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Euler's Method and the S-I-R Model of Disease
In this Euler's method activity, students determine algebraic expressions for quantities in given terms. Next, they use Euler's Method to identify the information. Lastly, they confirm their calculations through the S-I-R method. This one-page activity contains three multi-step problems. | 677.169 | 1 |
Course Info
Page Content
CSC
202:
Discrete Structures for Computer Science
This course introduces basic mathematical tools essential for solving problems in computer science. The mathematical topics are presented through their applications in computer science: logic and set theory in the context of relational databases, arithmetic, number theory, and algebra through cryptography; graph theory and discrete structures through puzzles and games. Optional subjects include fractals (recursion and rewriting systems), computer graphics (linear algebra), and Artificial Intelligence (first-order logic). | 677.169 | 1 |
I couldn't find books in English for learning mathematics and physics
1. The problem statement, all variables and given/known data
I'm a student of a correspondance university in Japan. I want to learn physics and mathematics in English,but I can't find which books are suitable to study from very basic mathematics and Physics.
"Very basic", I mentioned, means on a level like pupils at school in the U.K. or the U.S. etc.
I study a little mathematics and physics in Japanese, but it`s no so high level and it`s difficult for me to understand because of difficult `kanji`. So, I think it`s a good way to study them in English. Please tell me how to find good textbooks or recommnded textbook for learning matematics and physics.
2. Relevant equations
I`m very sorry, but I can`t understand what I should write on this space.
3. The attempt at a solution
I searched books by Inter Net. I used amazon site or directly using search engine like google.But, I couldn`t find any books I want. Perhaps, I can't understand how other countries people learn these subjects. So, if I could I want to know steps to reach college level skills of mathematics and physics. Maybe this question is too obscure, but really want to study mathematics and physics in English and understand this world further. I'll be vary glad to answer me.
A very basic (but very good) book is "Algebra" by Gelfand. He has other books on trigonometry and geometry that are worth reading. He also has very advanced math books, so be sure not to get those yet.
Thank you for replying. How much level is "Algebla"by Gelfand? For high school student or even yonger student? I might the name of author of the book. Some of his books might be translated in Japanese. However, I want to know about English textbooks, so thank you for telling me about the book.
By the way I want to know what kind of textbooks for learning mathematics and physics are used by from elementary school to highschool students in the U.S. or from keystage1-4 to even higher level students in the U.K. or other countries.
If I want to understand mathematics and physics thoroughly, I think I should learn from early education in English,because I was taught almost all subject in Japanese. Is this idea correct?
Staff: Mentor
Shouldn't matter what language you learn math in, it is ideas that count, not how you call them, and ideas are always the same. Sure, you will need familiarity with the language used, but in my experience it is a matter of learning specific meanings of words you will most likely already know from English, and learning specific names of objects mathematicians use - so it is not different from learning English as a second language.
1. The problem statement, all variables and given/known data
I'm a student of a correspondence university in Japan. I want to learn physics and mathematics in English,but I can't find which books are suitable to study from very basic mathematics and Physics. | 677.169 | 1 |
Euclidean and Non-Euclidean Geometry: An Analytic Approach
This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic. The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical formulae. However, the book not only provides students with facts about and an understanding of the structure of the classical geometries, but also with an arsenal of computational techniques for geometrical investigations. The aim is to link classical and modern geometry to prepare students for further study and research in group theory, Lie groups, differential geometry, topology, and mathematical physics. The book is intended primarily for undergraduate mathematics students who have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments. Some familiarity with linear algebra and basic mathematical functions is assumed, though all the necessary background material is included in the appendices. Citește tot Restrânge
Recenzii
"An elegant geometry text...the whole very crisply printed and illustrated. Good exercises and helpful references." SciTech Book News "...a thoughtful, carefully crafted textbook..." Science Books & Films "The exposition is lucid; the body of the text and exercises are thoughtfully organized...The book should be brought to the attention of instructors wishing for a fresh outlook..." American Scientist "In his introduction the author expresses the hope that he can instill good working attitudes that will help students go on to research in group theory, Lie groups, differential geometry and topology. The naturalness and sophistication of his development go far to fulfilling his aim...The book is produced to a very high standard. Both graphics and text are exceptionally clear." The Mathematical Gazette | 677.169 | 1 |
COURSES OFFERED BY SYLVAN LEARNING
It strives to reach out to the needs of every student individually and guide them accordingly on the base of their current level of math skills. Covered topics include Addition and Subtraction, Numbers, Decimals, Problm solving and much moreIt strives to reach out to the needs of every student individually of 11th grade and guide them accordingly on the base of their current level of math skills. Topics like Algebra 1 &2, Gemotry, Pre-Calculus etc will be covered here.
It strives to reach out to the needs of every student individually of grade 10 and guide them accordingly on the base of their current level of math skills. Areas like Algebra 1, 2, Gemoetry, PreCalculus etc. will be covered.
It strives to reach out to the needs of every student individually of grade 9 and guide them accordingly on the base of their current level of math skills. Areas like Algebra 1& 2, Integrated Math, Geometry, etc will be covered.
It strives to reach out to the needs of every student individually of grade 8 and guide them accordingly on the base of their current level of math skills. You get to practice Geometry, Algebra, Probabilit, Statistics etc.
It strives to reach out to the needs of every student individually of grade 6 and guide them accordingly on the base of their current level of math skills. You get to practice Addition and Subtraction, Geometry, Algebra, Integers etc.
It strives to reach out to the needs of every student individually of grade 5 and guide them accordingly on the base of their current level of math skills. It covers different sections like Addition/Subtraction, Numbers, Problem Solvint, Fractions etc.
It strives to reach out to the needs of every student individually of grade 1 and guide them accordingly on the base of their current level of math skills. You will learn about Addition.Subtraction, Numbers, Fractions, Measurement, etc.
It strives to reach out to the needs of every kid individually of Kindergarten and guide them accordingly on the base of their current level of math skills. Your kids will learn about numbers and shapes. | 677.169 | 1 |
International Mathematics for the Middle Years 5 Teacher's Resource CD by Smith McSeveny, Conway, Wilkes (9780733985935)
International Mathematics for the Middle Years has been developed with the international student in mind. This series is particularly beneficial to students studying the International Baccalaureate MYP. All examples and exercises take an international viewpoint, giving students an opportunity to learn Mathematics with a global perspective.
The content is appropriate for international curricula and will meet the needs of all middle school students studying Mathematics.
The Teacher's Resource CD contains:
Practice tests with answers provide additional asessment opportunities.
Unit Plans in editable Microsoft Word format with ideas for linking to the Areas of Interaction.
Sample projects.
Sample guiding questions.
An electronic version of the coursebook | 677.169 | 1 |
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Why do card tricks work? How can magicians do astonishing feats of mathematics mentally? Why do stage "mind-reading" tricks work? As a rule, we simply accept these tricks and "magic" without recognizing that...
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The National 4 Mathematics Course enables learners to select and apply straightforward mathematical skills in a variety of mathematical and real-life situations. Learners interpret, communicate and manage information in mathematical form.
Numeracy
The general aim of this Unit is to develop learners' numerical and information handling skills. | 677.169 | 1 |
Maths Formulas For Class 8/10/12 Vedic/Basic Free List PDF Download
Maths Formulas
Maths is a most difficult subject for some students because solving mathematical problems acquires lots of knowledge and formulas. Well here on this page we are providing you the Maths Formulas of Class 8,9,10, 11 &12 and shortcuts for Learning Maths Formulas. Now most of the Bank, Government and Private Organizations conduct various competitive exams in order to recruit well talented and deserving aspirants for filing up the vacant positions. All these organizations ask many quantities and aptitude questions in these exams which create problem for the job seekers. So now you don't need to worry more and download the maths formulas PDF of Vedic/Basic Free List for Class 8,10 and 12 and start your preparation accordingly.
When we talk about Mensuration, Trigonometry and calculus, it becomes difficult for solving maths but through the below stated Maths Formulas, you can solve them very easily. When you understand the logic behind every problem and formulas, solving any kind of maths problem becomes easier. All the Formulas Of Maths, maths formulas list, Eamcet Maths Formulas PDF, SSC CGL Maths Formulas, Maths Formulas For IIT JEE PDF, basic formulas in maths for aptitude test, maths formulas for class 8 pdf, integration formulas in maths, maths important formulas, important maths formulas, all maths formulas free download, formulas in maths, maths formulas for class 12 PDF free download, maths formulas for competitive exams PDF and Jee Main Maths Formulas PDF is well mentioned below on this page which is designed by the team members of recruitmentresult.com
Maths Formulas
Shortcut To Solve Maths Question
122= 12+2/22= 14/4 = 144
132= 13+3/32= 16/9 = 169
Question: You can now Multiply 11 with Any Number of 3 Digits. Want to know the trick…
Well here is the trick, let's take an example
i.e. 352*11 = 3—-(3+5)—-(5+2)—-2 = 3872
How you can apply rule?
First of all, you have to Insert the sum of first and second digits
Then sum of second and third digits between the two terminal digits of the number
Question: Which of the following expressions is in the sum-of-products (SOP) form?
(A + B)(C + D)
(A)B(CD)
AB(CD)
AB + CD
Answer: AB + CD
Combination:
Number of combinations of n different things taken r at a time = ⁿCr = n!r!(n−r)!n!r!(n−r)!
ⁿP₀ = r!∙ ⁿC₀.
ⁿC₀ = ⁿCn = 1
ⁿCr = ⁿCn – r
ⁿCr + ⁿCn – 1 = n+1n+1Cr
If p ≠ q and ⁿCp = ⁿCq then p + q = n.
ⁿCr/ⁿCr – 1 = (n – r + 1)/r.
The total number of combinations of n different things taken any number at a time = ⁿC₁ + ⁿC₂ + ⁿC₃ + …………. + ⁿC₀ = 2ⁿ – 1.
The total number of combinations of (p + q + r + . . . .) things of which p things are alike of a first kind, q things are alike of a second kind r things are alike of a third kind and so on, taken any number at a time is [(p + 1) (q + 1) (r + 1) . . . . ] – 1.
If you have any query then asks freely by commenting on the underneath provided box, we will definitely solve it as soon as possible. You are recommended to keep in touch with this web portal and avail the latest information. You can also join us on Facebook or Google Plus to get latest updates and may subscribe to our free mail services to retrieve latest messages and recent information in your message box directly. | 677.169 | 1 |
This manual contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Publisher:
Cengage Learning, Inc
Publication date:
May 18, 2011
ISBN: 9780840053886
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$56 manual contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer | 677.169 | 1 |
Practical Mathematics by A.H. Bell
Publisher: Blackie And Son Limited1919 ISBN/ASIN: 140674610X Number of pages: 356
Description: Practical mathematics including trigonometry and an introduction to the calculus - a textbook for higher elementary secondary and technical schools. The processes of Algebra are established in a practical way. Academic treatment has been avoided as far as possible.
