope, y-intercept, and equation for the line that passes through those two points.
14) (-9, 2) and (6, -4)
15) (-8, 2) and (-5, 10)
16) Explain how you found the slope for the graph in problem 14.
17) Explain how you found the y-intercept for the graph in problem 15.
18) Challenge: Find the point (x, y) where the two lines in problems 10 and 11 would cross if they were on the same graph.
We found that the 75% of the students who had developed a solid conceptual understanding of functions in sixth-grade were able to pick up immediately where they had left off and build on what they had learned. In fact, the seventh-grade math teachers found that they had to do very little teaching in order to get students to generalize from real-world contexts to abstract x's and y's, or to introduce fractional rates of change. However, the 25% of the students who were shaky after sixth-grade got left in the dust.
I left Holliston after the 2009-10 school year, but the plan was to develop a new functions unit in eighth-grade. The 75% of the students with a solid conceptual understanding should have easily been able to extend what they knew to linear equations in standard and point-slope forms. This means that all of the eighth-grade Algebra I students (instead of less than half) should have mastered linear functions before moving on to Algebra II in ninth-grade, saving the high school math teachers 3-4 weeks of re-teaching time. And the students that didn't take Algebra I should have been much better prepared for Algebra I in ninth-grade. All of this should have then opened the door to a district-wide conversation about collaboration and curriculum and instruction.
Within the middle school math department, I was hoping that the success of this conceptual approach to functions would lead to a willingness to work together and to try new instructional strategies. Up until this point, getting teachers to try some new curriculum hadn't been too difficult, but getting them to accept coaching was like pulling teeth. And new curriculum was substantially less effective without the necessary coaching and shifts in practices. My next goal would have been to analyze what we were doing and then to try to increase the percentage of students developing solid conceptual understanding. If you've been reading my blog, then you know that the first step is to design a curriculum that breaks down tasks to intuitive subtasks. I think we did that in the sixth- and seventh-grade functions units. The next step is figuring out how to help students learn how to perform those tasks. We got to 75%, which is excellent for our first try, but now we needed to push that up to 95%+. Doing that would mean challenging even deeper core beliefs and forcing more changes in how we operated as a middle school and district. | 677.169 | 1 |
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With this program, you can create a wide variety of worksheets filled with fraction and whole number problems. A large array of options allows you to customize the types of problems to include, the ranges of numbers to use, and the way to format the final...
Algematics can actually do your high school algebra, one step at a time! You can finish your work without waiting to ask your teacher. Enter equations and expressions straight out of your book or from your homework sheet. Simply point and click to...
If you have problems to do those metric conversions, especially for complicated units, Metric Converter is the solution. Easily make conversions between units like distance, speed, area, temperature, density, color code and more. Various other options to...
Matrix is a Windows application designed to perform matrix operations for linear algebra students. All calculations are done with unlimited precision rational numbers and, the results are displayed as formatted fractions. The window scrolls leaving a...
Air and Exhaust Gas Properties is an MS Windows application that calculates the properties of air/exhaust gases within a wide range of pressure and temperature. It may be used for analyzing a number of energy conversion units used in power plants and... | 677.169 | 1 |
Imagine a Geometry classroom with no blackboard, whiteboard, or any other place to construct figures or make drawings. Imagine teaching Algebra without graphs or tables of values. There is no doubt that mathematics education is unique in its utter reliance on mathematical representations. The dependency is so complete that most of us forget that the objects of mathematics are purely in our minds (or "between" the minds of those engaged in mathematical discourse). Instead, we mistake the representations for the objects rather than the representatives of those objects. Technology blurs the line even further by endowing representations with properties of the underlying objects that the previous generations of paper and pencil representations did not have.
Suppose I wish to introduce students to the topic of solving trigonometric equations. I start with the equation sin(x)=sqrt(3)/2, where "sqrt" is used to denote the square root function. I wish my students to understand first that there are an infinite number of solutions, second that these solutions fall into two branches, and third that there is a conventional way of representing these two branches. After that, I can concentrate on the solution steps required to obtain the conventional representations.
Using the HP Prime graphing calculator, I enter sin(X)=sqrt(3)/2 in Symbolic view (first figure below) and then press Plot to see the graph (second figure below).
Figure 1 Figure 2
The graphical representation is a set of vertical lines, arranged in groups of two. The vertical lines suggest solutions of the form X=C, where C is a real nummber. The lines appear in pairs, suggesting two sets of solutions. I can zoom out to see that the pairs appear to extend indefinitely, suggesting that the number of solutions is infinite. In just a few keystrokes, I have created a representation that has a high degree of mathematical fidelity with respect to my first two objectives. In the figures below, I use a pinch gesture to zoom out and then back in to focus on just 2 or 3 of the pairs.
Figure 3 Figure 4
In Figure 4, the cursor is on X=1.04719755 or X=pi/3, the first line to the right of the y-axis. Tapping on the second line, I see X=2.0943951 (not shown here). It is not difficult to establish that this second value is twice the first, or X=2*pi/3. I have a start on my two solutions. Now I need to see how far apart these line pairs are. The Numeric view shown below displays the first 4 pairs of solutions numerically.
If I examine every other value, starting with the first, I see 1.047..., 7.330..., 13.6 | 677.169 | 1 |
Pages
Friday, August 7, 2015
Introduction to General Relativity by Lewis Ryder
A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics.
Student-friendly style, over 100 illustrations, and several exercises with solutions available online
Mathematical calculations are worked out in detail to help develop an intuitive understanding of the subject.
Includes topics that overlap with other areas of fundamental physics, to help the reader assess how GR relates to other physics subjects. | 677.169 | 1 |
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Composition of Functions Comic Book Style Doodle Notes and Practice
Your students will love this new Comic Book style Doodle Note resource which can be used as homework, assessment, or enrichment. A fun and engaging format to practice their skills with Composition of Functions plus it is NO Prep for you.
It has been shown that doodling and working with fun themed activities help students engage with the material, and be more at ease in the learning environment. This is especially true in mathematics.
Included are notes and practice problems using both common notations, parentheses and dot. Students practice a variety of problems using up to four functions. They then embellish the handout using their imagination and creativity. This resource does not include domains, but you could easily add that as an enhancement | 677.169 | 1 |
Trigonometric Functions - A guide for teachers - course
Familiarity with the material in the modules, Introduction to Trigonometry and Further Trigonometry.
Knowledge of basic coordinate geometry.
Introductory graphs and functions.
Facility with simple algebra, formulas and equations.
MOTIVATION
In the module, Further Trigonometry, we saw how to use points on the unit circle to extend the definition of the trigonometric ratios to include obtuse angles. That same construction can be extended to angles between 180° and 360° and beyond. The sine, cosine and tangent of negative angles can be defined as well. Once we can find the sine, cosine and tangent of any angle, we can use a table of values to plot the graphs of the functions y = sin x, y = cos x and y = tan x. In this module, we will deal only with the graphs of the first two functions. The graphs of the sine and cosine functions are used to model wave motion and form the basis for applications ranging from tidal movement to signal processing which is fundamental in modern telecommunications and radio-astronomy. This provides a breathtaking example of how a simple idea involving geometry and ratio was abstracted and developed into a remarkably powerful tool that has changed the world. | 677.169 | 1 |
4.88
9.00
Course Description
1. Roles and connections between matrices and vectors, linear equation solutions, linear algebra and vector calculus.
2. Vectors and vector space for interpreting matrix rank and the different solutions to linear equations
3. Vector calculus, including Jacobian, divergence, Green's and Stokes' theorems
4. Special matrices such as triangular, diagonal, and orthogonal matrices
5. Gauss elimination and Gauss-Jordan method and their relationship with
elementary matrices for different types of matrix factorization and decomposition | 677.169 | 1 |
The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development. This volume's BTE9780486458069
Descrizione libro Dover Publications. Paperback. Condizione libro: New. Paperback. 144 pages. Dimensions: 8.3in. x 5.3in. x 0.4in.The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetics underlying concepts and their logical development. This volumes This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback. Codice libro della libreria 9780486458069 | 677.169 | 1 |
for Algebra 2 Unit 1 Expressions, Equations and Inequalities. Interactive notebook foldables (color and black and white versions), notes, practice problems, task cards, and more are included.
Topics include:
- Real Number System & Algebra 2 Mega Teacher Resource Bundle which includes my entire library of Algebra 2 resources/units
- Expressions, Equations and Inequalities Foldable Only bundle for interactive notebooks all the foldables and notes for Algebra 2 Unit 1 Expressions, Equations and Inequalities. Interactive notebook foldables in color and black and white, practice problems, notes, and answer keys included.
Topics include:
- Real Number System and Expressions, Equations and Inequalities bundle - Unit 1 for Algebra 2
- Algebra 2 Mega Teacher Resource Bundle which includes my entire library of Algebra 2 resources/units foldable set is a quick review of Absolute value equations. This Is meant for students familiar with the concept. This is the third lesson in Algebra 2 unit 1.
This resource includes
- Color-coded graphic organizers
- Black-line master graphic organizers
- Color coded notes with examples.
- Practice problems
- Answer key for practice problems
Algebra 2 Unit 1 Topics include:
- Real Number System and Properties of Real Numbers (FREE)
- Evaluate and simplify algebraic expressions and equations
- Absolute value equations
- Solve linear inequalities and multi-step inequalities
- Solve absolute value inequalities
Related bundles include:
- Algebra 2 Mega Teacher Resource Bundle which includes my entire library of Algebra 2 resources/units
- Expressions, Equations and Inequalities Foldable Only bundle for interactive notebooks
- Algebra 2 unit 1 teacher resource an entire lesson to get your class used to the levels of the manipulation needed to survive the A level maths course. The powerpoint contains differentiated lesson objectives and examples which are worked through slowly. This can be printed and given to the students.
The excel document can be printed off and directed differentiation can let students select the level of work which will challenge them. This is the first in what potentially could be the entire a level course
Colouring + Polynomials = Awesomeness.
This includes a complete class set of 36 worksheets / colouring pages that combine to create the football player mosaic. Every single worksheet is different, so you can count on hearing great math dialog between your students.
Assessment is easy using the full answer key AND complete solution manual. You can make this available to students to self-assess their work (using the task for review or practice), or keep it for your own assessment purposes if you're markin' for keeps.
Common Core Standards: HSF-IF.A.2, HSF-BF.B.4
MATH INVOLVED:
◾Students evaluate polynomial functions for inputs in their domains
◾Each problem gives a set of 1 to 3 functions like this:
{h(x)= -5x-3, m(x)= 5x²+22x-1, j(x)= -3x²-5x}
And are asked to evaluate something like this:
h^-1(-5) + m(-3) - j(1)
◾Inverse functions are used in linear cases randomly about 40% of the time
◾Students gain practice with exponent and integer laws (yes senior students still make those mistakes!) and general order of operation skills when simplifying a multi-step expression.
I'd love to hear your feedback and your students' response to this task.
These tasks ensure individual accountability (since every worksheet is different) while harnessing collaborative motivation (since every worksheet is needed for the mosaic). The end result will look spectacular hanging up in your classroom, and your students will be proud!
It's simple!
1. Calculate the answers.
2. Colour the squares.
3. Cut out your section.
4. Combine with the class!
INCLUDED:
◾ .pdf and .docx versions of everything
◾ Complete 36-sheet class set of worksheets
◾ Teaching Tips page, for smooth implementation :)
◾ Complete answer keys for all worksheets
◾ Complete Solution Key showing steps taken to reach each answer
◾ 'Colour-Range' answer key for quick at-a-glance assessment. e.g. Blue [0, 2), Red [-4, -2) ...
◾ Every worksheets contains its answers randomized at the bottom, helping the student self-assess his/her work
The student buy-in factor is HUGE with these worksheets; they all want to see the finished picture come together!
Leave the picture a secret or show it for motivation… it's your call.
All my "Colouring by…" worksheets use standard pencil-crayon colours found in the Crayola 24 pack. For best results, use the exact colour name match (and encourage quality colouring!). Perhaps a class set of pencil crayons would be a fun math department investment!
~CalfordMath
These several worksheets of various difficulty on linear equations (one variable equations), they go from fairly difficult to monstrously difficult! Very detailed step-by-step solutions are provided with explanations of each step. Have a look and let me know what you think!
I hope you find these useful. You can get more FREE worksheets on many topics, mix and match, with detailed step-by-step solutions at No signup, ads, just great questions and solutions
This study guide provides an in-depth review of terms, concepts, and example problems related to limits and continuity. Topics include estimating and evaluating limits, resolving indeterminate form, infinite limits and limits at infinity, continuity, and Intermediate Value Theorem. Although specifically designed for AP Calculus AB, this resource could be used with almost any calculus course. Students can quickly familiarize themselves with the related concepts and go through corresponding examples with solutions. Targeted review assignments could correspond to this study guide, further allowing students to determine their grasp of the included concepts. [Page numbers correspond to Larson Seventh Edition] | 677.169 | 1 |
Tuesday 03 WORKSHOP - Introduction to logarithms : What are logarithms and what are they used for?
We start to demystify the logarithm by exploring its definition with simple examples and finding its place in your mathematical toolkit. The rules for manipulating logarithm expressions (the "log laws") are introduced and used in standard test style questions. After that we demonstrate | 677.169 | 1 |
Homework tasks
Standards Documents High School Mathematics Standards Coordinate Algebra and Algebra I Crosswalk Analytic Geometry and Geometry Crosswalk bowie state university essay question Resources. RnKey Node. Aplia significantly improves outcomes and elevates thinking by increasing student effort and engagement. Veloped by teachers, Aplia assignments connect concepts to. Tch the EARLY CHANNEL, Now on YouTube. Release notes; Node. Manual Documentation; NPM Node package registry; Awesome Node. Veloped by teachers, Aplia assignments connect concepts to. Re are some. GreatSchools is a free, mobile friendly nonprofit parenting site devoted to helping parents guide their children to success in and out of the classroom. Ganization Map. View Prior MissionsHow to Get Your Homework Done Fast. Download The EARLY Basic Base Construction Instructions. Get Moving With BB. Aplia significantly improves outcomes and elevates thinking by increasing student effort and engagement. A curated list of delightful Node.. Ing homework can be both time consuming and frustrating, and life is more than just homework. OneView portal replaces paper forms The new OneView portal is our electronic forms website allowing the majority of student forms to be completed and submitted digitally.
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R K 12 kids, teachers and parents. Veloped by teachers, Aplia assignments connect concepts to. Udyladder is an online english literacy mathematics learning tool? Ds activity games. Aplia significantly improves outcomes and elevates thinking by increasing student effort and engagement. Ds activity games. Get your work space set, your schedule organized, and your studying done with the help of this article. Used by over 70,000 teachers 1 million students at home and school. Udyladder is an online english literacy mathematics learning tool? Standards Documents High School Mathematics Standards Coordinate Algebra and Algebra I Crosswalk Analytic Geometry and Geometry CrosswalkUsed by over 70,000 teachers 1 million students at home and school. Highlights New GSE Grade 6 Support Materials for Remediation New GSE Grade 7 Support Materials for Remediation New GSE Grade 8 Support Materials for? Standards Documents High School Mathematics Standards Coordinate Algebra and Algebra I Crosswalk Analytic Geometry and Geometry CrosswalkHaving trouble getting a handle on all of your homework. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum.
How to write a good executive summary
Ease suggest songs you would REALLY like to hear at Prom. Download The EARLY Basic Base Construction Instructions. View Prior Missions
Student Council is compiling a list of songs that students would like to hear at Prom.
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Ncil and paper versions as well as computer based versions are. .
Astronomy Interactives.
Welcome to Dade Middle School! Is site provides ranking tasks for teaching introductory astronomy! An educational family made up of teachers, students, parents, and community members, Dade Middle School exists to use the.
IPI) is a diversified industrial cleaner processing company. Here's how to help them hit the books and develop good study habits . E company 10
The nagging, the battles, the lost papers—do you dread school work as much as the kids do. | 677.169 | 1 |
Algebrator
Softmath
Pre-Algebra to College
Algebrator is a desktop application for solving a large range of math equations frequently covered in algebra classes. Think of it like a calculator for algebra, or an 'algebrator'. The program's interface is a simple sheet of lined paper, with a selection of math icons, functions, and a toolbar at the top. Simply enter an equation and click on the 'solve all' button, and the program spits out the answer, step-by-step. Don't understand a step? Highlight it, then click on the 'explain' button. Need to graph your answer? Algebrator does that, too.
Algebrator handles not only what one might think of as classic algebra problems, but also simpler and more complex problems. Examples range from finding least common multipliers and greatest common factors, to synthetic division, quadratic equations, and solving matrices. But beware: the user has to be able to correctly translate word problems into equations for Algebrator to help.
Educational Value
Algebrator is impressive in its problem-solving capability. It handled every problem we threw at it, with ease. However, its teaching ability is limited. Explanations are accurate, but tend to be somewhere between cryptic and terse. As a result, if you were struggling to understand a concept, the explanations might prove to be of limited value. In addition, our testers felt that the step-by-step answers didn't always use the methodology they would have used, or expected. While this makes no difference to the ultimate answer, it may make a difference to students whose teachers expect them to follow a prescribed methodology.
Kid Appeal
Programs like Algebrator are in a special category, in that they are tools where the user provides much of the content. So in that respect, they are somewhat like spreadsheets or word processors, with the addition of a modicum of explanatory material. As a result, Kid Appeal is not measured by hours of use, or number of sessions. Rather, the appeal is measured by the frequency with which a student returns to the tool for use. On that basis, Algebrator offers promise for students who have a basic understanding of principles, but need some help, or wish to check their answers.
Ease of Use / Install
Algebrator installed quickly and without difficulty on SuperKids' test machines. The user interface is generally intuitive, and when questions arose, the built-in manual and video demos were adequate.
Best for... / Bottom-Line
Algebrator is best for students with a good general understanding of basic math principles, but need some help, or wish to check their answers. It will be of limited utility for struggling students, or those trying to teach themselves new material. | 677.169 | 1 |
Similar
Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails.
The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition.
· New edition of the "Classic" book on the topic · Wonderful introduction to a rich research area · Leading author in the field of algorithmic graph theory · Beautifully written for the new mathematician or computer scientist · Comprehensive treatment
This textbook is an introduction to algebra via examples. The book moves from properties of integers, through other examples, to the beginnings of group theory. Applications to public key codes and to error correcting codes are emphasised. These applications, together with sections on logic and finite state machines, make the text suitable for students of computer science as well as mathematics students. Attention is paid to historical development of the mathematical ideas. This second edition contains new material on mathematical reasoning skills and a new chapter on polynomials has been added. The book was developed from first-level courses taught in the UK and USA. These courses proved successful in developing not only a theoretical understanding but also algorithmic skills. This book can be used at a wide range of levels: it is suitable for first- or second-level university students, and could be used as enrichment material for upper-level school studentsThis clearly written, mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NP-complete problems, more. All chapters are supplemented by thought-provoking problems. A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering. "Mathematicians wishing a self-contained introduction need look no further." — American Mathematical Monthly.
An introduction to the mathematical theory of multistage decision processes, this text takes a "functional equation" approach to the discovery of optimum policies. Written by a leading developer of such policies, it presents a series of methods, uniqueness and existence theorems, and examples for solving the relevant equations. The text examines existence and uniqueness theorems, the optimal inventory equation, bottleneck problems in multistage production processes, a new formalism in the calculus of variation, strategies behind multistage games, and Markovian decision processes. Each chapter concludes with a problem set that Eric V. Denardo of Yale University, in his informative new introduction, calls "a rich lode of applications and research topics." 1957 edition. 37 figures bookCovering all the mathematical techniques required to resolve geometric problems and design computer programs for computer graphic applications, each chapter explores a specific mathematical topic prior to moving forward into the more advanced areas of matrix transforms, 3D curves and surface patches. Problem-solving techniques using vector analysis and geometric algebra are also discussed.
This easy-to-follow textbook introduces the mathematical language, knowledge and problem-solving skills that undergraduate students need to enter the world of computer and information sciences. The language is in part qualitative, with concepts such as set, relation, function and recursion/induction; but it is also partly quantitative, with principles of counting and finite probability. Entwined with both are the fundamental notions of logic and their use for representation and proof. In ten chapters on these topics, the book guides the student through essential concepts and techniques.
The extensively revised second edition provides further clarification of matters that typically give rise to difficulty in the classroom and restructures the chapters on logic to emphasize the role of consequence relations and higher-level rules, as well as including more exercises and solutions.
Topics and features: teaches finite mathematics as a language for thinking, as much as knowledge and skills to be acquired; uses an intuitive approach with a focus on examples for all general concepts; brings out the interplay between the qualitative and the quantitative in all areas covered, particularly in the treatment of recursion and induction; balances carefully the abstract and concrete, principles and proofs, specific facts and general perspectives; includes highlight boxes that raise common queries and clear away confusions; provides numerous exercises, with selected solutions, to test and deepen the reader's understanding.
This clearly-written text/reference is a must-read for first-year undergraduate students of computing. Assuming only minimal mathematical background, it is ideal for both the classroom and independent study.
From the Rosetta Stone to public-key cryptography, the art and science of cryptology has been used to unlock the vivid history of ancient cultures, to turn the tide of warfare, and to thwart potential hackers from attacking computer systems. Codes: The Guide to Secrecy from Ancient to Modern Times explores the depth and breadth of the field, remaining accessible to the uninitiated while retaining enough rigor for the seasoned cryptologist.
The book begins by tracing the development of cryptology from that of an arcane practice used, for example, to conceal alchemic recipes, to the modern scientific method that is studied and employed today. The remainder of the book explores the modern aspects and applications of cryptography, covering symmetric- and public-key cryptography, cryptographic protocols, key management, message authentication, e-mail and Internet security, and advanced applications such as wireless security, smart cards, biometrics, and quantum cryptography. The author also includes non-cryptographic security issues and a chapter devoted to information theory and coding. Nearly 200 diagrams, examples, figures, and tables along with abundant references and exercises complement the discussion.
Written by leading authority and best-selling author on the subject Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times is the essential reference for anyone interested in this exciting and fascinating field, from novice to veteran practitioner.
Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This lively introductory text focuses on a problem-solving approach to the subject.
Key features of Number Theory: Structures, Examples, and Problems:
* A rigorous exposition starts with the natural numbers and the basics.
* Important concepts are presented with an example, which may also emphasize an application. The exposition moves systematically and intuitively to uncover deeper properties.
* Topics include divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, quadratic residues, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems are covered.
* Unique exercises reinforce and motivate the reader, with selected solutions to some of the problems.
* Glossary, bibliography, and comprehensive index round out the text.
Written by distinguished research mathematicians and renowned teachers, this text is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles, from advanced high school students to undergraduates, their instructors, and general readers at all levels.
Across the Board is the definitive work on chessboard problems. It is not simply about chess but the chessboard itself--that simple grid of squares so common to games around the world. And, more importantly, the fascinating mathematics behind it. From the Knight's Tour Problem and Queens Domination to their many variations, John Watkins surveys all the well-known problems in this surprisingly fertile area of recreational mathematics. Can a knight follow a path that covers every square once, ending on the starting square? How many queens are needed so that every square is targeted or occupied by one of the queens?