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A Course of Elementary Mathematics by John Radford Young - Wm. H. Allen The preparation necessary for the profitable study of the following course of Mathematics is a knowledge of common Arithmetic, and some acquaintance with Geometry, as taught in Euclid's Elements. We shall commence with a treatise on Algebra. (5541 views) | 677.169 | 1 |
Sequencing Math DNA: Differences, Nth Terms, and Algebraic Sequences
byLawrence E. Yee
A sequence is an ordered list of numbers. Sequences can arise from visual patterns, recursive equations, explicit equations, and graphically. At the high school level, students work with arithmetic (linear) sequences, geometric (exponential) sequences, and quadratic sequences, as well as other types of sequences. With each type of sequence, students are expected to identify key features in order to write an equation that models the number pattern. Many students struggle significantly with writing an equation for a sequence, even when given the general formulas. This unit emphasizes the difference sequence and initial terms of an original sequence to determine a general equation for particular classes of sequences. Also, structures of difference tables, second difference tables, and ratio tables are introduced as tools to assist students with analyzing sequences in order to connect the numerical patterns with their algebraic representations. | 677.169 | 1 |
eTextbook
Math for Weldersis a combination text and workbook that provides numerous practical exercises designed to allow welding students to apply basic math skills. Major areas of instructional content include whole numbers, common fractions, decimal fractions, measurement, percentage, and the metric system. Answers to odd-numbered practice problems are listed in the back of the text. | 677.169 | 1 |
File Search: Math 1300 pdf formulas | Apps for Windows bittorrent.am :: 468 Mb
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The Everything Guide to Pre-Algebra
4.11 - 1251 ratings - Source
Master the building blocks of mathematics! Not everyone is born a math whiz. Sometimes, all you need is a little extra help and practice to improve your comprehension. If you're a student encountering complex math for the first time, a parent wanting to help with homework, or an adult returning to school, The Everything Guide to Pre-Algebra is perfect for you. This essential guide uses simple explanations, step-by-step examples, and lots of review exercises to cover all the pre-algebra basics, including: Rational and irrational numbers Fractions, decimals, and percents Variables and functions Expressions and equations Number properties Inequalities Absolute values Plane geometry With unique study strategies and proven test-taking tips, The Everything Guide to Pre-Algebra will help boost your math knowledge--and your confidence--one right answer at a time.A Helpful Practice Guide Through the Pre-Algebra Basics - in Plain English! Jane
Cassie ... Multiplication tables will help you, even if you are allowed to use a
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The Everything Guide to Pre-Algebra
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Mathematical Methods Exam 2 VCE Units 3 & 4 provides a comprehensive practice exam book for VCE Mathematical Methods Units 3 & 4. The Exam 2 Specialist Mathematics Notes VCE Units 3 & 4 is designed to be the most comprehensive, easy to use study guide for VCE Specialist Mathematics Units 3 & 4. There are notes at the beginning of each section which summarise the main definitions, formulas and techniques required for each section of the course. There are over 400 questions graded into three levels of difficulty for each Area of Study including technology-free and technology-assumed questions which are clear... View more info
94 Designed to help students in their preparation for exams to maximise results and overall performance, the Neap Maths Methods Units 3&4 smartstudy® Exams Guide contains: ◾4 ... View more info
A+ Further Mathematics Notes VCE Units 3 & 4 provides a comprehensive study guide that is designed to enhance the skills and confidence of students studying VCE Further Mathematics Units 3 & 4. There are notes at the beginning of each section which summarise the main definitions, formulas and techniques for each section of the course and over 400 questions graded into three levels of difficulty for each Area of Study. The graduated difficulty allows students to use the b... View more info | 677.169 | 1 |
This course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We'll cover some basic statistics of data sets, such as mean values and variances, we'll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we'll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction.
At the end of this course, you'll be familiar with important mathematical concepts and you can implement PCA all by yourself. If you're struggling, you'll find a set of jupyter notebooks that will allow you to explore properties of the techniques and walk you through what you need to do to get on track. If you are already an expert, this course may refresh some of your knowledge.
This examples and exercises require:
1. Some ability of abstract thinking
2. Good background in linear algebra (e.g., matrix and vector algebra, linear independence, basis)
3. Basic background in multivariate calculus (e.g., partial derivatives, basic optimization)
4. Basic knowledge in python programming and numpy
From the lesson
Principal Component Analysis
We can think of dimensionality reduction as a way of compressing data with some loss, similar to jpg or mp3. Principal Component Analysis (PCA) is one of the most fundamental dimensionality reduction techniques that are used in machine learning. In this module, we use the results from the first three modules of this course and derive PCA from a geometric point of view. Within this course, this module is the most challenging one, and we will go through an explicit derivation of PCA plus some coding exercises that will make us a proficient user of PCA. | 677.169 | 1 |
function, known as "factoring", is a key concept to algebra on the GMAT--make sure you spend time with it.
The general rule, known as the distributive property (I think? I'm so bad with names!), is that a(b + c) = a x b + a x c
Keep working through our math workbook--this property may not be covered in detail (I'm not familiar with the specific text) but the rules used will be tested again and again. Once you've learned to go both ways--multiplying both terms inside parentheses by a factor, or pulling a common factor of two terms out of parentheses--you'll have mastered a powerful tool for taking on GMAT mathematics.
_________________
MBA Acceptance Rate by Undergraduate Major Many applicants may wonder if their undergraduate major impacts their chance of getting into business school. Admissions data suggests that your college major... | 677.169 | 1 |
Synopsis
This book provides a basic, initial resource, introducing science and engineering students to the field of optimization. It covers three main areas: mathematical programming, calculus of variations and optimal control, highlighting the ideas and concepts and offering insights into the importance of optimality conditions in each area. It also systematically presents affordable approximation methods. Exercises at various levels have been included to support the learning process.
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A Modern Introduction to Linear Algebra
4.11 - 1251 ratings - Source
Useful Concepts and Results at the Heart of Linear Algebra A one- or two-semester course for a wide variety of students at the sophomore/junior undergraduate level A Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory. The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix. The author goes on to cover finite-dimensional real vector spaces, infinite-dimensional spaces, linear transformations, and complex vector spaces. The final chapter presents Hermitian and normal matrices as well as quadratic forms. Taking a computational, algebraic, and geometric approach to the subject, this book provides the foundation for later courses in higher mathematics. It also shows how linear algebra can be used in various areas of application. Although written in a qpencil and paperq manner, the text offers ample opportunities to enhance learning with calculators or computer usage. Solutions manual available for qualifying instructorsI provide proofs for virtually all results, leaving some others (or parts of others) as
exercises. ... provided more details and more a#39;a#39;between the linesa#39;a#39; explanation
than is customary in mathematical proofs, even in textbooks. ... There will be
anInstructora#39;s Solutions Manual, containing worked-out solutions to all exercises.
Title
:
A Modern Introduction to Linear Algebra
Author
:
Henry Ricardo
Publisher
:
CRC Press - 2009-10 | 677.169 | 1 |
Numerical Mathematics and Computing
4.11 - 1251 ratings - Source
Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. NUMERICAL MATHEMATICS AND COMPUTING, 7th Edition also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.Solve it by Gaussian elimination with scaled partial pivoting using procedures
Gauss and Solve. ... follows: (i) skip elimination if a, q, , k : 0 and (ii) add an error
parameter ierr to the parameter list and perform error checking (e.g., on division
by zero or a row of zeros). Test the modified code on linear systems of varying
sizes.
Title
:
Numerical Mathematics and Computing
Author
:
E. Cheney, David Kincaid
Publisher
:
Cengage Learning - 2012 | 677.169 | 1 |
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Mathematics AS & A Level at King George V College
Course description
At KGV you will study Pure Maths, including algebraic methods for solving equations and calculus - the mathematics of changing quantities. Your
programme of study will include opportunities for learning Statistics, which includes methods that are used for analysing data in Economics, Psychology and the other Sciences; Mechanics, the study of motion and forces, which is the basis for Engineering and
Ballistics; and Decision Maths, which involves solving problems using algorithms in much the same way that computer programs work. You will leave KGV with a broad transferable skill set including problem-solving, critical thinking and the ability to work independently.
Many Maths students go on to study Maths, Engineering, Sciences and Medicine at University and work in Finance, Sciences and Engineering.
Course content
First Year
Pure Core 1: You will use algebra to investigate and solve topics such as quadratic equations; intersection of lines/curves: equations of perpendicular lines; gradients of curves with calculus and extending this to finding stationary points
areas under curves.
Pure Core 2: You will extend your Core 1 knowledge and use algebra to solve topics such as solving quadratic trigonometric equations ; sums of series; transformations; exponentials and logarithms and more calculus involving terms expressed
in negative or fractional powers.
Applied (Statistics 1 or Mechanics 1): Statistics applies your Pure knowledge to draw sensible conclusions from data collected in an uncertain world. You will solve problems like finding the chance of starting a game with a double six by
probability or finding whether the results of a medical trial are significant by using Binomial or Normal distributions. Mechanics applies your Pure knowledge to understand the way in which things move by studying the forces that act upon them. You will study
topics like velocity and acceleration; collisions; friction between surfaces slowing motion down and the angle a free kick should be projected at to win the match.
Second Year
Pure Core 3: Calculus plays a big part of second year Pure and you will learn how to use differentiate and integrate lots of different functions and extend your skills in algebra and trigonometry further.
Pure Core 4: You will develop your algebraic skills further again in this module particularly with trigonometry and calculus and be introduced to geometry with vectors
Applied: Most students choose Decision 1 as their applied option in the second year to broaden their experience. This involves solving organisational problems by using algorithms like finding the shortest path from college to home or the
minimum amount of tv cables to connect a number of houses. It may be possible to take Stats 2 or Mechanics 2 as an alternative if there is sufficient demand.
Entry requirements
A-level Maths at KGV is ideally suited to students who enjoyed Maths at high school and like the satisfaction of mastering new methods for solving
problems. A good mathematician can apply techniques and knowledge they have already learned to new situations. The pace of learning can be challenging and students need to be dedicated and curious about learning new ideas.
Future opportunities
Maths A level is a bonus to any degree course as it shows you have logical and problem solving skills plus the determination to overcome obstacles. In the last few years, quite a number of our students have gone on to a Maths degree (at Oxford, Warwick,
Leeds, Manchester and Birmingham) to give them more career choices rather than specialising in Engineering or Finance at this stage at university.
In addition there are always a sizeable number of our Maths students going on to Engineering, Medicine, Science and Finance courses. Some examples of destinations of our A* Maths students are Mechanical Engineering at Durham University , Natural Sciences
at Cambridge University, Physics at the University of Manchester , Chemical Engineering at Leeds University, Medicine at Liverpool University , Pharmacy at Cardiff University, Genetics at Sheffield University, Zoology at Liverpool University, Accounting and
Finance at University of Manchester .
Further information
You will have the opportunity to take part in local and national maths competitions such as the UKMT Senior Challenge and the LMS Pop quiz. Trips
to interesting, relevant talks such as the Maths Inspiration lectures, are also planned. Each week, every Maths class has an additional subject session where extra help on difficult topics is available to everyone.
How to apply
If you want to apply for this course, you will need to contact King George V College directly. | 677.169 | 1 |
"a Canadian perspective", this text has a more quantitative bias than Arnold's, and a less opulent presentation. Its case studies offer useful worked examples, and it would be a helpful secondary text, especially for students who want computational practice.
Who is it for? Postgraduates and second- and third-year undergraduates.
Presentation: Straightforward.
Would you recommend it? Yes | 677.169 | 1 |
—all explained in simple, clear, and yes, funny illustrations. Never again will you order the Poisson Distribution in a French restaurant!
--all explained in simple, clear, and yes, funny illustrations. Never again will you order the Poisson Distribution in a French restaurant!
The Cartoon Introduction to Statistics is the most imaginative and accessible introductory statistics course you'll ever take. Employing an irresistible cast of dragon-riding Vikings, lizard-throwing giants, and feuding aliens, the renowned illustrator Grady Klein and the award-winning statistician Alan Dabney teach you how to collect reliable data, make confident statements based on limited information, and judge the usefulness of polls and the other numbers that you're bombarded with every day. If you want to go beyond the basics, they've created the ultimate resource: "The Math Cave," where they reveal the more advanced formulas and concepts. Timely, authoritative, and hilarious, The Cartoon Introduction to Statistics is an essential guide for anyone who wants to better navigate our data-driven world.
Uses the story in graphic novel format of a teenage girl and her tutor to introduce statistics and provide everyday examples, a text explanation, exercises, and a summary, including using Microsoft Excel to do calculations.
A complete—and completely enjoyable—new illustrated guide to calculus Master cartoonist Larry Gonick has already given readers the history of the world in cartoon form. Now, Gonick, a Harvard-trained mathematician, offers a comprehensive and up-to-date illustrated course in first-year calculus that demystifies the world of functions, limits, derivatives, and integrals. Using clear and helpful graphics—and delightful humor to lighten what is frequently a tough subject—he teaches all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education—a valuable supplement for any student, teacher, parent, or professional.
Have you ever asked yourself: Are spliced genes the same as mended Levis? Watson and Crick? Aren't they a team of British detectives? Plant sex? Can they do that? Is Genetic Mutation the name of one of those heavy metal bands? Asparagine? Which of the four food groups is that in? Then you need The Cartoon Guide to Genetics to explain the important concepts of classical and modern genetics—it's not only educational, it's funny too!
Do you think that the Ozone Hole is a grunge rock club? Or that the Food Web is an on-line restaurant guide? Or that the Green Revolution happened in Greenland? Then you need The Cartoon Guide to the Environment to put you on the road to environmental literacy. The Cartoon Guide to the Environment covers the main topics of environmental science: chemical cycles, life communities, food webs, agriculture, human population growth, sources of energy and raw materials, waste disposal and recycling, cities, pollution, deforestation, ozone depletion, and global warming—and puts them in the context of ecology, with discussions of population dynamics, thermodynamics, and the behavior of complex systems.