Each main topic is treated in depth from its historical conception through to its status today. Many beautiful solutions have emerged for basic chessboard problems since mathematicians first began working on them in earnest over three centuries ago, but such problems, including those involving polyominoes, have now been extended to three-dimensional chessboards and even chessboards on unusual surfaces such as toruses (the equivalent of playing chess on a doughnut) and cylinders. Using the highly visual language of graph theory, Watkins gently guides the reader to the forefront of current research in mathematics. By solving some of the many exercises sprinkled throughout, the reader can share fully in the excitement of discovery.
Showing that chess puzzles are the starting point for important mathematical ideas that have resonated for centuries, Across the Board will captivate students and instructors, mathematicians, chess enthusiasts, and puzzle devotees.
Designed for advanced high school students, undergraduates, graduate students, mathematics teachers, and any lover of mathematical challenges, this two-volume set offers a broad spectrum of challenging problems — ranging from relatively simple to extremely difficult. Indeed, some rank among the finest achievements of outstanding mathematicians. Translated from a well-known Russian work entitled Non-Elementary Problems in an Elementary Exposition, the chief aim of the book is to acquaint the readers with a variety of new mathematical facts, ideas, and methods. And while the majority of the problems represent questions in higher ("non-elementary") mathematics, most can be solved with elementary mathematics. In fact, for the most part, no knowledge of mathematics beyond a good high school course is required. Volume One contains 100 problems, with detailed solutions, all dealing with probability theory and combinatorial analysis. Topics include the representation of integers as sums and products, combinatorial problems on the chessboard, geometric problems on combinatorial analysis, problems on the binomial coefficients, problems on computing probabilities, experiments with infinitely many possible outcomes, and experiments with a continuum of possible outcomes. Volume Two contains 74 problems from various branches of mathematics, dealing with such topics as points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, nondecimal counting, theory of primes, and more. In both volumes the statements of the problems are given first, followed by a section giving complete solutions. Answers and hints are given at the end of the book. Ideal as a text, for self-study, or as a working resource for a mathematics club, this wide-ranging compilation offers 174 carefully chosen problems that will test the mathematical acuity and problem-solving skills of almost any student, teacher, or mathematician.
Discrete mathematics is quickly becoming one of the most important areas of mathematical research, with applications to cryptography, linear programming, coding theory and the theory of computing. This book is aimed at undergraduate mathematics and computer science students interested in developing a feeling for what mathematics is all about, where mathematics can be helpful, and what kinds of questions mathematicians work on. The authors discuss a number of selected results and methods of discrete mathematics, mostly from the areas of combinatorics and graph theory, with a little number theory, probability, and combinatorial geometry. Wherever possible, the authors use proofs and problem solving to help students understand the solutions to problems. In addition, there are numerous examples, figures and exercises spread throughout the book.
László Lovász is a Senior Researcher in the Theory Group at Microsoft Corporation. He is a recipient of the 1999 Wolf Prize and the Gödel Prize for the top paper in Computer Science. József Pelikán is Professor of Mathematics in the Department of Algebra and Number Theory at Eötvös Loránd University, Hungary. In 2002, he was elected Chairman of the Advisory Board of the International Mathematical Olympiad. Katalin Vesztergombi is Senior Lecturer in the Department of Mathematics at the University of Washington.
It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples.
Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors.
The book covers key foundation topics:
o Taylor series methods
o Runge--Kutta methods
o Linear multistep methods
o Convergence
o Stability
and a range of modern themes:
o Adaptive stepsize selection
o Long term dynamics
o Modified equations
o Geometric integration
o Stochastic differential equations
The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via
Mathematics and mathematical modelling are of central importance in computer science, and therefore it is vital that computer scientists are aware of the latest concepts and techniques.
This concise and easy-to-read textbook/reference presents an algorithmic approach to mathematical analysis, with a focus on modelling and on the applications of analysis. Fully integrating mathematical software into the text as an important component of analysis, the book makes thorough use of examples and explanations using MATLAB, Maple, and Java applets. Mathematical theory is described alongside the basic concepts and methods of numerical analysis, supported by computer experiments and programming exercises, and an extensive use of figure illustrations.
Topics and features: thoroughly describes the essential concepts of analysis, covering real and complex numbers, trigonometry, sequences and series, functions, derivatives and antiderivatives, definite integrals and double integrals, and curves; provides summaries and exercises in each chapter, as well as computer experiments; discusses important applications and advanced topics, such as fractals and L-systems, numerical integration, linear regression, and differential equations; presents tools from vector and matrix algebra in the appendices, together with further information on continuity; includes definitions, propositions and examples throughout the text, together with a list of relevant textbooks and references for further reading; supplementary software can be downloaded from the book's webpage at
This textbook is essential for undergraduate students in Computer Science. Written to specifically address the needs of computer scientists and researchers, it will also serve professionals looking to bolster their knowledge in such fundamentals extremely well.
Following the recent updates to the 2013 ACM/IEEE Computer Science curricula, Discrete Structures, Logic, and Computability, Fourth Edition, has been designed for the discrete math course that covers one to two semesters. Dr. Hein presents material in a spiral medthod of learning, introducing basic information about a topic, allowing the students to work on the problem and revisit the topic, as new information and skills are established. Written for prospective computer scientist, computer engineers, or applied mathematicians, who want to learn about the ideas that inspire computer science, this edition contains an extensive coverage of logic, setting it apart from similar books available in the field of Computer Science.
Research on distributions associated with sorting algorithms has grown dramatically over the last few decades, spawning many exact and limiting distributions of complexity measures for many sorting algorithms. Yet much of this information has been scattered in disparate and highly specialized sources throughout the literature. In Sorting: A Distribution Theory, leading authority Hosam Mahmoud compiles, consolidates, and clarifies the large volume of available research, providing a much-needed, comprehensive treatment of the entire emerging distributional theory of sorting.
Mahmoud carefully constructs a logical framework for the analysis of all standard sorting algorithms, focusing on the development of the probability distributions associated with the algorithms, as well as other issues in probability theory such as measures of concentration and rates of convergence. With an emphasis on narrative rather than technical explanations, this exceptionally well-written book makes new results easily accessible to a broad spectrum of readers, including computer professionals, scientists, mathematicians, and engineers. Sorting: A Distribution Theory: * Contains introductory material on complete and partial sorting * Explains insertion sort, quick sort, and merge sort, among other methods * Offers verbal descriptions of the mechanics of the algorithms as well as the necessary code * Illustrates the distribution theory of sorting using a broad array of both classical and modern techniques * Features a variety of end-of-chapter exercises
Newly enlarged, updated second edition of a valuable text presents algorithms for shortest paths, maximum flows, dynamic programming and backtracking. Also discusses binary trees, heuristic and near optimums, matrix multiplication, and NP-complete problems. 153 black-and-white illus. 23 tables. Newly enlarged, updated second edition of a valuable, widely used text presents algorithms for shortest paths, maximum flows, dynamic programming and backtracking. Also discussed are binary trees, heuristic and near optimums, matrix multiplication, and NP-complete problems. New to this edition: Chapter 9 shows how to mix known algorithms and create new ones, while Chapter 10 presents the "Chop-Sticks" algorithm, used to obtain all minimum cuts in an undirected network without applying traditional maximum flow techniques. This algorithm has led to the new mathematical specialty of network algebra. The text assumes no background in linear programming or advanced data structure, and most of the material is suitable for undergraduates. 153 black-and-white illus. 23 tables. Exercises, with answers at the ends of chapters.
This is a unique type of book; at least, I have never encountered a book of this kind. The best description of it I can give is that it is a mystery novel, developing on three levels, and imbued with both educational and philosophical/moral issues. If this summary description does not help understanding the particular character and allure of the book, possibly a more detailed explanation will be found useful. One of the primary goals of the author is to interest readers—in particular, young mathematiciansorpossiblypre-mathematicians—inthefascinatingworldofelegant and easily understandable problems, for which no particular mathematical kno- edge is necessary, but which are very far from being easily solved. In fact, the prototype of such problems is the following: If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors? More than half a century ago it was established that the least number of colors needed for such a coloring is either 4, or 5, or 6 or 7. Well, which is it? Despite efforts by a legion of very bright people—many of whom developed whole branches of mathematics and solved problems that seemed much harder—not a single advance towards the answer has been made. This mystery, and scores of other similarly simple questions, form one level of mysteries explored. In doing this, the author presents a whole lot of attractive results in an engaging way, and with increasing level of depth.
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come.
Problem-solving competitions for mathematically talented sec ondary school students have burgeoned in recent years. The number of countries taking part in the International Mathematical Olympiad (IMO) has increased dramatically. In the United States, potential IMO team members are identified through the USA Mathematical Olympiad (USAMO), and most other participating countries use a similar selection procedure. Thus the number of such competitions has grown, and this growth has been accompanied by increased public interest in the accomplishments of mathematically talented young people. There is a significant gap between what most high school math ematics programs teach and what is expected of an IMO participant. This book is part of an effort to bridge that gap. It is written for students who have shown talent in mathematics but lack the back ground and experience necessary to solve olympiad-level problems. We try to provide some of that background and experience by point out useful theorems and techniques and by providing a suitable ing collection of examples and exercises. This book covers only a fraction of the topics normally rep resented in competitions such as the USAMO and IMO. Another volume would be necessary to cover geometry, and there are other v VI Preface special topics that need to be studied as part of preparation for olympiad-level competitions. At the end of the book we provide a list of resources for further study.
Introduction to Enumerative and Analytic Combinatorics fills the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The book first deals with basic counting principles, compositions and partitions, and generating functions. It then focuses on the structure of permutations, graph enumeration, and extremal combinatorics. Lastly, the text discusses supplemental topics, including error-correcting codes, properties of sequences, and magic squares.
Strengthening the analytic flavor of the book, this Second Edition:
Features a new chapter on analytic combinatorics and new sections on advanced applications of generating functions Demonstrates powerful techniques that do not require the residue theorem or complex integration Adds new exercises to all chapters, significantly extending coverage of the given topics
Introduction to Enumerative and Analytic Combinatorics, Second Edition makes combinatorics more accessible, increasing interest in this rapidly expanding field.
Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University
The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications.
Features:
* Balances presentation of the mathematics with applications to signal processing * Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox
A Wavelet Tour of Signal Processing: The Sparse Way, Third Edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering.
Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company.Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4 Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox Balances presentation of the mathematics with applications to signal processing
Graph theory is an invaluable tool for the designer of algorithms for distributed systems.
This hands-on textbook/reference presents a comprehensive review of key distributed graph algorithms for computer network applications, with a particular emphasis on practical implementation. Each chapter opens with a concise introduction to a specific problem, supporting the theory with numerous examples, before providing a list of relevant algorithms. These algorithms are described in detail from conceptual basis to pseudocode, complete with graph templates for the stepwise implementation of the algorithm, followed by its analysis. The chapters then conclude with summarizing notes and programming exercises.
Topics and features: introduces a range of fundamental graph algorithms, covering spanning trees, graph traversal algorithms, routing algorithms, and self-stabilization; reviews graph-theoretical distributed approximation algorithms with applications in ad hoc wireless networks; describes in detail the implementation of each algorithm, with extensive use of supporting examples, and discusses their concrete network applications; examines key graph-theoretical algorithm concepts, such as dominating sets, and parameters for mobility and energy levels of nodes in wireless ad hoc networks, and provides a contemporary survey of each topic; presents a simple simulator, developed to run distributed algorithms; provides practical exercises at the end of each chapter.
This classroom-tested and easy-to-follow textbook is essential reading for all graduate students and researchers interested in discrete mathematics, algorithms and computer networks.
What is the maximum number of pizza slices one can get by making four straight cuts through a circular pizza? How does a computer determine the best set of pixels to represent a straight line on a computer screen? How many people at a minimum does it take to guard an art gallery?