If you have ever suspected that "heavy water" is the title of a bootleg Pink Floyd album, believed that surface tension is an anxiety disorder, or imagined that a noble gas is the result of a heavy meal at Buckingham Palace, then you need The Cartoon Guide to Chemistry to set you on the road to chemical literacy. You don't need to be a scientist to grasp these and many other complex ideas, because The Cartoon Guide to Chemistry explains them all: the history and basics of chemistry, atomic theory, combustion, solubility, reaction stoichiometry, the mole, entropy, and much more—all explained in simple, clear, and yes, funny illustrations. Chemistry will never be the same! WhereasWhen you took statistics in school, your instructor gave you specially prepared datasets, told you what analyses to perform, and checked your work to see if it was correct. Once you left the class, though, you were on your own. Did you know how to create and prepare a dataset for analysis? Did you know how to select and generate appropriate graphics and statistics? Did you wonder why you were forced to take the class and when you would ever use what you learned? That's where "Stats with Cats" can help you out. The book will show you: How to decide what you should put in your dataset and how to arrange the data.How to decide what graphs and statistics to produce for your data.How you can create a statistical model to answer your data analysis questions. The book also provides enough feline support to minimize any stress you may experience. Charles Kufs has been crunching numbers for over thirty years, first as a hydrogeologist, and since the 1990s as a statistician. He is certified as a Six Sigma Green Belt by the American Society for Quality. He currently works as a statistician for the federal government and he is here to help you.
From the medicine we take, the treatments we receive, the aptitude and psychometric tests given by employers, the cars we drive, the clothes we wear to even the beer we drink, statistics have given shape to the world we inhabit. For the media, statistics are routinely 'damning', 'horrifying', or, occasionally, 'encouraging'. Yet, for all their ubiquity, most of us really don't know what to make of statistics. Exploring the history, mathematics, philosophy and practical use of statistics, Eileen Magnello - accompanied by Bill Mayblin's intelligent graphic illustration - traces the rise of statistics from the ancient Babylonians, Egyptians and Chinese, to the censuses of Romans and the Greeks, and the modern emergence of the term itself in Europe. She explores the 'vital statistics' of, in particular, William Farr, and the mathematical statistics of Karl Pearson and R.A. Fisher.She even tells how knowledge of statistics can prolong one's life, as it did for evolutionary biologist Stephen Jay Gould, given eight months to live after a cancer diagnoses in 1982 - and he lived until 2002. This title offers an enjoyable, surprise-filled tour through a subject that is both fascinating and crucial to understanding our world. statistical analysis, Head First's brain-friendly formula helps you get a firm grasp of statistics so you can understand key points and actually use them. Learn to present data visually with charts and plots; discover the difference between taking the average with mean, median, and mode, and why it's important; learn how to calculate probability and expectation; and much more. Head First Statistics is ideal for high school and college students taking statistics and satisfies the requirements for passing the College Board's Advanced Placement (AP) Statistics Exam. With this book, you'll: Study the full range of topics covered in first-year statistics Tackle tough statistical concepts using Head First's dynamic, visually rich format proven to stimulate learning and help you retain knowledge Explore real-world scenarios, ranging from casino gambling to prescription drug testing, to bring statistical principles to life Discover how to measure spread, calculate odds through probability, and understand the normal, binomial, geometric, and Poisson distributions Conduct sampling, use correlation and regression, do hypothesis testing, perform chi square analysis, and more Before you know it, you'll not only have mastered statistics, you'll also see how they work in the real world. Head First Statistics will help you pass your statistics course, and give you a firm understanding of the subject so you can apply the knowledge throughout your life. | 677.169 | 1 |
Iteration
Abstract
The activities in this unit transform the traditional mathematical content of the typical secondary curriculum into a new, dynamic, visual, geometric world. Here we see things constantly change as we search for patterns in iterative behaviors. Graphical iteration produces paths that staircase or spiral in to specific points that serve as attractors and out from other points that play the role of repellers. Some intervals compress through graphical iteration so that errors are reduced, while others expand through iteration, causing small errors to explode into large ones. Underlying these characteristics is the key question: When is the iteration behavior predictable and when is it unpredictable? | 677.169 | 1 |
aim of this textbook is to impart the necessary mathematical competency for understanding modern developments in the age of the internet. It includes an introduction to elementary arithmetic with elliptical curves, which helps explain standard applications in cryptography. All important propositions are accompanied by complete proofs, and thus, the book presumes little prior knowledge. | 677.169 | 1 |
The Department of Mathematics
Written by simon on 15 March 2017.
The Department of Mathematics at Egerton University, Njoro Campus, is composed of three main branches of Mathematics namely: Applied Maths, Pure Maths and Statistics. The department has over 23 academic staff, One secretary, Three Clerical staff, Four Ph.D research students and eleven M.Sc. students. The department is one of the busiest departments in the University, teaching over 4,000 undergraduate students in a year and graduating not less than 3 postgraduate students per year!Mathematics is a living and growing subject in which new results are continually being discovered, theorems proved, examples constructed and whole new theories developed. Even in the "purest" areas of Mathematics, the most vital research topics are often those that stem from real-life scientific problems. We also solve real problems using mathematical techniques. Conversely, it frequently happens that progress in Mathematics research, even when undertaken without regard to possible applications, eventually turns out to have relevance to other parts of science. Students who have enjoyed and excelled in Mathematics at the undergraduate level will find an additional excitement on reaching the frontiers of the subject and becoming involved with areas of current research. | 677.169 | 1 |
Preliminaries: Background and Notation
Summary
In this chapter we review some elements of linear algebra, function spaces, Fourier series, and Fourier transforms that are essential to a proper understanding of wavelet analysis. In the process, we introduce some notation that will be used throughout the book. All readers are advised to read this chapter carefully before proceeding further. | 677.169 | 1 |
comprises a rigorous and self-contained treatment of initial-value problems for ordinary differential equations. It additionally develops the basics of control theory, which is a unique feature in the current textbook literature. The following topics are particularly emphasised: existence, uniqueness and continuation of solutions, continuous dependence on initial data, flows, qualitative behaviour of solutions, limit sets, stability theory, invariance principles, introductory control theory, feedback and stabilization. The last two items cover classical control theoretic material such as linear control theory and absolute stability of nonlinear feedback systems. It also includes an introduction to the more recent concept of input-to-state stability. Only a basic grounding in linear algebra and analysis is assumed. Ordinary Differential Equations will be suitable for final year undergraduate students of mathematics and appropriate for beginning postgraduates in mathematics and in mathematically oriented engineering and science.Read more...
Abstract:
This book comprises a rigorous, self-contained treatment of initial-value problems for ordinary differential equations. It also includes an introduction to mathematical control theory as well as an introduction to the concept of input-to-state stability.Read more...
Reviews
Editorial reviews
Publisher Synopsis
From the book reviews:"This new textbook on ordinary differential equations offers a surprisingly fresh approach to the subject by incorporating an introduction to control theory into a rigorous treatment of initial value problems. The work is written at an intermediate level and aimed at students of mathematics and mathematically-oriented engineering. ... The book is characterized throughout by strong writing, clear and complete proofs, good examples and plenty of exercises." (William J. Satzer, MAA Reviews, January, 2015)"The text concerns itself primarily with systems of linear differential equations. Logemann and Ryan immediately give examples of applications that are referred to and expanded on later in the text. This approach, as well as the dispersal of exercises throughout the text (as opposed to only at the end of each chapter), keeps students engaged. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty." (J. T. Zerger, Choice, Vol. 52 (5), January, 2015)"This book is one of several recent textbooks on ordinary differential equations intended for use in an undergraduate (and not only) course. If one wants to have a self-contained book dealing with the basic theory of initial value problems ... . then he/she must take a look at this very interesting book written by two experts on the field, Hartmut Logemann and Eugene Ryan. ... the book consists of six chapters plus an appendix." (George Karakostas, zbMATH, Vol. 1300, 2015)Read more... | 677.169 | 1 |
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In some classes, all it will require to move an exam is note having, memorization, and remember. Nevertheless, exceeding inside a math class requires another variety of work. You can't simply demonstrate up for just a lecture and enjoy your instructor "talk" about geometry and . You understand it by accomplishing: being attentive at school, actively finding out, and fixing math complications – regardless if your teacher hasn't assigned you any. In the event you end up having difficulties to perform well inside your math class, then take a look at finest internet site for fixing math problems to determine the way you may become a better math pupil.
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Math classes observe a natural progression – each one builds on the understanding you've acquired and mastered from your earlier study course. When you are obtaining it hard to abide by new ideas in class, pull out your aged math notes and evaluate past materials to refresh by yourself. Ensure that you satisfy the prerequisites in advance of signing up to get a class.
Review Notes The Night Ahead of Course
Hate any time a teacher phone calls on you and you have neglected tips on how to address a selected dilemma? Stay away from this instant by reviewing your math notes. This tends to make it easier to ascertain which concepts or issues you'd prefer to go over at school another day.
The considered doing research each individual night may seem annoying, however, if you want to achieve it is actually essential that you continually apply and master the problem-solving strategies. Make use of your textbook or on-line guides to operate by means of best math complications over a weekly foundation – even if you've got no research assigned.
Utilize the Nutritional supplements That come with Your Textbook
Textbook publishers have enriched modern day publications with more content (like CD-ROMs or on the net modules) that can be utilized to support students acquire more follow in . Many of these elements could also include things like a solution or explanation guidebook, which may assist you with operating via math issues by yourself.
Browse Forward To stay Ahead
If you need to minimize your in-class workload or the time you expend on homework, make use of your spare time right after college or on the weekends to study ahead to your chapters and ideas that could be lined another time you happen to be at school.
Review Old Tests and Classroom Examples
The do the job you do in school, for homework, and on quizzes can present clues to what your midterm or closing exam will look like. Make use of your outdated checks and classwork to make a personal review guideline on your future test. Glance on the way your instructor frames issues – this is often possibly how they may look on the take a look at.
Learn how to Function From the Clock
This is the well-known research tip for people using timed examinations; particularly standardized exams. For those who have only forty minutes for a 100-point check, you'll be able to optimally commit four minutes on just about every 10-point problem. Get information and facts regarding how prolonged the examination are going to be and which kinds of thoughts might be on it. Then strategy to attack the easier thoughts initially, leaving yourself enough time for you to shell out over the additional difficult ones.
Improve your Means to obtain math homework aid
If you are acquiring a tough time comprehending principles in class, then make sure to get assist beyond course. Inquire your folks to produce a examine group and pay a visit to your instructor's office hours to go more than difficult challenges one-on-one. Show up at analyze and overview classes once your instructor announces them, or use a non-public tutor if you need one.
Talk To Oneself
If you are examining issues for an exam, check out to elucidate out loud what tactic and techniques you used to obtain your alternatives. These verbal declarations will come in useful through a examination any time you must recall the actions you need to just take to find a resolution. Get extra observe by hoping this tactic which has a friend.
Use Review Guides For Excess Follow
Are your textbook or course notes not helping you understand everything you should be learning in class? Use analyze guides for standardized exams, including the ACT, SAT, or DSST, to brush up on outdated material, or . Analyze guides typically occur equipped with extensive explanations of ways to clear up a sample dilemma, , so you can often come across in which would be the far better purchase mathtroubles. | 677.169 | 1 |
blends much of the best aspects of calculus reform with the reasonable goals and methodology of traditional calculus. Readers benefit from an innovative pedagogy and a superb range of problems. Modeling is a major theme -- qualitative and quantitative problems demonstrate an extremely wide variety of mathematical, engineering, scientific, and social models. This book addresses topics such as continuity, the mean value theorm, l'Hopital's rule, parametric equations, polar coordinates, sequences, and series. Differential equations are integrated and coverage is expanded including an introduction to slope fields. Suitable for professionals in engineering, science, and math | 677.169 | 1 |
FOREWORD
I take great pleasure in recommending to all readers - Romanians or from abroad
- the book of professors Titu Andreescu and Dorin Andrica. This book is the fruit of a
prodigious activity of the two authors, well-known creators of mathematics questions
for Olympiads and other mathematical contests. They have published innumerable
original problems in various mathematical journals.
The book is organized in six chapters: algebra, number theory, geometry,
trigonometry, analysis and comprehensive problems. In addition, other fields of math
ematics found their place in this book, for example, combinatorial problems can be
found in the last chapter, and problems involving complex numbers are included in
the trigonometry section. Moreover, in all chapters of this book the serious reader can
find numerous challenging inequality problems. All featured problems are interesting,
with an increased level of difficulty; some of them are real gems that will give great
satisfaction to any math lover attempting to solve or even extend them.
Through their outstanding work as jury members of the National Mathematical
Olympiad, the Balkan Mathematics Contest (BMO) , and the International Math
ematical Olympiad (IMO) , the authors also supported the excellent results of the
Romanian contestants in these competitions. A great effort was given in preparing
lectures for summer and winter training camps and also for creating original problems
to be used in selection tests to search for truly gifted mathematics students. To support
the claim that the Romanian students selected to represent the country were really
the ones to deserve such honor, we note that only two mathematicians of Romanian
origin, both former IMO gold-medalists, were invited recently to give conferences at
the International Mathematical Congress: Dan Voiculescu (Zurich, 1994) and Daniel
Tataru (Beijing, 2002). The Romanian mathematical community unanimously recog
nized this outstanding activity of professors Titu Andreescu and Dorin Andrica. As a
consequence, Titu Andreescu, at that time professor at Loga Academy in Timi§oara
and having students on the team participating in the IMO, was appointed to serve
as deputy leader of the national team. Nowadays, Titu's potential, as with other Ro
manians in different fields, has been fully realized in the United States, leading the
USA team in the IMO, coordinating the training and selection of team contestants
and serving as member of several national and regional mathematical contest juries.
One more time, I strongly express my belief that the 360 mathematics problems
featured in this book will reveal the beauty of mathematics to all students and it will
be a guide to their teachers and professors.
Professor loan Tomescu
Department of Mathematics and Computer Science
University of Bucharest
Associate member of the Romanian Academy
FROM THE AUTHORS
This book is intended to help students preparing for all rounds of Mathematical
Olympiads or any other significant mathematics contest. Teachers will also find this
work useful in training young talented students.
Our experience as contestants was a great asset in preparing this book. To this we
added our vast personal experience from the other side of the " barricade" , as creators
of problems and members of numerous contest committees.
All the featured problems are supposed to be original. They are the fruit of our
collaboration for the last 30 years with several elementary mathematics journals from
all over the world. Many of these problems were used in contests throughout these
years, from the first round to the international level. It is possible that some problems
are already known, but this is not critical. The important thing is that an educated
- to a certain extent - reader will find in this book problems that bring something
new and will teach new ways of dealing with key mathematics concepts, a variety of
methods, tactics, and strategies.
The problems are divided in chapters, although this division is not firm, for some
of the problems require background in several fields of mathematics.
Besides the traditional fields: algebra, geometry, trigonometry and analysis, we
devoted an entire chapter to number theory, because many contest problems require
knowledge in this field.
The comprehensive problems in the last chapter are also intended to help under足
graduate students participating in mathematics contests hone their problem solving
skills. Students and teachers can find here ideas and questions that can be interesting
topics for mathematics circles.
Due to the difficulty level of the problems contained in this book, we deemed it
appropriate to give a very clear and complete presentation of all solutions. In many
cases, alternative solutions are provided.
As a piece of advice to all readers, we suggest that they try to find their own
solutions to the problems before reading the given ones. Many problems can be solved
in multiple ways and pertain to interesting extensions.
4
This edition is significantly different from the 2002 Romanian edition. It features
more recent problems, enhanced solutions, along with references for all published
problems.
We wish to extend our gratitude to everyone who influenced in one way or another
the final version of this book.
We will gladly receive any observation from the readers.
The authors
Chapter 1
ALGEBRA
6
PROBLEMS
C2 , ,cn }.
be a set of n characters { Cl '
We call word a string of at
most m characters, m ::; n, that does not start nor end with Cl .
How many words can be formed with the characters of the set C?
1. Let
C
. . •
2. The numbers 1, 2,
, 5n are divided into two disjoint sets. Prove that these
sets contain at least n pairs
such that the number
is also an
element of the set which contains the pair.
. . .
P(x) aoxn + al xn-1 . + an
be a polynomial with integral coefficients such that an =1= 0 (mod p). Prove that
if there are n + 1 integers 0' 1 , 0'2 , , an+ 1 such that P (ar)
0 (mod p) for all
1, 2, . . . , n + 1, then there exist i,j with i i j such that a i aj (mod p).
52. Determine all polynomials P with real coefficients such that pn (x) P(xn )
=
P has at least a zero with the absolute value less than 1.
54. Find all polynomials P of degree n having only real zeros Xl , X2 , . . . , Xn such
that
n 1
2
tt P(x) - Xi XPnI (X) ,
Prove that the polynomial
=
for all nonzero real numbers x.
55. Consider the polynomial with real coefficients
P(x) aoxn + alxn- l + . . . + an ,
=
16
1.
an
and
f:. O.
Prove that if the equation
equation
ALGEBRA
P(x) 0 has all of its roots real and distinct, then the
x2 PII (x) 3xP'(x) P(x) 0
has the same property.
=
+
+
=
56. Let R�f and R�) be the sets of polynomials with real coefficients having
no multiple zeros and having multiple zeros of order respectively. Prove that if
P(x) R�j and P(Q(x)) R�j , then Q'(x) R�]-l).
57. Let P(x) be a polynomial with real coefficients of degree at least 2. Prove
with real coefficients
Prove that if the equation has all of its roots real, then
(n
2:: 2n O . Is the reciprocal true?
59. Solve the equation
+
= n
n
58. Consider the equation
a a2
A
C,
- 1. The number that we seek is Nl N2 Nm·
{I, 2,...,k}, 1 � k � We need to find out the number of functions
Clearly, Nl
f
P has at least two nonreal zeros.
- l)ar
m.
Ak =
m.
k -+ A, k = 2,n with the properties
Let
f:
>
then
1. Let Nk be the number of words having exactly k characters from the set
1�k�
+ ... +
+
m +
a2n_2X2 - 2 x
n
+
= n
+
+m =
0,
+ ... +
= n
+ n
n
O
n + n
n + ... + n -
n
=
(Dorin Andrica)
1
=
0,
2.
Suppose, for the sake of contradiction, that there are two sets A and B such
that AU B = {I, 2,...,5n}, An B = 0 and the sets contain together less than n pairs
>
with the desired property.
Let k be a given number, k = 1, n. If k and 2k are in the same set A or B the same can be said about the difference 2k - k = k. The same argument is applied
for 4k and 2k. Consider the case when k and 4k are elements of A and 2k is an
element of B. If 3k is an element of A, then 4k - 3k = k E A, so let 3k E B. Now if
5k E A, then 5k - 4k = k E A and if 5k E B, then 5k - 3k = 2k E B; so among the
numbers k, 2k, 3k,4k,5k there is at least a pair with the desired property. Because
k = 1 , 2,... , n, it follows that there are at least n pairs with the desired property.
Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 75,
Problem 3698)
(x, y), x y,
-
(Dorin Andrica,
3. Note that
(1)
17
1.
18
1.2.
ALGEBRA
and furthermore
for all integers ml, m2 2:: 1 .
Suppose that for all integers k 2:: 1 we have I[k](x) iBecause there are n! permutations, it follows that for k
such that
positive integers nl >
Let Sn be the symmetric group of order n and Hn the cyclic
subgroup generated by a. It is clear that Hn is a finite group and therefore there is
integer h such that a[h) is identical permutation.
Notice that
if =
1, n
l
I k ](X) =
otherwIse
and then
To conclude, we have found that
(i ) If
then the equation has the solution
( ii ) If f:. c and f:. there is no solution.
( iii ) If f:. and
then
is the only solution.
Revista Matematica Timi§oara (RMT ) , No.
Problem
Let a > b be two positive integers such that
�-�> 1.
Then between numbers a and b there is at least a perfect cube.
Proof. Suppose, for the sake of contradiction, that there is no perfect cube between
a and b. Then there is an integer c such that
c3 � b < a � (c + 1) 3 .
Lemma.
as desired.
Observe that the inequality is strict because the numbers �
=
are
�
distinct.
In order to prove the first inequality we apply the AM-GM inequality in the form
Therefore
>
=
,.�
_
=
i
_
as desired.
�
31
1.2. SOLUTIONS
This means
so
which is false. 0
Now we can easily check that for
If 2:
then
2n 2
Therefore, it is sufficient to prove that k is an integer. Let us denote
v'3)
v'3) where is a positive integer. Clearly,
is an integer for
all
We will prove that
Moreover, the numbers
divides
satisfy the relation
which is false, since the greatest integer function is not periodical.
Revista Matematica Timi§oara ( RMT ) , No.
Problem
(Dorin Andrica,
3430)
1(1978), pp. 89,
42. Considering the determinant
=
Take now
12 (0) +12 (�) 2/(t)
=
and because the left-hand side is positive, it follows that is positive and
for all Then is a constant function, as claimed.
pp.
Revista Matematica Timi§oara ( RMT ) , No.
Problem
Problem
pp.
Gazeta Matematica ( GM-B ) , No.
On the other hand, multiplying the first column by
the second by
and adding all these to the fourth, we obtain on the last column
the third by
the numbers abed, bade, edab, debao Because all those numbers are divisible by the
prime number p, it follows that p divides � and therefore p divides at least one of the
numbers a b e d, a b - e - d, a - b e - d, a - b - e d.
10
+++ +
(Titu Andreescu)
+
+
x
x
Because the quadratic polynomials h ( ) and t2 ( ) have zeros of the same
nature, it follows that their discriminants have the same sign, hence
divides the determinant.
Adding the first and second columns and subtracting the last two columns implies
that
divides the determinant.
Analogously we can check that
and
divide the determinant,
and taking into account that it has degree 4 in each of the variables, the determinant
equals
x +y - z- v
where
..\
x-y+z-v x-y-z+v
..\(x + y + z + v) (x + y - z - v) (x - y + z - v)(x - y - z + v),
is a constant.
Consequently,
(pIP2
+ 4ql q2 )2 - 4(plq2 + P2Ql )2 2:: O.
Note now that the left-hand side of the inequality is the discriminant of the
quadratic polynomial and the conclusion follows.
Revista Matematica Timi§oara (RMT), No.
pp.
Problem
Gazeta Matematica (GM-B), No.
pp.
Problem
(Titu Andreescu,
3267;
t
5(1979), 191,
1(1978), 63,
17740)
46. Because the quadratic polynomial T has nonreal zeros, the discriminant
�
is negative.
a bserve that
= b2 e2 - 4a (b3 + e3 - 4abe)
=
0,
� ( b2 - 4ae) (e2 - 4ab) <
where � l
b2 - 4ae and �2 e2 - 4ab are the discriminants of the quadratic
polynomials Tl and T2 • Hence exactly one of the numbers � l and � 2 is negative and
since a > the conclusion follows .
a of order k. Since P(Q(a)) = 0, we have
m
ao II(Q(a) - Xi) = 0,
i=l
and so there is an integer 1 � �
such that Q(a) - xp = O. Observe that
=
Q(a) - xp f:. 0, for all j f:. otherwise Xj xp , which is false. Hence Q(x) - Xj has
the multiple zero a of order k and so QI(X) = (Q(x) - xp )' = QI(X) has a multiple
zero of order k - 1. This concludes the proof.
(Dorin Andrica, Romanian Mathematical Olympiad - final round, 1978; Revista
Matematica Timi§oara (RMT) , No. 2(1978), pp. 67, Problem 3614)
57. Assume by way of contradiction that P(x) has less than two nonreal zeros.
As a polynomial with real coefficients P(x) cannot have only one nonreal zero, hence
all of its are real. Let Xl , X2 , . . . ,X n be the zeros of P(x).
Then
PI(X) - n _1_
P(x) � X - Xi
has a multiple zero
16. Let p be a prime and let a be a positive real number such that pa2
Prove that
[n0i - �] = [n0i+ �]
for all integers
<
=
26. Prove that for all positive integers n the equation
1
4'
n [ y'1 -a2avp] + 1.
x2 + y2 + Z2 59n
=
has integral solutions.
27. Let n be a positive integer. Prove that the equations
2::
17. Let n be an odd positive integer. Prove that the set
18. Find all positive integers and n such that (:) = 1984.
19. Solve in nonnegative integers the equation
m
x2 + 8y2 + 6xy - 3x - 6y
and
xn + y n + zn + u n = vn+ 1
have infinitely many solutions in distinct positive integers.
contains an odd number of odd numbers.
20. Solve in integers the equation
are both primes.
x2 (y - z) + y2 (z - x) + Z2 (x - y) = 2.
n 6 between and 1 there is a perfect square.
13. Prove that for all positive integers n the number n ! is not a perfect square.
14. Prove that if n is a perfect cube then n2
cannot be a perfect cube.
2::
v
23. Find all triples (x, y, z) of integers such that
12. Let (Un) n�l be the Fibonacci sequence:
Prove that for all integers
p be a positive integer and let Xl be a positive real number. Define the
(Xn ) n�l by
Xn+l Vp2 1xn pvx; 1
for all positive integers n. Prove that among the first terms of the sequence there
50.
Let
sequence
=
are at least
+
+
+
m
[�] irrational numbers.
51. Define the sequence (Xn )n�o by
1) Xn = 0 if and only if n = 0 and
2) Xn+l X[�] (-l) nx[�] for all n 2:: O.
=
Find
2 3
52. Define the sequence (an ) n �O by ao = 0, al = 1, a2 = 2, a3
=
6 and
n divides an for all n > O.