Discrete mathematics has the answer to these—and many other—questions of picking, choosing, and shuffling. T. S. Michael's gem of a book brings this vital but tough-to-teach subject to life using examples from real life and popular culture. Each chapter uses one problem—such as slicing a pizza—to detail key concepts about counting numbers and arranging finite sets. Michael takes a different perspective in tackling each of eight problems and explains them in differing degrees of generality, showing in the process how the same mathematical concepts appear in varied guises and contexts. In doing so, he imparts a broader understanding of the ideas underlying discrete mathematics and helps readers appreciate and understand mathematical thinking and discovery.
This book explains the basic concepts of discrete mathematics and demonstrates how to apply them in largely nontechnical language. The explanations and formulas can be grasped with a basic understanding of linear equations.
Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite.
An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.
The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and computational models. The P versus NP Question asks whether or not finding solutions is harder than checking the correctness of solutions. An alternative formulation asks whether or not discovering proofs is harder than verifying their correctness. It is widely believed that the answer to these equivalent formulations is positive, and this is captured by saying that P is different from NP. Although the P versus NP Question remains unresolved, the theory of NP-completeness offers evidence for the intractability of specific problems in NP by showing that they are universal for the entire class. Amazingly enough, NP-complete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics and science are NP-complete.
Fortran continues to be the premier language used in scientific and engineering computing since its introduction in the 1950s. Fortran 2003 is the latest standard version and has many excellent modern features that assist programmers in writing efficient, portable and maintainable programs that are useful for everything from 'hard science' to text processing.
The Fortran 2003 Handbook is the definitive and comprehensive guide to Fortran 2003, the latest standard version of Fortran. This all-inclusive volume offers a reader-friendly, easy-to-follow and informal description of Fortran 2003, and has been developed to provide not only a readable explanation of features, but also some rationale for the inclusion of features and their use. Experienced Fortran 95 programmers will be able to use this volume to assimilate quickly those features in Fortran 2003 that are not in Fortran 95 (Fortran 2003 contains all of the features of Fortran 95).
Features and benefits:
• The complete syntax of Fortran 2003 is supplied.
• Each of the intrinsic standard procedures is described in detail.
• There is a complete listing of the new, obsolescent, and deleted features.
• Numerous examples are given throughout, providing insights into intended uses and interactions of the features.
• IEEE module procedures are covered thoroughly.
• Chapters begin with a summary of the main terms and concepts described.
• Models provide the reader with insight into the language.
Key Topics:
• Fortran Concepts and Terms
• Language Elements and Source Form
• Data Types
• Block Constructs and Execution Control
• I/O Processing and Editing
• Interoperability with C
• Standard Intrinsic Procedures
This highly versatile and authoritative handbook is intended for anyone who wants a comprehensive survey of Fortran 2003, including those familiar with programming language concepts but unfamiliar with Fortran. It offers a practical description of Fortran 2003 for professionals developing sophisticated application and commercial software in Fortran, as well as developers of Fortran compilers.
All authors have been heavily involved in the development of Fortran standards. They have served on national and international Fortran standard development committees, and include a chair, convenors and editors of the Fortran 90, 95, and 2003 standards. In addition, Walt Brainerd is the owner of The Fortran Company, Tucson, AZ, USA.
A second edition of a book is a success and an obligation at the same time. We are satis ed that a number of university courses have been orga› nized on the basis of the rst volume of Comprehensive Mathematics for Computer Scientists. The instructors recognized that the self›contained presentation of a broad specturm of mathematical core topics is a rm point of departure for a sustainable formal education in computer sci› ence. We feel obliged to meet the valuable feedback of the responsible in› structors of such courses, in particular of Joel Young (Computer Science Department, Brown University) who has provided us with numerous re› marks on misprints, errors, or obscurities. We would like to express our gratitude for these collaborative contributions. We have reread the entire text and not only eliminated identi ed errors, but also given some addi› tional examples and explications to statements and proofs which were exposed in a too shorthand style. A second edition of the second volume will be published as soon as the errata, the suggestions for improvements, and the publisher's strategy are in harmony.
Data mining essentially relies on several mathematical disciplines, many of which are presented in this second edition of this book. Topics include partially ordered sets, combinatorics, general topology, metric spaces, linear spaces, graph theory. To motivate the reader a significant number of applications of these mathematical tools are included ranging from association rules, clustering algorithms, classification, data constraints, logical data analysis, etc. The book is intended as a reference for researchers and graduate students. The current edition is a significant expansion of the first edition. We strived to make the book self-contained and only a general knowledge of mathematics is required. More than 700 exercises are included and they form an integral part of the material. Many exercises are in reality supplemental material and their solutions are included.
Text Mining with MATLAB provides a comprehensive introduction to text mining using MATLAB. It's designed to help text mining practitioners, as well as those with little-to-no experience with text mining in general, familiarize themselves with MATLAB and its complex applications.
The first part provides an introduction to basic procedures for handling and operating with text strings. Then, it reviews major mathematical modeling approaches. Statistical and geometrical models are also described along with main dimensionality reduction methods. Finally, it presents some specific applications such as document clustering, classification, search and terminology extraction.
All descriptions presented are supported with practical examples that are fully reproducible. Further reading, as well as additional exercises and projects, are proposed at the end of each chapter for those readers interested in conducting further experimentation.
A group of 100 prisoners, all together in the prison dining area, are told that they will be all put in isolation cells and then will be interrogated one by one in a room containing a light with an on/off switch. The prisoners may communicate with one another by toggling the light switch (and that is the only way in which they can communicate). The light is initially switched off. There is no fixed order of interrogation, or interval between interrogations, and the same prisoner may be interrogated again at any stage. When interrogated, a prisoner can either do nothing, or toggle the light switch, or announce that all prisoners have been interrogated. If that announcement is true, the prisoners will (all) be set free, but if it is false, they will all be executed. While still in the dining room, and before the prisoners go to their isolation cells (forever), can the prisoners agree on a protocol that will set them free?
At first glance, this riddle may seem impossible to solve: how can all of the necessary information be transmitted by the prisoners using only a single light bulb? There is indeed a solution, however, and it can be found by reasoning about knowledge.
This book provides a guided tour through eleven classic logic puzzles that are engaging and challenging and often surprising in their solutions. These riddles revolve around the characters' declarations of knowledge, ignorance, and the appearance that they are contradicting themselves in some way. Each chapter focuses on one puzzle, which the authors break down in order to guide the reader toward the solution.
For general readers and students with little technical knowledge of mathematics, One Hundred Prisoners and a Light Bulb will be an accessible and fun introduction to epistemic logic. Additionally, more advanced students and their teachers will find it to be a valuable reference text for introductory course work and further study It is a valuable source of the latest techniques and algorithms for the serious practitioner It provides an integrated treatment of the field, while still presenting each major topic as a self-contained unit It provides a mathematical treatment to accompany practical discussions It contains enough abstraction to be a valuable reference for theoreticians while containing enough detail to actually allow implementation of the algorithms discussed Now in its third printing, this is the definitive cryptography reference that the novice as well as experienced developers, designers, researchers, engineers, computer scientists, and mathematicians alike will use.
This book deals with the analysis of the structure of complex networks by combining results from graph theory, physics, and pattern recognition. The book is divided into two parts. 11 chapters are dedicated to the development of theoretical tools for the structural analysis of networks, and 7 chapters are illustrating, in a critical way, applications of these tools to real-world scenarios. The first chapters provide detailed coverage of adjacency and metric and topological properties of networks, followed by chapters devoted to the analysis of individual fragments and fragment-based global invariants in complex networks. Chapters that analyse the concepts of communicability, centrality, bipartivity, expansibility and communities in networks follow. The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.
The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems (apart from some general comments at the beginning of each chapter). In the second part, a hint is given for each exercise, which contains the main idea necessary for the solution, but allows the reader to practice theechniques by completing the proof. In the third part, a full solution is provided for each problem. This book will be useful to those students who intend to start research in graph theory, combinatorics or their applications, and for those researchers who feel that combinatorial techniques mightelp them with their work in other branches of mathematics, computer science, management science, electrical engineering and so on. For background, only the elements of linear algebra, group theory, probability and calculus are needed.
This volume offers a collection of non-trivial, unconventional problems that require deep insight and imagination to solve. They cover many topics, including number theory, algebra, combinatorics, geometry and analysis. The problems start as simple exercises and become more difficult as the reader progresses through the book to become challenging enough even for the experienced problem solver. The introductory problems focus on the basic methods and tools while the advanced problems aim to develop problem solving techniques and intuition as well as promote further research in the area. Solutions are included for each problem.
Mathematics plays a key role in computer science, some researchers would consider computers as nothing but the physical embodiment of mathematical systems. And whether you are designing a digital circuit, a computer program or a new programming language, you need mathematics to be able to reason about the design -- its correctness, robustness and dependability. This book covers the foundational mathematics necessary for courses in computer science.
The common approach to presenting mathematical concepts and operators is to define them in terms of properties they satisfy, and then based on these definitions develop ways of computing the result of applying the operators and prove them correct. This book is mainly written for computer science students, so here the author takes a different approach: he starts by defining ways of calculating the results of applying the operators and then proves that they satisfy various properties. After justifying his underlying approach the author offers detailed chapters covering propositional logic, predicate calculus, sets, relations, discrete structures, structured types, numbers, and reasoning about programs.
The book contains chapter and section summaries, detailed proofs and many end-of-section exercises -- key to the learning process. The book is suitable for undergraduate and graduate students, and although the treatment focuses on areas with frequent applications in computer science, the book is also suitable for students of mathematics and engineering.
This book constitutes the refereed proceedings of the 14th Algorithms and Data Structures Symposium, WADS 2015, held in Victoria, BC, Canada, August 2015. The 54 revised full papers presented in this volume were carefully reviewed and selected from 148 submissions. The Algorithms and Data Structures Symposium - WADS (formerly Workshop on Algorithms And Data Structures), which alternates with the Scandinavian Workshop on Algorithm Theory, is intended as a forum for researchers in the area of design and analysis of algorithms and data structures. WADS includes papers presenting original research on algorithms and data structures in all areas, including bioinformatics, combinatorics, computational geometry, databases, graphics, and parallel and distributed computing.
R is a powerful and free software system for data analysis and graphics, with over 5,000 add-on packages available. This book introduces R using SAS and SPSS terms with which you are already familiar. It demonstrates which of the add-on packages are most like SAS and SPSS and compares them to R's built-in functions. It steps through over 30 programs written in all three packages, comparing and contrasting the packages' differing approaches. The programs and practice datasets are available for download.
The glossary defines over 50 R terms using SAS/SPSS jargon and again using R jargon. The table of contents and the index allow you to find equivalent R functions by looking up both SAS statements and SPSS commands. When finished, you will be able to import data, manage and transform it, create publication quality graphics, and perform basic statistical analyses.
This new edition has updated programming, an expanded index, and even more statistical methods covered in over 25 new sections.
Geometric Fundamentals of Robotics provides an elegant introduction to the geometric concepts that are important to applications in robotics. This second edition is still unique in providing a deep understanding of the subject: rather than focusing on computational results in kinematics and robotics, it includes significant state-of-the art material that reflects important advances in the field, connecting robotics back to mathematical fundamentals in group theory and geometry.
Key features:
* Begins with a brief survey of basic notions in algebraic and differential geometry, Lie groups and Lie algebras
* Examines how, in a new chapter, Clifford algebra is relevant to robot kinematics and Euclidean geometry in 3D
* Introduces mathematical concepts and methods using examples from robotics
* Solves substantial problems in the design and control of robots via new methods
* Provides solutions to well-known enumerative problems in robot kinematics using intersection theory on the group of rigid body motions
* Extends dynamics, in another new chapter, to robots with end-effector constraints, which lead to equations of motion for parallel manipulators
Geometric Fundamentals of Robotics serves a wide audience of graduate students as well as researchers in a variety of areas, notably mechanical engineering, computer science, and applied mathematics. It is also an invaluable reference text.