53. Let X l X2 X3 1 and Xn+3 Xn + Xn+I Xn+2 for all positive integers n .
Prove that for any positive integer there is an integer k > 0 such that divides
Xk ·
54. Let (an ) n�O be the sequence defined by ao 0, al 1 and
an+l - 3an + an- l ( _ l) n
2
for all integers n O. Prove that an is a perfect square for all n 2:: O.
55. Let al a2 97 and
an+l anan-l V(a; - l)(a;_ 1 - 1), n > 1.
Prove that
a) 2 + 2an is a perfect square.
b) 2 J2 2an is a perfect square.
56. Let k � 2 be an integer. Find in closed form for the general term an of the
sequence defined by ao 0 and an - a [ �] 1 for all n O.
57. Let ao al 3 and an+l tan - an- l for n 2:: 1. Prove that an - 2 is a
perfect square for all n 2:: 1.
58. Let a and {3 be nonnegative integers such that 0'2 + 4{3 is not a perfect square.
Define the sequence (x n ) n �O by
=
m
m
=
=
=
>
=
=
=
+
+
+
=
=
=
=
=
>
(4n
+
yields all odd positive integers less than or equal to
Prove that
=
=
±1 ± ± ± . . . ±
+
=
55
60. Prove that for different choices of signs + and - the expression
Xn in closed form.
=
2. 1 .
for all integers n 2:: 0, where X l and X2 are positive integers.
Prove that there is no positive integer no such that
X� Xno-IXno+I '
59. Let n > 1 be an integer. Prove that there is no irrational number a such that
\/'a + Ja2 - 1 + \fa - Ja2 - 1 is rational.
PROBLEMS
1) If n -k (mod p), then the right-hand side of (1) is divisible by p2p , but the
left-hand side is not.
2) n -q (mod p), q f. k and q {O, 1, . . ,p - I}, then the left-hand side of
(1) is divisible by p. On the other hand
(n + k)2p (q - k)2 (mod p) � 0 (mod p),
because 0 < Iq - kl < p.
Both cases end up in contradictions, so the problem is solved.
(Dorin Andrica)
==
Denote by Pk the k-th prime number. Then
U
v
Pk is a
V
solution of equation x2
U for any integer k > O
Let and be two arbitrary primes, � Then
for some positive integer k. Because
(k+ 1) 2 - k 2 , it follows that all quadruples
{ x� -� �z =: 21
{ y: -= z: =: l=�
{ xxy --- yzz === l-1l
Since in the first case any two factors do not add up to the third, we only have
three POSS b i e
so (
Its complementary set has only even numbers, any two of them not being relatively
Let be the complementary set of a subset with
elements
prime. Hence �
of the set
Define D
E
If n D 0, then C U D has
elements but u D C
which is false.
Hence n D f. 0, so there is an element E n D. It follows that E and
+ E and since and + are relatively prime we are done.
Revista Matematica Timi§oara (RMT) , No.
pp.
Problem
(1)
But
(2)
and so
(3)
by summing up the relations
and (2).
Because p is a positive integer, it follows that Jp2 + 1 is irrational. From (3)
we deduce that among any three consecutive terms of the sequence there is at least
an irr�tional term. Hence there are at least [;] irrational terms among the first m
terms of the sequence.
(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 68,
Problem 4139; Gazeta Matematica ( GM-B ) , No. 6(1980), pp. 281, Problem C: 48)
51. Setting n = 0 and n = 1 yields = x� and X2 = x�, hence = X2 = 1.
Xn
lXn+ 1 - P
X +1
( I)
Xl
From the given condition we obtain
as
a4
Xl
a2
_
a4
a3
as
al
a6
an
a6
(Fn )n > l
an
an+4
a
nFn
3 Fn+3
2 Fn+2
I)Fn+1
I)Fn+ 1
n+ 1
n+
n
n+
X
Xt
.
rm3 + 1 ,
2.
80
2. 2 .
NUMBER THEORY
by the Pigeonhole Principle that at least two triples are equal. Let p be the smallest
number such that triple
is equal to another triple
p < q :::; m3 • We claim that p =
Assume by way of contradiction that p 1. Using the hypothesis we have
The last equation has no rational roots since the discriminant = a2 + 4/3 is not
a perfect square. This is a contradiction and hence the problem is solved.
Alternative solution. From the condition of the problem we obtain
aXn = Xn+l - /3Xn- l , n 2:: 1.
(1)
d
(2)
(a? + 4(3)x�o = (Xno+l + /3xno - d2
which is false, since a2 + 4/3 is not a square. This completes the proof.
(Dorin Andrica)
",
58. Note that all terms of the sequence are positive integers.
Assume by way of contradiction that there is a positive integer no such that
1. Triangle ABC has side lengths equal to c. Find a necessary and sufficient
condition for the angles of the triangle such that
c2 can be the side lengths of
a triangle.
2. Prove that the triangle whose side lengths are equal to the length of the medians
of a triangle ABC has area equal to 3/4 of the area of triangle [ABC].
3. In triangle ABC the median AM meets the internal bisector BN at P. Let Q
be the point of intersection of lines CP and AB. Prove that triangle B NQ is isosceles.
4. In triangle ABC the midline parallel to AB meets the altitudes from A and B
at points D and E . The midline parallel to AC intersects the altitudes from A and C
at points F and G.
Prove that DCI I BFI I GE.
5. Let M be a point in the interior of triangle ABC. Lines AM, BM, CM intersect
sides BC,CA,AB at points A',B', C', respectively. Denote by
the areas of triangles MA' B, MA' C, MB' C, MB' A, MC' A and MC' B, respectively.
Prove that if
a, b,
2 2
a ,b ,
81 , 82 , 83 , 84 , 85 , 86
81
82
+
83
84
+
85
-3
86 -
,
then M is the centroid of the triangle ABC.
6. Let M be a point in the interior of triangle ABC and let M, P, Q be three
colinear points on sides AB, BC and line CA.
Prove that if
SMAN
8MBN
+
5MBP
8MCP
=
2J
SMAQ
8McQ
,
then MP is skew-parallel with AC.
7. Let c and be the sides lengths and the area of a triangle AB C. Prove
that if P is a point interior to triangle ABC such that
a , b,
Al A�, A2 A� in the same ratios they divide the
AlBl Cl and A2 B2 C2 are similar.
BlBL B2 B�,
9. On side BC of a triangle ABC consider points M and N such that ifAjJ
CAN.
Prove that
MB + NB 2 AB
MC NC 2: AC
10. Let M be a point on the hypotenuse BC of a right triangle ABC and let
points N, P be the feet of the perpendiculars from M to AB and AC.
Find the position of M such that the length NP is minimal.
11. Let ABC be an equilateral triangle and let P be a point in its interior. Let the
lines AP, BP, CP meet the sides BC, CA, AB at the points AI, Bl, Cl respectively.
and 12 divide the internal bisectors
then triangles
internal bisectors
==
Prove that
S
be the set of all triangles
ABC for which
3
=6
5 ( AP1 + BQ1 + CR1 ) - min{P, BQ,
CR}
r'
P,
where T i s the imradius and Q , R are the points of tangency of the incircle with
sides
respectively. Prove that all triangles in S are isosceles and similar
to one another.
AB, BC, CA,
13.
ABC
ABC.
Let
be a triangle inscribed in a circle of radius R, and let P be a point
Prove that
in the interior of
PB and PC are the sides of an obtuse triangle. Assume that in this triangle the
obtuse angle opposes the side congruent to PA. Prove that BAlJ is acute.
17. Triangle ABC has the following property: there is an interior point P such
that PAJj = 10°, PEA = 20°, PcA 30°, and PAC 40°. Prove that the triangle
ABC is isosceles.
18. Let ABC be a triangle such that max{A,B} = C + 30°. Prove that ABC is
right angled if and only if !!. J3 + 1.
T
19. Prove that in the interior of any triangle ABC there is a point P such that
circum
radii of triangle PAB,PBC,PCA are equal.
the
20. The incircle of the triangle ABC touches the sides AB, BC, CA at the
points C', A', B', respectively. Prove that the perpendiculars from the midpoints of
A'B', B'C', C'A' to AB, BC, CA, respectively, are concurrent.
21. Let AlA2 A3 be a non-isosceles triangle with the incentre I. Let Ci , i = 1,2,3,
be the smaller circle through I tangent to AiAi+ l and Ai Ai+ 2 (the addition of indices
being mod 3). Let Bi , i = 1,2,3, be the second point of intersection of Ci+ l and Ci+ 2 .
Prove that the circumcenters of the triangles AlBlI, A2 B2 I, A3 B3 I are collinear.
22. On sides AB and AC of a triangle ABC consider points B' and C' such that
AB' + AC' k is constant.
B ' B C' C
Find the locus of the intersection point of lines BC' and C B'.
23. Let ABC be an equilateral triangle of side length 1 . Find the locus of points
P such that
· PB · PC
max{PA,PB,PC} = PA · PB +2PA
PB · PC + PC · PA - 1 .
=
=
=
=
24. Prove that in any acute triangle
Ja2 b2 - 4S2 + Ja2 c2 - 4S2 = a2 .
25. Prove that a triangle in which
r,;.-
. a
yT
=
=
89
16. Let P be a point in the plane of triangle ABC such ' that the segments PA,
for any triangle.
33. Let ABC be a triangle. Prove that A � i if and only if
(p - b)(p - c) � 4bc '
34. Three equal circles of radii r are given such that each one passes through the
centers of the other two.
Find the area of the common region.
35. Let ABCD be a nonisosceles trapezoid with bases AB and CD. Prove that
then the diagonals of the trapezoid are perpendicular.
37. Prove that if in a trapezoid with perpendicular diagonals the altitude is equal
to the midline, then the trapezoid is isosceles.
38. Let ABCD be a trapezoid with bases AB and CD. Prove that
39. Prove that a trapezoid whose difference of the diagonal lengths is equal to
the difference of nonparallel side lengths is isosceles.
40. Let ABCD be a cyclic quadrilateral. Prove that
lAB - CDI + l AD - BCI 2:: 21AC - BDl ·
41. Prove that a right trapezoid whose altitude length is equal to the geometrical
mean of the lengths of its bases has perpendicular diagonals.
42. Let E and F be the projections of the vertices A and B of a trapezoid ABCD
on line CD. Let M and N be the projections of E and F onto BD and AC, respectively
and let P and Q be projections of E and F onto BC and AD, respectively.
Prove that the quadrilateral MNPQ is cyclic.
43. Let ABCD be a convex quadrilateral that is not a parallelogram and let M
and N be the midpoints of the diagonals AC and BD. Prove that numbers
AB + CD, BC +AD, AC +BD, 2MN
can be the side lengths of a cyclic quadrilateral.
44. Let ABCD be a cyclic quadrilateral and let lines AB and CD meet at point E.
Point F is the reflection of C across E. Prove that lines AF and BD are perpendicular
if and only if lines AB and CD are perpendicular.
45. Find all cyclic quadrilaterals having the side lengths odd integers and the
area a prime number.
46. A cyclic quadrilateral has the side lengths a, b, c, d, the diagonal lengths f
and the semiperimeter p. Prove that
e,
3.
92
GEOMETRY
47. A cyclic quadrilateral has the side lengths
and
e I.
Prove that if max{l l Ib - dl} ::; 1, then
a,
b, d and the diagonal lengths
c,
a-c ,
Ie
-
93
3 . 1 . PROBLEMS
II ::; vi
48. A cyclic quadrilateral has area S and semiperimeter p. Prove that if S = (�) 2 ,
then the quadrilateral is a square.
49. Let ABCD be a convex quadrilateral and let P be the intersection point of
its diagonals.
Prove that
SPAB + SPCD = SPBC + SPDA
if and only if P is the midpoint of AC or BD .
50. Let M be a point on the circumcircle of the cyclic quadrilateral ABCD and
let points A', B', C', D' be the projections of M onto AB, BC, CD, DA, respectively.
Prove that
(i) lines A' B' , C'D' and AC are concurrent
(ii) lines B' C', D' A' and BD are concurrent.
51. A convex quadrilateral ABCD with area 2002 contains a point P in its interior
such that PA = 24, PB = 32, PC = 28, and PD = 45. Find the perimeter of ABCD .
52. Find the locus of points P in the plane of a square ABC such that
max(PA,PC) = v'21 (PB + PD) .
53. Let P be the set of all quadrialterals with the same diagonal lengths and let
)'1 (P) and A (P) be the lengths of the segments determined by the midpoints of two
opposite sides2 of a quadrilateral p E P .
Prove that for all p E P the sum A� (P) + A� (P) is constant and find the value of
this constant.
54. Let ABCDEFGHIJKL be a regular dodecagon and let R be the circumra
dius.