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From a Review of the First Edition:
"The majority of textbooks dealing with this subject cover various topics in kinematics, dynamics, control, sensing, and planning for robot manipulators. The distinguishing feature of this book is that it introduces mathematical tools, especially geometric ones, for solving problems in robotics. In particular, Lie groups and allied algebraic and geometric concepts are presented in a comprehensive manner to an audience interested in robotics. The aim of the author is to show the power and elegance of these methods as they apply to problems in robotics."
Unique in its approach, Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo provides a brief introduction to Monte Carlo methods along with a concise exposition of reliability theory ideas. From there, the text investigates a collection of principal network reliability models, such as terminal connectivity for networks with unreliable edges and/or nodes, network lifetime distribution in the process of its destruction, network stationary behavior for renewable components, importance measures of network elements, reliability gradient, and network optimal reliability synthesis.
Solutions to most principal network reliability problems—including medium-sized computer networks—are presented in the form of efficient Monte Carlo algorithms and illustrated with numerical examples and tables. Written by reliability experts with significant teaching experience, this reader-friendly text is an excellent resource for software engineering, operations research, industrial engineering, and reliability engineering students, researchers, and engineers.
Stressing intuitive explanations and providing detailed proofs of difficult statements, this self-contained resource includes a wealth of end-of-chapter exercises, numerical examples, tables, and offers a solutions manual—making it ideal for self-study and practical use.
Ernst Zermelo (1871-1953) is best-known for the statement of the axiom of choice and his axiomatization of set theory. However, he also worked in applied mathematics and mathematical physics. His dissertation, for example, promoted the calculus of variations, and he created the pivotal method in the theory of rating systems.
This biography attempts to shed light on all facets of Zermelo's life and achievements. Personal and scientific aspects are kept separate as far as coherence allows, in order to enable the reader to follow the one or the other of these threads. The description of his personality owes much to conversations with his late wife Gertrud. The presentation of his work explores motivations, aims, acceptance, and influence. Selected proofs and information gleaned from unpublished notes and letters add to the analysis.
All facts presented are documented by appropriate sources. The biography contains more than 40 photos and facsimiles, most of them provided by Gertrud Zermelo and published here for the first time.
Putnam and Beyond takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis in one and several variables, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research.
* Each chapter systematically presents a single subject within which problems are clustered in every section according to the specific topic.
* The exposition is driven by more than 1100 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors.
* Complete solutions to all problems are given at the end of the book. The source, author, and historical background are cited whenever possible.
This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for self-study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to to expand their mathematical horizons.
A quantitative study of the efficiency of computer methods requires an in-depth understanding of both mathematics and computer science. This monograph, derived from an advanced computer science course at Stanford University, builds on the fundamentals of combinatorial analysis and complex variable theory to present many of the major paradigms used in the precise analysis of algorithms, emphasizing the more difficult notions. The authors cover recurrence relations, operator methods, and asymptotic analysis in a format that is terse enough for easy reference yet detailed enough for those with little background. Approximately half the book is devoted to original problems and solutions from examinations given at Stanford.
There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the 1970s. Consideration of the patterns in question has been extremely interesting from the combinatorial point of view, and it has proved to be a useful language in a variety of seemingly unrelated problems, including the theory of Kazhdan—Lusztig polynomials, singularities of Schubert varieties, interval orders, Chebyshev polynomials, models in statistical mechanics, and various sorting algorithms, including sorting stacks and sortable permutationsQuantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems.
Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included.
This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography.
The book opens with a short introduction to Indian music, in particular classical Hindustani music, followed by a chapter on the role of statistics in computational musicology. The authors then show how to analyze musical structure using Rubato, the music software package for statistical analysis, in particular addressing modeling, melodic similarity and lengths, and entropy analysis; they then show how to analyze musical performance. Finally, they explain how the concept of seminatural composition can help a music composer to obtain the opening line of a raga-based song using Monte Carlo simulation.
The book will be of interest to musicians and musicologists, particularly those engaged with Indian music.
Now fully updated in a third edition, this is a comprehensive textbook on combinatorial optimization. It puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. The book contains complete but concise proofs, also for many deep results, some of which have not appeared in print before. Recent topics are covered as well, and numerous references are provided. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added.
Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. The first part of this book presents the relevant aspects of the theory of matrix algebra for applications in statistics. This part begins with the fundamental concepts of vectors and vector spaces, next covers the basic algebraic properties of matrices, then describes the analytic properties of vectors and matrices in the multivariate calculus, and finally discusses operations on matrices in solutions of linear systems and in eigenanalysis. This part is essentially self-contained.
The second part of the book begins with a consideration of various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes the special properties of those matrices. The second part also describes some of the many applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. The brief coverage in this part illustrates the matrix theory developed in the first part of the book. The first two parts of the book can be used as the text for a course in matrix algebra for statistics students, or as a supplementary text for various courses in linear models or multivariate statistics.
The third part of this book covers numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes accurate and efficient algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. This part is essentially self-contained, although it assumes some ability to program in Fortran or C and/or the ability to use R/S-Plus or Matlab. This part of the book can be used as the text for a course in statistical computing, or as a supplementary text for various courses that emphasize computations.
The book includes a large number of exercises with some solutions provided in an appendix.
its capabilities in the process. Lattice is a powerful and elegant high level data visualization system that is sufficient for most everyday graphics needs, yet flexible enough to be easily extended to handle demands of cutting edge research. Written by the author of the lattice system, this book describes it in considerable depth, beginning with the essentials and systematically delving into specific low levels details as necessary. No prior experience with lattice is required to read the book, although basic familiarity with R is assumed.
The book contains close to150 figures produced with lattice. Many of the examples emphasize principles of good graphical design; almost all use real data sets that are publicly available in various R packages. All code and figures in the book are also available online, along with supplementary material covering more advanced topics.
Graph theory continues to be one of the fastest growing areas of modern mathematics because of its wide applicability in such diverse disciplines as computer science, engineering, chemistry, management science, social science, and resource planning. Graphs arise as mathematical models in these fields, and the theory of graphs provides a spectrum of methods of proof. This concisely written textbook is intended for an introductory course in graph theory for undergraduate mathematics majors or advanced undergraduate and graduate students from the many fields that benefit from graph-theoretic applications.
* Comprehensive index and bibliography, with suggested literature for more advanced material
New to the second edition:
* New chapters on labeling and communications networks and small-worlds
* Expanded beginner's material in the early chapters, including more examples, exercises, hints and solutions to key problems
* Many additional changes, improvements, and corrections throughout resulting from classroom use and feedback
Striking a balance between a theoretical and practical approach with a distinctly applied flavor, this gentle introduction to graph theory consists of carefully chosen topics to develop graph-theoretic reasoning for a mixed audience. Familiarity with the basic concepts of set theory, along with some background in matrices and algebra, and a little mathematical maturity are the only prerequisites.
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From a review of the first edition:
"Altogether the book gives a comprehensive introduction to graphs, their theory and their application...The use of the text is optimized when the exercises are solved. The obtained skills improve understanding of graph theory as well... It is very useful that the solutions of these exercises are collected in an appendix."
This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.
Abu?erover?owoccurswheninputiswrittenintoamemorybu?erthatisnot large enough to hold the input. Bu?er over?ows may allow a malicious person to gain control over a computer system in that a crafted input can trick the defectiveprogramintoexecutingcodethatisencodedintheinputitself.They are recognised as one of the most widespread forms of security vulnerability, and many workarounds, including new processor features, have been proposed to contain the threat. This book describes a static analysis that aims to prove the absence of bu?er over?ows in C programs. The analysis is conservative in the sense that it locates every possible over?ow. Furthermore, it is fully automatic in that it requires no user annotations in the input program. Thekeyideaoftheanalysisistoinferasymbolicstateforeachp- gram point that describes the possible variable valuations that can arise at that point. The program is correct if the inferred values for array indices and pointer o?sets lie within the bounds of the accessed bu?er. The symbolic state consists of a ?nite set of linear inequalities whose feasible points induce a convex polyhedron that represents an approximation to possible variable valuations. The book formally describes how program operations are mapped to operations on polyhedra and details how to limit the analysis to those p- tionsofstructuresandarraysthatarerelevantforveri?cation.Withrespectto operations on string bu?ers, we demonstrate how to analyse C strings whose length is determined by anul character within the string.
Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version | 677.169 | 1 |
- Build the critical thinking and problem solving skills you need at school, at work, and at home - Maximize your time using the 20 easy steps for effective critical thinking and problem solving - Learn everything from recognizing a problem to fol
Express Review Guides: Algebra I helps students acquire practical skills with fast, targeted lessons. All key topics are covered. Readers will benefit from math tips, strategies for avoiding common pitfalls, sidebars of math definitions, and a detailed glossary. In addition, pre- and posttests help students gauge both their weak areas as well as their progress.
Become an effective critical thinker in just 20 minutes a day! Whether at work, at school, or at home, critical thinking skills are essential for success. Learning to think critically will improve your decision-making and problem-solving skills, giving you the tools you need to tackle the tough decisions and choices you face. This book teaches you the skills to recognize and define problems and sort out unnecessary information to make smart decisions.
These fully revised new editions of LearningExpress's best-selling Skill Builders series offer a unique review of basic academic skills in a fast, easy-to-learn format. Each LearningExpress book focuses on practical applications and provides a built-in incentive-oriented study plan in the "20 Minutes a Day" concept. Students will find these self-study programs a valuable tool for improving the critical thinking and reasoning skills that lead to success at work and in the classroom. For Study Skills, College Survival Skills, and developmental courses.
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The new, third edition of this popular title helps students and adult learners sharpen essential vocabulary and spelling skills. Vocabulary and Spelling Success, 3rd edition is a vital tool for any student who needs to pass the GED, any proficiency exam, school/placement exam, job readiness test, civil service exam, the ASVAB, and law enforcement exams.
This newly updated and revised Grammar Success in 20 Minutes a Day helps students write and speak without mistakes, and master English and grammar on standardized tests--and all it takes is 20 minutes a day!The Critical Thinking Toolkit is a comprehensive compendium that equips readers with the essential knowledge and methods for clear, analytical, logical thinking and critique in a range of scholarly contexts and everyday situations. Takes an expansive approach to critical thinking by exploring concepts from other disciplines, including evidence and justification from philosophy, cognitive biases and errors from psychology, race and gender from sociology and political science, and tropes and symbols from rhetoric Follows the proven format of The Philosopher's Toolkit and The Ethics Toolkit with concise, easily digestible entries, "see also" recommendations that connect topics, and recommended reading lists Allows readers to apply new critical thinking and reasoning skills with exercises and real life examples at the end of each chapter Written in an accessible way, it leads readers through terrain too often cluttered with jargon Ideal for beginning to advanced students, as well as general readers, looking for a sophisticated yet accessible introduction to critical thinking
The Psychology of Prejudice and Discrimination provides a comprehensive and compelling overview of what psychological theory and research have to say about the nature, causes, and reduction of prejudice and discrimination. It balances a detailed discussion of theories and selected research with applied examples that ensure the material is relevant to students. Newly revised and updated, this edition addresses several interlocking themes, such as research methods, the development of prejudice in children, the relationship between prejudice and discrimination, and discrimination in the workplace, which are developed in greater detail than in other textbooks. The first theme introduced is the nature of prejudice and discrimination, which is followed by a discussion of research methods. Next comes the psychological underpinnings of prejudice: the nature of stereotypes, the conditions under which stereotypes influence responses to other people, contemporary theories of prejudice, and how values and belief systems are related to prejudice. Explored next are the development of prejudice in children and the social context of prejudice. The theme of discrimination is developed via discussions of the nature of discrimination, the experience of discrimination, and specific forms of discrimination, including gender, sexual orientation, age, ability, and appearance. The concluding theme is the reduction of prejudice. An ideal core text for junior and senior college students who have had a course in introductory psychology, it is written in a style that is accessible to students in other fields including education, social work, business, communication studies, ethnic studies, and other disciplines. In addition to courses on prejudice and discrimination, this book is also adapted for courses that cover topics in racism and diversity.Advocates a new model for delivering healthcare services based on the author's efforts to get care for his wife at a medical center, discussing five strategies for addressing key problems and building necessary skills to produce change.
Through specific examples, real-life scenarios, and diagrams, this book vividly conveys the most fundamental and effective tactics for boosting reading proficiency while enhancing student and teacher performance.
Presents lessons in critical reading skills, including distinguishing between fact and opinion, defining words in context, perspective, tone, and drawing conclusions to prepare the student for standardized tests | 677.169 | 1 |
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ECE 3250 Mathematics of Signal and System Analysis
Course description
Course aims to deepen students' working knowledge of mathematical tools relevant to ECE applications. While the course emphasizes fundamentals, it also provides an ECE context for the topics it covers, which include foundational material about sets and functions; modular arithmetic and public-key cryptography; inner products, orthogonal representations, and Fourier analysis; LTI systems as mappings on function spaces; sampling and interpolation; singular-value decomposition; and, as time permits, an introduction to wavelets and elementary convex analysis.
Outcome 1: Deepen their understanding of fundamental concepts from real analysis and linear algebra to which they have been exposed in their calculus and differential equation courses by putting them to work in an engineering context.
Outcome 2: Achieve a sophisticated understanding of fundamental signals and systems concepts, a few of which they have been exposed to on an elementary level in ECE 2200, by learning the mathematics behind them.
Outcome 3: Attain an appreciation of the central role that advanced mathematics plays in modeling, analysis, and design of engineering systems.
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The Cornell University School of Continuing Education and Summer Sessions strives to offer valuable educational opportunities in many formats for any person, in any study, at any time, and in any place. | 677.169 | 1 |
Mathematics
Pupils will follow a course based upon the New National Curriculum designed for students working towards the achievement of the new GCSE grade 4 by the end of Year 7. Achievement of the new GCSE Grade 5 will be aimed for at the end of Year 8.
The structure of the course is such that each assessment objective: A01: "Use and apply standard techniques", A02: "Reason, interpret and communicate mathematically" and AO3: "Solve problems within mathematics and in other contexts" is encountered throughout the course. | 677.169 | 1 |
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B> This comprehensive book covers all aspects of MATLAB presented within an easy-to-follow "learn while doing" tutorial format. This book provides a complete reference to all MATLAB functions and graphics properties in one volume and provides numerous examples demonstrating the usefulness of MATLAB in solving real-world problems. It includes over 100 MATLAB M-files that illustrate the use of MATLAB in performing practical tasks and provides tools for Fourier series; interpolation; curve fitting; optimization; data analysis; graphical data analysis; array manipulation; plotting data; GUI construction; and GUI control of axes, line and surface properties. An essential reference for any scientific or technical professional who needs an in-depth understanding of the uses of MATLAB | 677.169 | 1 |
3.4 NOTES APPLICATIONS OF QUADRATICS
1) Find the max value of y = -2x2 + 8x 5
2) Suppose x represents one of two positive numbers whose sum is 45. For
what two such numbers is the product equal to 504?
3) A golf ball is hit so that its height h in feet af
Section 2.1
Graph of basic functions
Continuous:_
Increasing, decreasing, constant: _
Determine the largest interval of the domain over which the function is
continuous.
Determine intervals of increasing, decreasing, constant, and domain/range.
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She sets up notes for you, that you can print out or not. Its your choice, ofc there is homework and the homework is basically your practice work, she has like 12 quizzes for the whole semester, she has a fast pace but, she will help you, she doesn't say no indirectly, unless she was going to teach your question, during class. she is a great teacher, and I love how she teaches.
Course highlights:
I learned Imaginary and complex problems, matrixes, and the parents.
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6-8 hours
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Always do your homework! Do not think you will be able to pass her class without any practice. | 677.169 | 1 |
Graphing Calculator Explortation {Introduction Lesson}
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Introduction to the Graphing Calculator
This is a great activity for Algebra I students who are using a graphing calculator for the first time or for 8th graders in preparation for the upcoming year. It is a great activity for 8th graders after state assessments because they have never used them before and find it very fascinating!
This activity is included in a Word Document if you would like to edit or change anything as well as a PDF for quick and easy printing.
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Researchers in fields ranging from biology and medicine to the social sciences, law, and economics regularly encounter variables that are discrete or categorical in nature. While there is no dearth of books on the analysis and interpretation of such data, these generally focus on large sample methods. When sample sizes are not large or the data are... more...
This book is a collection of research papers and surveys on algebra that were presented at the Conference on Groups, Rings, and Group Rings held in Ubatuba, Brazil. This text familiarizes researchers with the latest topics, techniques, and methodologies in several branches of contemporary algebra. With extensive coverage, it examines broad themes from... more...
This volume provides a selection of previously published papers and manuscripts of Uno Kaljulaid, an eminent Estonian algebraist of the last century. The central part of the book is the English translation of Kaljulaid's 1979 Candidate thesis, which originally was typewritten in Russian and manufactured in not so many copies. The thesis is devoted... more...
The monograph is written with a view to provide basic tools for researchers working in Mathematical Analysis and Applications, concentrating on differential, integral and finite difference equations. It contains many inequalities which have only recently appeared in the literature and which can be used as powerful tools and will be a valuable source... more...
Numbers deals with the development of numbers from fractions to algebraic numbers to transcendental numbers to complex numbers and their uses. The book also examines in detail the number pi, the evolution of the idea of infinity, and the representation of numbers in computers. The metric and American systems of measurement as well as the applications... more...
Shows how well-meant teaching strategies and approaches can in practice exacerbate underachievement in maths by making inappropriate demands on learners. As well as criticizing some of the teaching and grouping practices that are considered normal in many schools, this book also offers an alternative view of attainment and capability. more...
The book is the first to give a comprehensive overview of the techniques and tools currently being used in the study of combinatorial problems in Coxeter groups. It is self-contained, and accessible even to advanced undergraduate students of mathematics. The primary purpose of the book is to highlight approximations to the difficult isomorphism problem... more...
During the first decades of the last century Italian mathematics was considered to be the third national school due to its importance and the high level of its numerous - searchers. The decision to organize the 1908 International Congress of Mathematicians in Rome (after those in Paris and Heidelberg) confirmed this position. Qualified Italian universities... more...
The algebra of primary cohomology operations computed by the well-known Steenrod algebra is one of the most powerful tools of algebraic topology. This book computes the algebra of secondary cohomology operations which enriches the structure of the Steenrod algebra in a new and unexpected way. The book solves a long-standing problem on the algebra... more...
Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered... more... | 677.169 | 1 |
CLE 3102.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including testing cases, estimation, and then checking induced errors and the reasonableness of the solution. CU 3102.1.20 Estimate solutions to evaluate the reasonableness of results and to check technological computation.
CU 3102.2.4 Operate efficiently with both rational and irrational numbers.
Lesson 0-5: Multiplying and Dividing Rational Numbers
CU 3102.2.4 Operate efficiently with both rational and irrational numbers.
Lesson 0-8: Area
SPI 3102.4.1 Develop and apply strategies to estimate the area of any shape on a plane grid.
Lesson 0-11: Simple Probability and Odds
CU 3102.5.12 Use techniques (Venn Diagrams, tree diagrams, or counting procedures) to identify the possible outcomes of an experiment or sample space and compute the probability of an event. Lesson 0-12: Mean, Median, Mode, Range, and Quartiles
CU 3102.5.3 When a set of data is changed, identify effects on measures of central tendency, range, and inter-quartile range. CU 3102.5.4 Explore quartiles, deciles, and percentiles of a distribution.
SPI 3102.5.2 Identify the effect on mean, median, mode, and range when values in the data set are changed.
Lesson 0-13: Representing Data
CLE 3102.5.1 Describe and interpret quantitative information. CU 3102.5.2 Develop a meaning for and identify outliers in a data set and verify.
CU 3102.1.10 Use algebraic properties to develop a valid mathematical argument. CU 3102.3.3 Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions. CU 3102.4.1 Using algebraic expressions solve for measures in geometric figures as well as for perimeter, area, and volume.
CU 3102.1.10 Use algebraic properties to develop a valid mathematical argument. CU 3102.1.11 Use manipulatives to model algebraic concepts. CU 3102.1.20 Estimate solutions to evaluate the reasonableness of results and to check technological computation. CU 3102.2.1 Recognize and use like terms to simplify expressions. CU 3102.3.3 Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions.
CLE 3102.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including testing cases, estimation, and then checking induced errors and the reasonableness of the solution.
CU 3102.2.2 Apply the order of operations to simplify and evaluate algebraic expressions. CU 3102.3.3 Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions. CU 3102.3.12 Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions.
Lesson 1-6: Relations
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. CU 3102.1.12 Create and work flexibly among representations of relations (including verbal, equations, tables, mappings, graphs). CU 3102.1.13 Change from one representation of a relation to another representation, for example, change from a verbal description to a graph. CU 3102.3.15 Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents.
SPI 3102.3.6 Interpret various relations in multiple representations.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
Lesson 1-7: Functions
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems.
CU 3102.1.3 Understand and use mathematical symbols, notation, and common mathematical abbreviations correctly. CU 3102.1.12 Create and work flexibly among representations of relations (including verbal, equations, tables, mappings, graphs). CU 3102.3.16 Determine if a relation is a function from its graph or from a set of ordered pairs.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
Lesson 1-8: Logical Reasoning and Counterexamples
CLE 3102.1.3 Develop inductive and deductive reasoning to independently make and evaluate mathematical arguments and construct appropriate proofs; include various types of reasoning, logic, and intuition. CU 3102.1.10 Use algebraic properties to develop a valid mathematical argument.
Chapter 2 Linear Equations Lesson 2-1: Writing Equations1.6 Employ reading and writing to recognize the major themes of mathematical processes, the historical development of mathematics, and the connections between mathematics and the real world.
CLE 3102.3.4 Solve problems involving linear equations and linear inequalities. CU 3102.3.3 Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions. CU 3102.3.11 Solve multi-step linear equations with one variable. CU 3102.3.12 Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions.
Lesson 2-5: Solving Equations Involving Absolute Value CU 3102.3.12 Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions. CU 3102.3.14 Solve absolute value equations and inequalities (including compound inequalities) with one variable and graph their solutions on a number line.
Lesson 2-8: Literal Equations and Dimensional Analysis
CLE 3102.3.5 Manipulate formulas and solve literal equations.
CLE 3102.4.2 Apply appropriate units of measure and convert measures in problem solving situations. CU 3102.4.1 Using algebraic expressions solve for measures in geometric figures as well as for perimeter, area, and volume. CU 3102.4.5 Use dimensional analysis to convert rates and measurements both within a system and between systems and check the appropriateness of the solution.
SPI 3102.4.4 Convert rates and measurements.
Lesson 2-9: Weighted Averages
CLE 3102.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies. CU 3102.1.5 Use formulas, equations, and inequalities to solve real-world problems including time/rate/distance, percent increase/decrease, ratio/proportion, and mixture problems. CU 3102.1.6 Use a variety of strategies to estimate and compute solutions, including real-world problems.
SPI 3102.4.4 Convert rates and measurements.
Chapter 3 Linear Functions
Lesson Get Started on Chapter 3: Getting Started on Chapter 3 CU 3102.1.1 Develop meaning for mathematical vocabulary.
Lesson 3-1: Graphing Linear Equations SPI 3102.1.4 Translate between representations of functions that depict real-world situations. SPI 3102.3.8 Determine the equation of a line and/or graph a linear equation.
Lesson 3-2: Solving Linear Equations by Graphing
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. CU 3102.1.6 Use a variety of strategies to estimate and compute solutions, including real-world problems. CU 3102.1.20 Estimate solutions to evaluate the reasonableness of results and to check technological computation23 Determine the graph of a linear equation including those that depict contextual situations.