Prove that
AB + AF = 4 and AB2 +AC2 +AD2 +AE2 +AF2 = 10R2 .
AF AB
55. Prove that if inside a convex poligon there is a point such that the sum of the
squares of its distances to the vertices of the poligon is twice the area of the poligon,
then the poligon is a square.
56. Let Al A2 . . . An be a cyclic polygon and let P be a point on its circumcircle.
Let Pl, P2 , . .. ,Pn be the projections of P onto the sides of the polygon. Prove that
the product
n PA�
IIl P��
i=
is constant.
57. Let AlA2 . . . A2n be a cyclic polygon and let M be a point on
its circumcircle. Points Kl, K2 , . . . , K2n are the projections of M onto sides
AlA2 ,A2 A3 , . . . ,A2n Al and points Hl,H2 , . . . ,Hn are the projections of M onto
diagonals AlAn+ l,A2 An+2 , ' " ,An A2n . Prove that
l
are all situated in the interior of the faces. S is the total area and the inradius of
the tetrahedron, prove that
SBCD + SCDA + SDAB + SABC �
PAl PBl PCl PDl
When does the equality hold?
60. Let AlA2 A3 A4 be a tetrahedron, G its centroid, and A�, A�, A�, A� the points
where the circumsphere of AlA2 A3 A4 intersects GAl, GA2 , GA3 , GA4 respectively.
Prove that
If
r
>
-
and
r
1 1 1 1 1 1 1 1
GA� + GA� + GA� + GA4 ::; GAl + GA2 + GA3 + GA4 °
SOLUTIONS
1. The positive real numbers a2 , b2 , c2 can be the side lengths of a triangle if and
only if
2. Let A', B', C' be the midpoints of sides BC, CA, AB respectively. Construct
point M such that BC MC' is a parallelogram. Note that AC' C M and BB' MA' are
also parallelograms hence AM CC' and A'M BB' . Hence the triangle determined
by the medians is AA'M.
=
=
M
A
.c------ďż˝
A'
C
Let N be the intersection point of lines B' C' and AA'. Because AC' A' B' is a
parallelogram, we have
C'N B'N !B'M
2 '
B
to obtain a quadratic equation in three variables. Without loss of generality, we may
set
1. Then the solution proceeds as above.
USA Mathematical Olympiad,
x=
(Titu Andreescu,
13. Let a, b,
2000)
A, B, C be the side lengths and angles of triangle ABC. Let X, Y, Z
be the feet of the perpendiculars from P to lines BC, CA, AB, respectively. Recall
the inequality (the key ingredient in the proof of the Erdos-Mordell inequality) :
c,
PA sin A � P Y sin C
+ PZ sin B.
(1)
3.
104
3. 2.
G EOMETRY
105
SOLUTIONS
A
A
B'
Z
.
•M
.
C'
'N� " " ' : : :
I
.
p" .
B
This says that the length of Y Z is greater than equal to its projection onto BC,
the latter being equal to the sum of the lengths of the projections of PY and P Z
onto BC. In fact, since AYP Azp 90°, AZPY is cyclic with AP as a diameter
of its circumcircle. By the Extended Law of Sines, Y Z P A sin A. Let M and N be
the feet of perpendiculars from Z and Y to the line P X. Since 1fZj5 BxP 90° , .
PZBX is cyclic. Hence x:iPZ 13 and ZM PZ sin B. Similarly, YN PY sin C.
Thus (1) is equivalent to Y Z � Y N +M Z. Multiplying by 2R and using the Extended
Law of Sines, (1) becomes
We prove that I == H.
Assume by way of contradiction that points I and H are distinct. Let AI , Bl , Cl
be the feet of the altitudes from A, B , C and let A' , B' , C' be the projection of I onto
the sides BC, CA, AB, respectively. Hence at least two of the inequalities
IA + lA' � AAl ,
a · IA + b · IB + c · IC
abc
R'
Equality in the first step requires that Y Z be parallel to B C and so on. This
occurs if and only if P is the circumcenter of ABC. Equality in the second step
requires that a b c. Thus equality holds if and only if ABC is equilateral and P
is its center.
USA IMO Team Selection Test, 2000)
Because PA is the longest side of an obtuse triangle with side lengths
PA, PB, PC, we have PA VPB2 + PC2 , and hence
>
P A . BC
2:: ..jPB2 + PC2 . BC.
AQ 2
2:: AQ2 _ QC2
=
>
=
(Ap2 _ PQ 2 ) _ (Cp 2 _ PQ 2 )
AP2 - PC2
> BP2 2:: DQ2 .
=
l,
l,
o
P
o
O�--�O�----�Or---O
A
D
Q
c
+
+
It
=
+
=
z · z
C
=
z -
z
+
=
z ·
+
z
=
0
Applying the Lemma to points A, B , C, P gives
o � AB 2 + Bp2 + PC 2 + CA2 _ AP 2 - BC2
=
>
Let I be the ray AC minus the point A. Note that, since PA PC, Q lies on ray
D did not lie on then AQ would be less than or equal to DQ, a contradiction.
Thus, D lies on and angle BAC is acute.
B
l. If
=
c ,
=
=
2:: BC,
implying that angle BAC is acute.
With some careful argument, it can be proved that quadrilateral ABPC is
indeed convex. We leave it as an exercise for the reader.
Let D and Q be the feet of the perpendiculars from B and P to
line AC, respectively. Then DQ � BP. Furthermore, the given conditions imply that
Ap2 Bp2 + PC2 , which can be written as Ap2 - PC 2 BP 2 . Hence,
the circumcircle of ABPC, arcs AB and AC are bigger than arcs PC and PB,
respectively. Thus, IiPC > IfAiJ Because these two angles are supplementary, angle
BAC is acute.
Thus
-
If () =
-
--
JPA2 - PB2
PB
+ sin - 1
BAC � BAP + PAC < sin PA
PA
1
sin then
��,
sin(90° - ())
Hence
=
1
.
'\.
JPA2 - PB2
.
cos () = V'1 - sin2 () =
PA
. - 1 JPA2 _ PB2
. - 1 PB + sm
BAC < sm
PA
PA
and angle B AC is acute.
=
C '\.
1
B
B
As we mentioned at the end of the first solution, the conditions in the problem
imply that quadrilateral ABPC is indeed convex. Thus, the diagram on the right-hand
side is not possible, but this solution does not depend on this fact.
(
\
...-I
I
\
\
"-
I
\
'-...
""
,/
'\.
I
/
17.
A
/
A
All angles will be in degrees. Let x = F0B. Then PiiC
Law of Sines (or the trigonometric form of Ceva's Theorem) ,
P
I
,/
'-
\
(Titu Andreescu, USA Mathematical Olympiad, 2001)
P
/'
\
P
90° ,
C
Sixth solution.
109
3.2. SOLUTIONS
3. G EOMETRY
_
/
_
;'
B
\
P
C
'"
\
P A PB P C
PB PC PA
./
I
=
� !!.!!. �
sin
sin
sin
sin PAB sin PBC sin PCA
80 - x. By the
=
4 sin x sin 40 cos 10
sin(80 - x)
sin 20 sin x sin 40
sin 10 sin(80 - x) sin 30
B
\
A
,/
...--
\
=
=
x
/
Note that PA2 > PB 2 + PC2 . Regard P as fixed and A, B, C as free to rotate
on circles of radii PA, PB, PC about P, respectively. As A, B, C vary, IfAiJ will be
maximized when B and C are on opposite sides of line PA and Jfijp and ;[(j'p are
right angles, i.e. , lines AB and AC are tangent to the circles passing through B and
C.
Without loss of generality, we assume that PA > PB � PC. In this case, ABPC
is cyclic and AB 2 = P A2 - PB 2 > PC2 , and similarly AC 2 > PB 2 . Hence on
Construct in the exterior of triangle ABC three circles equal to the circum
circle ABC that pass through two vertices of the triangle. By the five-coin theorem
the circles will have a common point P, as desired (see Dorin Andrica, Csaba Varga,
Daniel Vacare�u, " Selected Topics and Problems in Geometry" , PLUS, Bucharest,
2002, pp. 51-56) .
Let H be the orthocenter of triangle ABC. The reflections
of H across the sides of the triangle are points of the circumcircle of triangle ABC.
Therefore the circum circles of HAB, HBC, HCA are equal to the circumcircle of
ABC and for P H the claim holds.
(
Revista Matematica Timi§oara (RMT), No. 2(1978) , pp. 74)
Alternative solution.
=
Dorin Andrica,
20.
Denote the midpoints of A'B', B'C' , C'A' by Co , Ao , Bo, respectively, and
the three perpendiculars in question by
,
Consider the centroid of triangle
A'B' C'.
lc, lA lB .
11 2
3.
C
B
A
C'
Since AoG : GA' BoG : GB' CoG : GC' 1 : 2, the dilatation h with center
G and coefficient -2 takes Ao, Bo, Co to A', B', C', respectively. Since dilatations carry
straight lines into parallel lines, h transforms le into the line through C' perpendicular
to AB. But C' is the point of tangency of the incircle and AB, so this line passes
through the incenter of triangle ABC. The same applies to the images of lA and
lB under h. Since the images of lA , lB , le under h are concurrent, so are lA , IB , le
themselves.
(Titu Andreescu, Romanian IMO Selection Test, 1986)
21. Because triangle A1 A2 A3 is not isosceles, it is not difficult to see that the
circumcenter of the triangles A 1 B 1 I, A2 B2 I, A3 B 3 1 are defined. We start with a
sim pIe lemma.
Lemma. Let ABC be a triangle with the incenter I. Let T be the circumcenter
of the triangle BIC. Then T lies on the internal bisector of the angle A.
Proof. Let us draw the external bisectors of the angles B and C as shown in the
figure below.
B
=
=
A
C
The lemma is proved.
..1
113
SOLUTIONS
=
They intersect at the excenter E, which lies on the internal bisector of the angle
A. Since BE BI and GE GI, the quadrilateral BEGI is cyclic with the center
of the circumscribed circle on IE . This center will be also the circumcenter of BIC.
..1
Let us prove the main statement. For =
we denote by the center of the
circle Ci and by the circumcenter of the triangle
Clearly,
lies on the
internal bisector of the angle
By the lemma, also lies on the same bisector. Thus
the triangles
and
are perspective from the point By Desargues'
theorem these triangles are perspective from a line. This is to say that if we denote
then
and
=
to be the point of intersection of the lines
are collinear. But since
is the perpendicular bisector
the points
of
and
is the perpendicular bisector of
these points are exactly
respectively.
the circumcenter of the triangles
A student not familiar with Desargues' theorem may proceed from the
point as follows. Applying Menelaus' theorem to the triangles
and to the triples of points
respectively, one
can, observing usual agreement about the signs, write:
Ti
0 1 02 03
Qi , i 1,2,3,
Q 1 , Q , Q3
Ail 0i+120i+2
Remark.
_
=
primes to denote images of points under the inversion we have the following " dual"
figure shown below.
B'2
Indeed, the image of the circle Ci is a straight line Bi+ 1 Bi+2 ' with these lines
forming the triangle Bi B�B� . The line AiAi + 1 will be transformed into the circle
ri+ 2 , with the side AiAi+ 1 becoming the arc AiAi+ 1 which does not contain I. Note
that all these circles have equal radii since the distances from I to the sides of AI A2 A3
were equal.
Let us note that if �I ' � 2 ' � 3 are three circles passing through the common point
I and no two of them touch, then their centres are collinear if and only if there is
another common point f:. I through which all these three circles pass.
We will use this observation for �i being the circumcircle of AiBiI. Since the
inversion takes :Ei to the line AiB�, the desired result is to show that the lines A� Bi ,
A� B� , A�B� are concurrent. For this, it suffices to show that the triangles A� A� A�
and Bi B�B� are homothetic, which is the same to say that their corresponding sides
are parallel. Since the radii of the circles r I , r 2 , r3 are equal, the triangle PI P2 P3
formed by their centre has its sides parallel to the corresponding sides of the triangle
Bi B�B�. The homothety of ratio
centred at I takes the triangle A� A� A� into
the triangle whose vertices are the midpoints of the triangle PI P2 P3 . Therefore the
corresponding sides of the triangles A� A�A� and PI P2 P3 are also parallel and the
result follows.
IMO 1997 Shortlist)
J
PC PA · PB + l · 1
-1- PA . 1 PC . 1 .
From the converse of the second theorem of Ptolemy it follows that P ACB is a
cyclic quadrilateral. Note that P cannot be A, B or C otherwise the denominator of
the right-hand side equals Hence the locus of point P is the circumcircle of triangle
ABC without the vertices A, B, C.