CU 3102.1.16 Understand and express the meaning of the slope and y-intercept of linear functions in real-world contexts. CU 3102.3.20 Understand that a linear equation has a constant rate of change called slope and represent slope in various forms.
Lesson 3-3: Rate of Change and Slope20 Understand that a linear equation has a constant rate of change called slope and represent slope in various forms. CU 3102.5.9 Determine an equation for a line that fits real-world linear data; interpret the meaning of the slope and y-intercept in context of the data.
SPI 3102.1.6 Determine and interpret slope in multiple contexts including rate of change in real-world problems.
Lesson 3-5: Arithmetic Sequences as Linear Functions
CLE 3102.1.3 Develop inductive and deductive reasoning to independently make and evaluate mathematical arguments and construct appropriate proofs; include various types of reasoning, logic, and intuition.
CLE 3102.3.1 Use algebraic thinking to analyze and generalize patterns. CU 3102.1.4 Write a rule with variables that expresses a pattern CU 3102.3.15 Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents.
SPI 3102.1.1 Interpret patterns found in sequences, tables, and other forms of quantitative information using variables or function notation.
SPI 3102.1.4 Translate between representations of functions that depict real-world situations. SPI 3102.3.1 Express a generalization of a pattern in various representations including algebraic and function notation.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
CLE 3102.1.3 Develop inductive and deductive reasoning to independently make and evaluate mathematical arguments and construct appropriate proofs; include various types of reasoning, logic, and intuition.
Lesson 3-6: Proportional and Non-proportional Relationships CU 3102.1.5 Use formulas, equations, and inequalities to solve real-world problems including time/rate/distance, percent increase/decrease, ratio/proportion, and mixture problems. CU 3102.3.23 Determine the graph of a linear equation including those that depict contextual situations. CU 3102.3.25 Find function values using f(x) notation or graphs.
SPI 3102.3.1 Express a generalization of a pattern in various representations including algebraic and function notation.
CU 3102.3.24 Interpret the changes in the slope-intercept form and graph of a linear equation by looking at different parameters, m and b in the slope-intercept form.
Lesson 4-1: Graphing Equations in Slope-Intercept Form21 Determine the equation of a line using given information including a point and slope, two points, a point and a line parallel or perpendicular, graph, intercepts. .CLE 3102.1.7 Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate problem solving, and to produce accurate and reliable models. CU 3102.1.14 Apply graphical transformations that occur when changes are made to coefficients and constants in functions. CU 3102.3.17 Recognize families of functions24 Interpret the changes in the slope-intercept form and graph of a linear equation by looking at different parameters, m and b in the slope-intercept form.
Lesson 4-2: Writing Equations in Slope-Intercept FormCULesson 4-3: Writing Equations in Point-Slope Form CU
Lesson 4-4: Parallel and Perpendicular Lines CU 3102.3.21 Determine the equation of a line using given information including a point and slope, two points, a point and a line parallel or perpendicular, graph, intercepts.
CU 3102.5.9 Determine an equation for a line that fits real-world linear data; interpret the meaning of the slope and y-intercept in context of the data.
CU 3102.5.10 Using technology with a set of contextual linear data to examine the line of best fit; determine and interpret the correlation coefficient. CU 3102.5.11 Use an equation that fits data to make a prediction.
SPI 3102.5.4 Generate the equation of a line that fits linear data and use it to make a prediction.
Lesson 5-2: Solving Inequalities by Multiplication and Division
CLE 3102.3.4 Solve problems involving linear equations and linear inequalities. CU 3102.1.17 Connect the study of algebra to the historical development of algebra. CU 3102.3.3 Justify correct results of algebraic procedures using extension of properties of real numbers to algebraic expressions. CU 3102.3.13 Solve multi-step linear inequalities with one variable and graph the solution on a number line.
Chapter 6 Systems of Linear Equations and Inequalities
Lesson 6-1: Graphing Systems of Equations
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods. CU 3102.3.27 Determine the number of solutions for a system of linear equations (0, 1, or infinitely many solutions). CU 3102.3.28 Solve systems of linear equations graphically, algebraically, and with technology.
SPI 3102.3.9 Solve systems of linear equations/inequalities in two variables.
SPI 3102.3.9 Solve systems of linear equations/inequalities in two variables.
Lesson 6-2: Substitution
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods. CU 3102.1.8 Recognize and perform multiple steps in problem solving when necessary3: Elimination Using Addition and Subtraction
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods. CU 3102.1.17 Connect the study of algebra to the historical development of algebra4: Elimination Using Multiplication
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods. Lesson 6-5: Applying Systems of Linear Equations
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods.
SPI 3102.3.9 Solve systems of linear equations/inequalities in two variables.
Lesson 6-8: Systems of Inequalities
CLE 3102.3.7 Construct and solve systems of linear equations and inequalities in two variables by various methods. CU 3102.3.29 Solve contextual problems involving systems of linear equations or inequalities and interpret solutions in context.
Lesson 7-4 Polynomials
CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. CU 3102.4.1 Using algebraic expressions solve for measures in geometric figures as well as for perimeter, area, and volume.
SPI 3102.3.2 Operate with polynomials and simplify results.
Lesson 7-7 Multiplying Polynomials
CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials.2 Operate with polynomials and simplify results. Lesson 7-8 Special Products3 Factor polynomials.
SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function.
Chapter 9 Quadratic and Exponential Functions
Lesson 9-1 Graphing Quadratic Functions
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. CU 3102.1.8 Recognize and perform multiple steps in problem solving when necessary. CU 3102.1.19 Recognize and practice appropriate use of technology in representations and in problem solving. CU 3102.3.25 Find function values using f(x) notation or graphs.
SPI 3102.3.11 Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situation. Lesson Extend 9-3 Extend: Graphing Technology Lab Systems of Linear and Quadratic Equations CU 3102.3.12 Recognize and articulate when an equation has no solution, a single solution, or all real numbers as solutions. Lesson 9-4 Solving Quadratic Equations by Completing the Square CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. CU 3102.1.11 Use manipulatives to model algebraic concepts
SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function.
Lesson 9-5 Solving Quadratic Equations by Using the Quadratic Formula.
CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots CU 3102.3.31 Determine the number of real solutions for a quadratic equation including using the discriminant and its graph.
SPI 3102.3.1 Express a generalization of a pattern in various representations including algebraic and function notation.
SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function.
Lesson 9-6 Exponential Functions
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems.
CLE 3102.3.9 Understand and use exponential functions to solve contextual problems. CU 3102.3.15 Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents. CU 3102.3.25 Find function values using f(x) notation or graphs. CU 3102.3.33 Recognize data that can be modeled by an exponential function.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.3.9 Understand and use exponential functions to solve contextual problems. CU 3102.3.33 Recognize data that can be modeled by an exponential function.
Lesson 9-8 Geometric Sequences as Exponential Functions
CLE 3102.3.1 Use algebraic thinking to analyze and generalize patterns Lesson 9-9 Analyzing Functions with Successive Differences and Ratios
CU 3102.5.11 Use an equation that fits data to make a prediction.
Chapter 10 Radical Functions and Geometry
Lesson 10-1 Square Root Functions
CLE 3102.3.6 Understand and use relations and functions in various representations to solve contextual problems. CU 3102.3.19 Explore the characteristics of graphs of various nonlinear relations and functions including inverse variation, quadratic, and square root function. Use technology where appropriate.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
Lesson Extend 10-1 Extend: Graphing Technology Lab Graphing Square Root Functions
Lesson 10-4 Radical Equations CLE 3102.1.4 Move flexibly between multiple representations (contextual, physical, written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution strategies. CU 3102.3.15 Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents.
SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value. Lesson 10-5 The Pythagorean Theorem CU 3102.4.2 Use the Pythagorean Theorem to find the missing measure in a right triangle including those from contextual situations. CU 3102.4.3 Understand horizontal/vertical distance in a coordinate system as absolute value of the difference between coordinates; develop the distance formula for a coordinate plane using the Pythagorean Theorem. 3102.4.4 Develop the midpoint formula for segments on a number line or in the coordinate plane.
SPI 3102.4.2 Solve contextual problems using the Pythagorean Theorem.
Lesson 10-6 The Distance and Midpoint Formulas
SPI 3102.4.3 Solve problems involving the distance between points or midpoint of a segment.
Lesson 10-7 Similar Triangles CU 3102.4.1 Using algebraic expressions solve for measures in geometric figures as well as for perimeter, area, and volume.
CLE 3102.1.6 Employ reading and writing to recognize the major themes of mathematical processes, the historical development of mathematics, and the connections between mathematics and the real world.
Lesson 11-2 Rational Functions
CLE 3102.3.3 Understand and apply operations with rational expressions and equations. CU 3102.1.14 Apply graphical transformations that occur when changes are made to coefficients and constants in functions. CU 3102.3.15 Determine domain and range of a relation and articulate restrictions imposed either by the operations or by the real life situation that the function represents. CU 3102.3.17 Recognize families of functions. CU 3102.3.25 Find function values using f(x) notation or graphs.
SPI 3102.3.4 Operate with, evaluate, and simplify rational expressions including determining restrictions on the domain of the variables. SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
Lesson 11-3 Simplifying Rational Expressions CLE 3102.3.3 Understand and apply operations with rational expressions and equations. CU 3102.3.8 Find the GCF of the terms in a polynomial. SPI 3102.3.4 Operate with, evaluate, and simplify rational expressions including determining restrictions on the domain of the variables.
SPI 3102.3.4 Operate with, evaluate, and simplify rational expressions including determining restrictions on the domain of the variables.
Lesson 11-8 Rational Equations CU 3102.1.1 Develop meaning for mathematical vocabulary. CU 3102.1.2 Use the terminology of mathematics correctly. CU 3102.1.8 Recognize and perform multiple steps in problem solving when necessary. CU 3102.1.19 Recognize and practice appropriate use of technology in representations and in problem solving SPI 3102.3.7 Determine domain and range of a relation, determine whether a relation is a function and/or evaluate a function at a specified rational value.
SPI 3102.5.2 Identify the effect on mean, median, mode, and range when values in the data set are changed.
Lesson 12-4 Permutations and Combinations
CLE 3102.5.3 Understand basic counting procedures and concepts of probability. CU 3102.1.3 Understand and use mathematical symbols, notation, and common mathematical abbreviations correctly. CU 3102.3.2 Explore patterns including Pascal's Triangle and the Fibonacci sequence. CU 3102.5.12 Use techniques (Venn Diagrams, tree diagrams, or counting procedures) to identify the possible outcomes of an experiment or sample space and compute the probability of an event.
Lesson 12-5 Probability of Compound Events
CLE 3102.5.3 Understand basic counting procedures and concepts of probability. CU 3102.5.12 Use techniques (Venn Diagrams, tree diagrams, or counting procedures) to identify the possible outcomes of an experiment or sample space and compute the probability of an event. CU 3102.5.13 Determine the complement of an event and the probability of that complement. CU 3102.5.14 Determine if two events are independent or dependent. CU 3102.5.15 Explore joint and conditional probability. CU 3102.5.18 Make informed decisions about practical situations using probability concepts.
Lesson Concepts and Skills 5: Venn Diagrams CU 3102.5.12 Use techniques (Venn Diagrams, tree diagrams, or counting procedures) to identify the possible outcomes of an experiment or sample space and compute the probability of an event.
CLE 3102.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including testing cases, estimation, and then checking induced errors and the reasonableness of the solution. CU 3102.1.4 Write a rule with variables that expresses a pattern. CU 3102.1.6 Use a variety of strategies to estimate and compute solutions, including real-world problems. CU 3102.1.7 Identify missing or irrelevant information in problems. CU 3102.1.20 Estimate solutions to evaluate the reasonableness of results and to check technological computation. CU 3102.3.2 Explore patterns including Pascal's Triangle and the Fibonacci sequence. Lesson Study Guide and Review | 677.169 | 1 |
Mathematics
Cambridge Scholars publishes a niche but important collection of mathematics books with a particular strength in the philosophy and history of mathematics. Titles will be of key interest to those working in the field of educational studies in mathematics, as well as specialised researchers and the general reader.
Udny Yule's seminal influence on time series analysis has long been recognized but much less recognized is that Yule was not only a wonderful expositor but that he had also published equally important research in an extraordinarily wide range of fields, from developing the theory of correlation and regression to providing mathemati...
This book represents a crucial resource for students taking a required statistics course who are intimidated by statistical symbols, formulae, and daunting equations. It will serve to prepare the reader to achieve the level of statistical literacy required not only to understand basic statistics, but also to embark on their advance...
How do we get new knowledge? Following the maverick tradition in the philosophy of science, Carlo Cellucci gradually came to the conclusion that logic can only fulfill its role in mathematics, science and philosophy if it helps us to answer this question. He argues that mathematical logic is inadequate and that we need a new logic,...
From Foundations to Philosophy of Mathematics provides an historical introduction to the most exciting period in the foundations of mathematics, starting with the discovery of the paradoxes of logic and set theory at the beginning of the twentieth century and continuing with the great foundational debate that took place in the 1920...
In this book, a wide range of problems concerning recent achievements in the field of industrial and applied mathematics are presented. It provides new ideas and research for scientists developing and studying mathematical methods and algorithms, and researchers applying them for solving real-life problems. The importance of the co...
Why should some essential properties of geometry (i.e., infinity, symmetry, and dimensionality) be both necessary and desirable in the way that they have been constructed—albeit with different modifications over time—since time immemorial?Contrary to the conventional wisdom in all history hitherto existing, the essential properties...
Why should mathematical logic be grounded on the basis of some formal requirements in the way that it has been developed since its classical emergence as a hybrid field of mathematics and logic in the 19th century or earlier? Contrary to conventional wisdom, the foundation of mathematic logic has been grounded on some false (or dog... | 677.169 | 1 |
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I made this as a review and warm-up for Middle School students in PreAlgebra and Algebra that struggle with inequalities, as it reviews the basics of what the signs mean and vocabulary.
Many developing students that I work with struggle with inequalities! Some of those students struggle because they were never taught how to read inequalities. They read 'x < y' as 'x eats the bigger one y?!?!?' (If you work with younger students as well, you might be interested in my worksheet "Less Than, Greater Than, and Equals Signs.")
In order to complete this sheet, students already need to be familiar with the concept of variables. This sheet covers the foundations of inequalities, which students then learn how to solve, graph and more around Algebra I. | 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 |
Matlab An Introduction with Applications
ISBN-10: 0470108770
ISBN-13: 9780470108772
Edition: 3rd MATLAB(r)! If you want a clear, easy-to-use introduction to MATLAB(r), this book is for you! The Third Edition of Amos Gilat's popular MATLAB(r), An Introduction with Applications requires no previous knowledge of MATLAB and computer programming as it helps you understand and apply this incredibly useful and powerful mathematical tool. Thoroughly updated to match MATLAB(r)'s newest release, MATLAB(r) 7.3 (R2007b), the text takes you step by step through MATLAB(r)'s basic features--from simple arithmetic operations with scalars, to creating and using arrays, to three-dimensional plots and solving differential equations. You'll appreciate the many features that make it easy to grasp the material and become proficient in using MATLAB(r), including: * Sample and homework problems that help you hands-on practice solving the kinds of problems you'll encounter in future science and engineering courses * New coverage of the Workspace Window and the save and load commands, Anonymous Functions, Function Functions, Function Handles, Subfunctions and Nested Functions * In-depth discussions of script files, 2-D and 3-D plotting, function files, programming (flow control), polynomials, curve fitting, interpolation, and applications in numerical analysis By showing you not just how MATLAB(r) works but how to use it with real-world applications in mathematics, science, and engineering, MATLAB(r), An Introduction with Applications, Third Edition will turn you into a MATLAB(r) master faster than you imagined | 677.169 | 1 |
Sobre este título:
This complete guide to numerical methods in chemical engineering is the first to take full advantage of MATLAB's powerful calculation environment. Every chapter contains several examples using general MATLAB functions that implement the method and can also be applied to many other problems in the same category.
The authors begin by introducing the solution of nonlinear equations using several standard approaches, including methods of successive substitution and linear interpolation; the Wegstein method, the Newton-Raphson method; the Eigenvalue method; and synthetic division algorithms. With these fundamentals in hand, they move on to simultaneous linear algebraic equations, covering matrix and vector operations; Cramer's rule; Gauss methods; the Jacobi method; and the characteristic-value problem. Additional coverage includes:
The numerical methods covered here represent virtually all of those commonly used by practicing chemical engineers. The focus on MATLAB enables readers to accomplish more, with less complexity, than was possible with traditional FORTRAN. For those unfamiliar with MATLAB, a brief introduction is provided as an Appendix.
Over 60+ MATLAB examples, methods, and function scripts are covered, and all of them are included on the book's CD
From the Inside Flap:
Preface This book emphasizes the derivation of a variety of numerical methods and their application to the solution of engineering problems, with special attention to problems in the chemical engineering field. These algorithms encompass linear and nonlinear algebraic equations, eigenvalue problems, finite difference methods, interpolation, differentiation and integration, ordinary differential equations, boundary-value problems, partial differential equations, and linear and nonlinear regression analysis.
MATLAB 2 is adopted as the calculation environment throughout the book. MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment. MATLAB is distinguished by its ability to perform all the calculations in matrix form, its large library of built-in functions, its strong structural language, and its rich graphical visualization tools. In addition, MATLAB is available on all three operating platforms: WINDOWS, Macintosh, and UNIX. The reader is expected to have a basic knowledge of using MATLAB. However, for those who are not familiar with MATLAB, it is recommended that they cover the subjects discussed in Appendix A: Introduction to MATLAB prior to studying the numerical methods. Several worked examples are given in each chapter to demonstrate the numerical techniques. Most of these examples require computer programs for their solution. These programs were written in the MATLAB language and are compatible with MATLAB 5.0 or higher. In all the examples, we tried to present a general MATLAB function that implements the method and that may be applied to the solution of other problems that fall in the same category of application as the worked example. The general algorithm for these programs is illustrated in the section entitled, "General Algorithm for the Software Developed in this Book." All the programs that appear in the text are included on the CD-ROM that accompanies this book. There are three versions of these programs on the CD-ROM, one for each of the major operating systems in which MATLAB exists: WINDOWS, Macintosh, and UNIX. Installation procedures, a complete list, and brief descriptions of all the programs are given in the section entitled "Programs on the CD-ROM" that immediately follows this Preface. In addition, the programs are described in detail in the text in order to provide the reader with a thorough background and understanding of how MATLAB is used to implement the numerical methods. It is important to mention that the main purpose of this book is to teach the student numerical methods and problem solving, rather than to be a MATLAB manual. In order to assure that the student develops a thorough understanding of the numerical methods and their implementation, new MATLAB functions have been written to demonstrate each of the numerical methods covered in this text. Admittedly, MATLAB already has its own built-in functions for some of the methods introduced in this book. We mention and discuss the built-in functions, whenever they exist. The material in this book has been used in undergraduate and graduate courses in the Department of Chemical and Biochemical Engineering at Rutgers University.
Basic and advanced numerical methods are covered in each chapter. Whenever feasible, the more advanced techniques are covered in the last few section s of each chapter. A one-semester graduate level course in applied numerical methods would cover all the material in this book. An undergraduate course (junior or senior level) would cover the more basic methods in each chapter. To facilitate the professor teaching the course, we have marked with an asterisk (*) in the Table of Contents those sections that may be omitted in an undergraduate course. Of course, this choice is left to the discretion of the professor.
Prentice Hall and the authors would like to thank the reviewers of this book for their constructive comments and suggestions. NM is grateful to Professor Jamal Chaouki of Ecole Polytechnique de Montréal for his support and understanding.
Alkis Constantinides Navid Mostoufi
Programs on the CD-ROM
Brief Description The programs contained on the CD-ROM that accompanies this book have been written in the MATLAB 5.0 language and will execute in the MATLAB command environment in all three operating systems (WINDOWS, Macintosh, and UNIX). There are 21 examples, 29 methods, and 13 other function scripts on this CD-ROM. A list of the programs is given later in this section. Complete discussions of all programs are given in the corresponding chapters of the text. MATLAB is a high-performance language for technical computing. It integrates computation,visualization, and programming in an easy-to-use environment. It is assumed that the user has access to MATLAB. If not, MATLAB may be purchased from: The MathWorks, Inc. 24 Prime Park Way Natick, MA 07160-1500 Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks mathworks The Student Edition of MATLAB may be obtained from: Prentice Hall PTR, Inc. One Lake Street Upper Saddle River, NJ 07458 prenhall An introduction to MATLAB fundamentals is given in Appendix A of this book.
Program Installation for WINDOWS.
To start the installation, do the following: Insert the CD-ROM in your CD-ROM drive (usually d: or e:) Choose Run from the WINDOWS Start menu, type d: (or e: ) and click OK. | 677.169 | 1 |
Module 12
SECTION 6.2: VOLUMES
The concept of volume provides the second application of a definite integral. Now
lim f ( xi ) xi
is replaced by
lim A( xi ) xi
where A( xi ) is the area of a cross-section of a solid. Examples of cross-sections are
shown in
Module 5
SECTION 3.4: THE CHAIN RULE
We have noted that the Power Rule can be applied to x 4 , x 2 , and x 4 , where each base
3
is x. But if the base is, say, x 3 + 1 as in ( x 3 + 1) , the same pattern will not give a
2
correct answer. The derivative of
Module 1
A PREVIEW OF CALCULUS
In this section in your text, the author presents an overview of calculus. Read it as a story
that exposes you to some of the ideas in calculus that will be covered in more detail later.
The major thread presented in the fiv
MATH 16A, SUMMER 2008, REVIEW SHEET FOR MIDTERM EXAM
BENJAMIN JOHNSON
The midterm exam will be held Thursday, July 17, from 8:10AM to 9:40AM in 3 Evans. The
exam will cover chapters 0, 1, and 2 of the textbook.
To do well on this exam you should be able t
Module 9
SECTION 4.7: OPTIMIZATION PROBLEMS
If you have been wondering where you might use calculus other than in graphing
functions, the answer is near at hand. We now look at some practical applications of
calculus, but first a word of warning. Most stu
Module 10
SECTION 5.1: AREAS AND DISTANCES
We now begin the study of the branch of calculus called integral calculus. Read page 360
carefully. In particular, it is not so easy to find the area of a region with curved sides.
We all have an intuitive idea o
Module 4
SECTION 1.5: EXPONENTIAL FUNCTIONS
We begin with an outline of the review section for exponential functions.
1. How do we interpret an irrational exponent? The power 32 means 3 times 3
1
and 3 2 is 3 but what does 3 2 mean?
2. When does the graph
Module 6
SECTION 1.6: INVERSE FUNCTIONS AND LOGARITHMS
The first part of this section can be skimmed with just a brief look at the different ways of
seeing the relationships between inverse functions. A major point is the algebraic method
for finding an i
Module 2
SECTION 2.1: THE TANGENT AND VELOCITY PROBLEMS
MORE ON NOTATION
In the discussion of the tangent line, it is important to have a clear concept of the tangent
y y1
line. Remember the slope of a line is 2
where y2 y1 is the length of a vertical
x2
Module 8
SECTION 4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
We start with a brief summary of this section.
1. A function is increasing when the derivative is positive. In a graph the curve is
rising.
2. A function is decreasing when the derivative i | 677.169 | 1 |
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