Revista Matematidl Timi§oara (RMT), No. 1(1985) , Problem
C7:3)
The three sectors with centers 0 1 , O2 , 03 which subtend the arcs
02 03 , 0 1 03 ,02 0 1 , respectively, cover the 2 surface
of area S and twice more
r
the surface of triangle 0 1 02 03 ( which is f3 ). On the other hand, the area of
these three circular sectors equals the area of a semicircle, which is
AB + CD
= EF,
2
which is the length of the midline and hence the length of the altitude. It follows that
IM and IN are also altitudes in triangles lAB and ICD therefore lAB and ICD
are isosceles. Thus ABCD is isosceles, as claimed.
Revista Matematidi Timi§oara (RMT), No. 1 (1978) , pp. 48,
Pro blem 28 17)
IM + IN =
where M and N are the midpoints of AC and BD, respectively.
From (1) and (2) ,
1 23
SOLUTIONS
AB BC
AD AB '
so the right triangles ABC and ABD are similar.
On the other hand we have
(4)
(3)
and (4) , -2AC · BD > -2AB · CD - 2AD · BC, in contradiction with
From
Ptolemy's Theorem. We are done.
The cyclicity is essential. The inequality fails if ABCD is a parallelogram.
Let E be the intersection of AC and BD. Then the triangles
ABE and DCE are similar, so if we let x = AE, y = BE, z = AB, then there exists
k such that kx = l)E, ky = CE, kz = CD. Now
By Sturm's theorem, we know that if 0
then there is a cyclic quadrilateral having side lengths
Denote by
the lengths of the segments
respectively. Without loss of generality assume that
Let be the midpoint of the side
The segments
and
in triangles
and
so
1
1
=
and
=
=
=
On the other hand, if 0 is the intersection point of the diagonals, we have
Relations ( 1 ) and
imply fiE : EaT , therefore
is cyclic.
It follows that BEG = 90° , hence
as desired.
In the right triangle
ii ) Assume that lies on the segment
median, so
AB .l CD,
DE.
C
Because
CTF, TE is the
Ere : EaT .
CTI IB D, we have
(2)
(3)
( 4)
Also,
ATEC
(5)
so from ( 3 ) , (4) , (5) , we obtain Ere : Mo. Hence
is cyclic, then AEC =
ArC = 90° , and
.l
as desired.
Conversely, consider that
i) If is on the segment
then ME == AcE On the other hand AcE ==
Aci5 == ABi5 so ME == ABi5 and
is cyclic. It follows that JiiF = liEF =
90° , hence
as desired.
ii ) If is on the segment
then
AB CD,
D
C
,
BD .l AF,
AB .l CD.
CE,
FBIE
DE,
.
,
(6)
1 26
11oreover, J.U515 = ACB15,
SO
AcE = ABl.
S
(7)
a, b, c, d
Let
be the side lengths of the quadrilateral and let
Because the quadrilateral is cyclic, we have
4
Note that this is the equality case in the AM-GM inequality, hence p - a
d, so the quadrilateral is a square.
p - It follows that a
Revista Matematica Timi§oara (RMT) , No.
pp.
Problem
p e = d.
(Titu Andreescu,
2136)
= p-b =
1(1977), 24,
=b=e=
4�. Observe that SPAB . SPCD = SPBC . SPDA, since both are equal to � PA .
PB· PC· PD · sin P. The numbers SPAB , SPCD and SPBC , SPDA have the same sum
and the same product, thus SPAB = SPBC and SPCD = SPDA or SPAB = SPDA and
SPBC = SPCD, i.e. P is the midpoint of AC or BD, desired.
as
We recall the Simpson's theorem: the projections of a point of the circumcircle of
a triangle onto the sides of the triangle are collinear. Applying this result to triangles
and
yields that
are collinear and
are collinear.
Hence the lines
and
meet at
as claimed.
and
we deduce that
(ii) From Simpson's Theorem for triangles
and
Since
is a point of
the conclusion follows.
is on the lines
Revista Matematidi Timi§oara (RMT), No.
pp. 54,
Problem
Romanian Regional Mathematical Contest " Grigore MoisH" ,
a..j2PA = aPB + aPD, hence
BD · PA = AD · PB +AB · PD.
From the converse of the Ptolemy ' s Theorem it follows that PDAB is a cyclic
quadrilateral, therefore P lies on the circumcircle of square ABCD.
Conversely, using the Ptolemy's Theorem we deduce that any point of the cir
cumcircle of square ABC D has the given property.
P
Then
38. Prove that the triangle ABC is equilateral if and only if
1
A
B
C
tg "2 + tg 2" + tg = 4S
2"
31. Solve the system of equations
{X
sin y +
VI
7r
x + y = 4'
-
x2 cos Y =
�
2
a cos A + b cos B + ccos C =
39. Let ABC be a triangle such that
Prove that triangle ABC has a right angle.
37r
x+y+Z= 4
tgx + tgy + tgz = 5
tgx . tgy . tgz = 1.
33. Prove that in any triangle
(a2 + b2 + c2 ).
sin2 B + sin2 C = 1 + 2 sin B sin C cos A.
32. Solve the system of equations
1
aA cos B + bA cos A = cA
a2A-1 cos B + b2A-1 cos A = C2A-1•
40. Let ABC be a triangle such that
(cot �r + (2 cot �r + (3 cot �r = ( �; r ,
abc
-2R2
where s and r denote its semiperimeter and its inradius, respectively. Prove that
triangle ABC is similar to a triangle whose side lengths are all positive integers
with no common divisor and determine these integers.
other by a positive number chosen so that the sum of the pair does not change. Each
such change decreases the product of the i'S. It follows that for a given sum of the
bi 'S, the minimum product is attained when all of the bi ' S are equal. In this case we
>
for each so
have bi n + 1'
made smaller by replacing bj and bk by two numbers closer together and with the
same sum. In particular, if bj < 0, then replacing bj and bk by their average reduces
the problem to the case where bi > 0 for all
We . may now successively replace the b/s by their arithmetic mean. As long as
the bi are not equal, one is greater than the mean and another one is less than the
mean. We can replace one of this pair by the arithmetic mean of all the bi'S, and the
Thus, is a local maximum for I, and no other extrema exist on the interval
(0, 1).
The only thing left is to check that 1 (1) � l(aJ(n + 1)). Note that the case x = 1
is also an extreme case with bo = b1 = . . . = bn - 1 = 1 . This case does not arise in
our problem, but we must check to be sure that on the interval 0 � x < 1, I (x) has
a minimum at x = aJ (n + 1). We have
1(1) = 2n (1 + a - n) (n + 1 - a) n+ 1 0,
24. Let d be the greatest common divisor of p and q. We prove that T = 27r is
d
the lowest positive period of the function f.
It is clear that
I(x + T1 ) = I (x)
for all real x, therefore T is a period of function I.
Suppose there are T1 > 0 and an integer A > 0 such that T = AT1 and l (x + T1 ) =
I (x) for all real x. Then I( T1 ) = 1(0) = 2, so
cos pT1 + cos qT1 = 2,
55. Let AI , A2 , A3 and A be the points of complex coordinates Zl , Z2 , Z3 and let
a)z3, a Hence point A lies on the line A A and the triangle
ZA =A aA3Z2 +has(1its- circumcenter
in the origin of the complex plane.
l Point
2 B is the foot of the altitude
from A l in the triangle A l A2 A3 . It follows that
Al A � Al B, so
E
where is the center of
Proof. The key idea is to show that the orthogonal projection of the vector sum
0Pi + � + . . . + OP� onto some line has length not less than 1 (see S. Savchev, T.
Andreescu, " Mathematical Miniatures" , The Mathematical Association of America,
2003, pp. 75) . Let n = 2k - 1. From the considerations of symmetry, the line
0
Rotate the polygon
. . . An such that the complex coordinates of its
vertices are the complex roots of unity of order n,
. . . , Cn Let be the complex
'
coordinate of point located on the circumcircle of the polygon and note that = 1.
The equality
It is technically convenient to consider I as an axis with positive direction deter
As is well known, the projection of the sum of several vectors is equal
mined by
to the sum of their projections. Hence it suffices to prove that the sum of the signed
-::::-=t -::::-=t
�
onto is
lengths
of the projections of
greater than or equal to 1. Denote the diameter of C by
and the orthogonal
= 1 and also
projections of and onto by
and
We have
--
because a >
Take an integer > �e . Then
an >
n
n
The interval (a , f3 ) has length greater than hence there is an integer m >
such that a < m <
or a < rm < as desired.
Revista Matematidi Timi§oara (RMT) , No.
pp.
Problem
c (a, b) such that F(c) = O. It follows that
m
f(c) �
a - c + b -2c
and the solution is complete.
(Dorin Andrica)
22. Note that g(x) = g(y) implies that g(g(x)) = g(g(y)) and hence x = y from
the given equation. That is, is injective. Since is also continuous, is either strictly
increasing or strictly decreasing. Moreover, cannot tend to a finite limit L as x
or else we'd have g(g(x)) - ag(x) = bx, with the left side bounded and the right side
unbounded. Similarly, cannot tend to a finite limit as x
Together with
monotonicity, this yields that is also surjective.
Pick Xo arbitrary, and define Xn for all n Z recursively by Xn+ l = g(xn ) for n >
0, and Xn - l = g - 1 (xn) for n < O. Let rl = (1+Ja 2 + 4b)j2 and r2 = (a-Ja2 + 4b)j2
be the roots of x2 - ax - b = 0, so that rl > 0 > r2 and 1 > I r ll > I r2 1 . Then there
exist C , C2 lR such that Xn clr� + c2 r� for all n E Z.
Suppose is strictly increasing. If C2 f:. 0 for some choice of xo, then Xn is
dominated by r� for n sufficiently negative. But taking Xn and X n+ 2 for n sufficiently
negative of the right parity, we get 0 < Xn < Xn+ 2 but g(xn) > g(Xn+ 2 ), contradiction.
Thus C2 = O. Since Xo = Cl and Xl Clrl, we have g(x) = rl X for all x. Analogously,
if is strictly decreasing, then C2 0 or else Xn is dominated by r� for n sufficiently
positive. But taking Xn and Xn+ 2 for n sufficiently positive of the right parity, we get
o < X n+ 2 < Xn but g(Xn+ 2 ) < g(xn), contradiction. Thus in that case, g(x) = r2 X for
all x.
(Titu Andreescu, The " William Lowell Putnam" Mathematical Competition,
hence there is
201
SOLUTIONS
then
for all real
f(y) f ( -y),
f(r) Ir l f (l),
=
y, hence
r.
f(x) I x l f (l)
(rn ) n->l
=
for all rational numbers
=
We prove that
for all real numbers Let be an arbitrary real
= Because
number and let
be a sequence of rational numbers with lim
Let g, :
be functions such that:
is a derivative;
is differentiable with a continuous derivative.
.
Then h is a derivative function.
Proof. Let H be an antiderivative of h and define u :
Then
u' (x) = g(x)h(x) + g' (x)H(x),
45.
= 1, . . . , p- = p- 2 and, for all n � p - 1,
is the least integer greater than
that does not form an arithmetic progression of length with any of the preceding
terms. Prove that, for all n,
is the number obtained by writing n in base - 1 and
reading it in base
32. The squares of a chessboard are randomly labeled from 1 to 64. On the first
63 there is a knight. After some moves, the 64's square, initially unoccupied, is also
unoccupied. Let n k be the square number of the knight who was initially on the k 's
square.
Prove that
63
L
k=l i nk - kl � 1984.
33. The Fibonacci sequence (Fn ) n2:: l is given by
Fl F2 1, Fn+2 = Fn+l +Fn ' n � 1.
Prove that
L'2n - F?n+2 +9 F?n-2 2 23n
for all n � 2.
34. Let n be a positive integer and let Nk be the number of increasing arithmetic
progressions with k terms from the set {I, 2, . . , n}.
=
a ,k such that In is continuous on lR*.
that In is a derivative function.
annsuch
q � 0 be integers. Find the numbers such that I : lR lR,
1 1
f (x) = { sm X cos x- ' ifif xx #= 0O
•.•
is a deri vati ve function.
.
p
_
.
q
Cp,q
{
0
G) ,
g(X) = fM(/),
x =O
p,q
-t
= Itllim-roo -lt iot I (x)dx = l iTo I(x)dx
O. Prove that
T
I : lR lR
: lR lR, g(x) = I /(x) 1 also a
60. If /l , /2 : lR lR are derivative functions then I = max{/l , /2 } is also a
59.
If
-t
is a derivative function, then is
derivative function?
derivative function?
=l
,
Y
I Y I -r oo Y
is a derivative function.
Let
-t
be a continuous function with a period T >
g(l)
lR,
=f.
.
-t
=l
Prove that
-t
n>
Prove that the function
n
233
PROBLEMS
-t
9
-t
SOLUTIONS
m functions h B B. Each of them can be extended in nn-m
ways to a function f : A A which satisfies f(B) � B.
nn -m .
Hence the required number is
(Dorin Andrica, Revista Matematidl Timi§oara (RMT) , No. 1-2(1981), 81,
Problem Cl: l )
2. Let Xq be a subset with q elements of X. Since Y has elements, it follows
that there are q! bijective functions
Xq Y. Each of them can be extended in
nn- q ways to a function f A A which satisfies Y � f(X).
The number of subsets Xq of X is (:) , hence the requested number is (:) nn - p .
Remark. If q consider (:)
1. There are
Since
=
it follows that f is bijective on There are
ways to a function f
and each of them can be extended in
desired number is
nn-k
---t
(Dorin Andrica)
4.
We consider two cases.
i) � m. Let
There is only one
be a subset with elements of the set
that has the property h (i) = i for all i E
This function can
---t
function h :
be extended in (m
ways to a function
---t B such that
f= i for all
iE
n
A \ Ak•
k
A
Ak Akk n- k
- l)
A.
f:A
Ak .
f(i)
Ak of A is (�), hence the desired number is (�)
l) n-k .
ii)
n. Let Bk be a subset with k elements of the set There is only one
function h : Bk Bk such that h(i) i for all i Bk . This function can be extended
in
- l) m- k ways to a function B B such that g(i) f= i for all i \
(m
The number of the subsets
m �
(m
---t
=
9
:
---t
235
E
B.
E
B
Bk •
23 6
mn-m ways to a function f : A
Moreover, each function 9 can be extended in
that clearly has exactly fixed points.
The problem reduces to finding the maximal value of
when
==
n and
2:: 0 are integers.
Note that if
then the absolute
is a triplet that maximizes
value of any difference of two numbers from this triplet is at most Indeed, assume
that
2:: and define
we may reduce the problem to the case when
Assume without loss of generality that
=
space are colored by three different colors. Find the maximum number of segments
AB such that A and B are different colors.
and
Romanian IMO Selection Test, 1982; Revista
Matematica Timi§oara ( RMT ) , No. 1 ( 1982), pp.
Problem
V5 + 1
V5 - 1
. easy to check that for these
two of them --- and the other two - --- . It 1S
2
2
numbers L a3 = O.
Romanian Mathematical Olympiad - final round, 2002)
(Titu Andreescu,
N
2 0 . We will say that a subset of
is p-progression-free if it does not con
tain an arithmetic progression of length p. Denote by b n the number obtained
by writing n in base p - 1 and reading it in base p. One can easily prove that
an = b n for all n = 0, 1, 2, . . . by induction, using the following properties of the
set B = {bo, b1 , . . . , b n , . . . } (whose proofs we postpone):
1 ° B is p-progression-free;
2° If b n -1 < a < b n for some n � 1, then the set {bo, b 1 , . . . , b n -1 , a} is not
p-progression-free.
Indeed, assume 1° and 2° hold. By the definition of ak and b k , we have a k = bk
for k = 0, 1, . . . , p - 2. Let al = b k for all k :::; n - 1, where n � p - 1. By 1°, the set
is p-progression-free, so an :::; b n - Also, the inequality an < b n is impossible, in view
of 2° . Hence an = b n and we are done.
So it suffices to prove 1° and 2°. Let us note first that B consists of all numbers
whose base p representation does not contain the digit p - 1 . Hence 1° follows from
the fact that if a, a + d, . . . , a + (p - l)d is any arithmetic progression of length p, then
all base p digits occur in the base p representation of its terms. To see this, represent
d in the form d = pm k, where gcd ( k,p) = 1 . Then d ends in m zeros, and the digit
o preceding them is nonzero. It is easy to see that if a is the ( m + l)st digit of a
(from right to left) , then the corresponding digits of a, a + d, . . . , a + (p - l)d are the
remainders of a, a + 0, . . . , a + (p - 1)0 modulo p, respectively. It remains to note that
a, a + 0, . . . , a + (p - 1 ) 0 is a complete set of residues modulo p, because 0 is relatively
prime to p. This finishes the proof of 1 ° .
We start proving 2° by the remark that b n- 1 < a < b n implies a (j. B. Since
B consists precisely of the numbers whose base p representations do not contain the
digit p - 1, this very digit must occur in the base p representation of a. Let d be the
249
6.2. SO LUTIONS
6. CO MPREHENSIVE PROBLEMS
248
number obtained from a be replacing each of its digits by 0 if the digit is not p - 1,
and by 1 if it is p - 1. Consider the progression
2 1 . i) For a bounded region M of the plane we denote by n (M ) the number of
points with nonnegative integral coordinates in M.
Function f is increasing and bijective, hence continuous . Consider the sets
1 . n 2 · n (k - l)n
-k-' -k-" ' " k
there are gcd(k, n) - 1 integers. Hence the open segment OAk does not contain points
with integral coordinates if and only if gcd( k, n) = 1 . There are <p(n) such numbers
and we are done.
Remark. An alternative version of this problem can be: " A hunter stays at the
point in a forest where the trees are placed at points with integral coordinates.
Deers stay at points Ao, Al, . . . ,A n -l . How many chances of success has the hunter?"
(Dorin Andrica)
0
3 1 . Assume by way of contradiction that the distance between any two points
is greater than or equal to 1. Then the spheres of radius 1/2 with centers at these
1981 points have disjoint interiors and are included in the cube of side 10 determined
by the six parallel planes to the given cube's faces and situated in the exterior at a
distance of 1/2. It follows that the sum of the volumes of the 1981 spheres is less than
the volume of the cube of side 10, hence
1981 ·
4rr o N
2
.
3 = 1981 � > 1000,
a contradiction. The proof is complete.
The pigeonhole principle does not help us here. Indeed, dividing each side
of the cube in [{f1981] = 12 congruent segments we obtain 123 = 1728 small cubes
of side � = � . In such a cube there will be two points from the initial 1981 points.
Remark.
12 4
3
�
The distance between them is less than J3 which is not enough, since 4 J3 > 1.
Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 68,
Problem 4627)
We prove that, for some labeling, = 1984. It is known that a knight
can pass through all the 64 squares of the board only once and then come back to the
initial square. Now label the squares from 1 to 64 in the order given by these knight
moves. The free position can be made successively 64, 1, 2, . . . , 63, 64, . . . so we can
reach the situation n l = 32, n 2 = 33, . . . , n3 2 = 63, n33 = 1, n 34 = 2, . . . , n6 3 = 31.
For this diagram we have = 1984.
Revista Matematica Timi§oara (RMT) , No. 2(1984), pp. 103,
Problem C6:6)
for all x
Consider a number x such that at least one of sin al x, sin a2X, . . . , sin anx is not
zero. Then the homogeneous system of linear equation (1) has a nontrivial solution,
hence the determinant is zero:
eR
a contradiction.
(Dorin Andrica)
46. We start with an useful lemma.
Lemma. If B is an irrational number, then the set
M = {mB + nl m,n integers}
is dense in JR.
Proof. We prove that in any open bounded interval J � JR\ {O} there is an element
of M, i.e . M -:f. 0. Let J be such an interval and without loss of generality consider
J (0, 00).
There is an integer n (J) such that
1
n(J) J (0, 1).
We consider two cases:
C
"I take great pleasure in recommending to all readers Romanians or from abroad - the book of professors Titu Andreescu
and Dorin Andrica. "
"All featu red problems are interesting, with an increased level
of difficulty; some of them are real gems that will give great
satisfaction to any math lover ... tďż˝
" One more time, I strongly express my belief that the 360
mathematics problems featured in this book will reveal the beauty of
mathematics to all students and it will be a guide to their teachers and
professors. "
Professor loan Tomescu
Department of Mathematics and Computer Science
University of Bucharest
Associate member of the Romanian Academy | 677.169 | 1 |
CREDIT EQUIVALENTS: MCR3UP - Functions, Grade 11, University Preparation (Enriched)Functions and Applications, Grade 11, University/College Preparation
MCF3MI
This course introduces basic features of the function by extending students' experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling real-world situations. Students will represent functions numerically, graphically, and algebraically; simplify expressions; solve equations; and solve problems relating to applications. Students will reason mathematically and communicate their thinking as they solve multi-step problems. MBF3CI - Foundations for College Mathematics is a similar course and you will not receive credit for taking both.
Foundations for College Mathematics
MBF3CICREDIT EQUIVALENTS: MBF3CE - eLearn - Foundations for College Mathematics is a similar course and you will not receive credit for taking both.
Mathematics for Work and Everyday Life, Grade 11, Workplace Preparation
MEL3EI
This course enables students to broaden their understanding of mathematics as it is applied in the workplace and daily life. Students will solve problems associated with earning money, paying taxes, and making purchases; apply calculations of simple and compound interest in saving, investing, and borrowing; and calculate the costs of transportation and travel in a variety of situations. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. | 677.169 | 1 |
על ידי Webrich Software Limited
תאור
TEAS Math Review app contains quick revision notes and tons of questions to help you preparing for the TEAS test and get into the nursing school of your choice.
This app covers 51 topics. Each topic has quick revision notes and multiple questions. Questions are similar to the ones asked in TEAS common entrance exams set by ATI. They vary in difficulty levels ranging from 1 to 10.
These 51 types are categorized as follows.
• Numbers & Operations
1. Order of Operations and PEDMAS
2. Multiplication and Division Problems
3. Decimals - Multiplication and Division 1
4. Decimals - Multiplication and Division 2
5. Approximation of Roots and Exponents
6. Improper Fractions and Mixed Numbers
7. Adding Fractions
8. Subtracting Fractions
9. Multiplying Fractions
10. Dividing Fractions
11. Fraction Problems in Words
12. Rational Numbers
13. Percent 1
14. Percent 2
15. Equivalent Percents Fractions and Decimals
16. Percent Problems in Words
17. Ratios and Fractions
18. Ratio Problems in Words
19. Direct Proportion
20. Best buys
21. Compound Interest
• Algebraic Applications
22. Addition and Subtraction of Algebraic Expressions
23. Multiplication of Algebraic Expressions
24. Algebraic Fractions
25. Translation into Algebraic Expressions
26. Evaluating Expressions 1
27. Evaluating Expressions 2
28. Linear Equations 1
29. Linear Equations 2
30. Linear Equations 3
31. Setting up Equations
32. Absolute Value
33. Inequalities
34. Solving Inequalities
35. Systems of Equations (Linear)
36. Setting up Systems of Equations
• Data Interpretation
37. Frequency Tables
38. Grouped Data
39. Two Way Tables
40. Box Plots
41. Histograms
42. Bar Graphs
43. Pie Charts
44. Line Graphs
45. Scatter Graphs
46. Basic Probability
47. Independent Events
• Measurement
48. Units and Measurements
49. Estimating measurements
50. Converting Units
51. Time and Timetables
DETAILED TEST RESULTS
On the completion of each test the user is given a summary of questions answered correctly, incorrectly and skipped along with the time taken on each question. Every question can be reviewed and all questions have an explanation of the answer.
PROGRESS METERMOCK TEST
After you have practiced all types of questions, you can take a Mock Test with 50 questions. The Mock Test has questions selected randomly from each topic. Scores for the mock tests taken in the past are displayed in a bar chart. This is a great way to track how you are progressing in mock tests. There is no limit on how many mock tests you can take.
FEATURE LIST
• Covers all 51 commonly found types of questions.
• 509 "ON" or "OFF" for sound effects.
• Option to choose reset the progress and start from scratch. | 677.169 | 1 |
PAP111/PHYS1A Tutorial Sheet 1 31stof August 2010 Claus-Dieter Ohl Math Math Math RefresherRefresherRefresherRefresherThis tutorial serves as a refresher for you to get started with the mathematics needed in the first course of your studies, Physics PAP111/PHYS1A.Some questions in the tutorials are simple, some are tough. Important is that you give all of them a try. In the tutorial the teaching assistants (TA) will discuss the questions with you. Apart from doing the homework, it is of second most importance to attend the tutorials (much more than anything else in your studies). Lecture may be more fun, but surely not as important as the a)homeworkand b)tutorial. Btw, some questions seem to be very easy, and some very hard. Sorry for that, I want to keep all of you entertained. At the end of this course I want to bring you all to a good level of understanding physical concepts. And (if you like it or not) mathematics is the language of physics. Therefore, LET'S GET STARTED:
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A proof of the technological know-how of numbers and area, with basic demonstrations of mathematics, algebra, trigonometry, arithmetic in tune and paintings, and together with a few tips and video games utilizing numbers.
Extra info for Math Contests for High School, Volume 2: School Years 1982-83 through 1990-91
Example text
A structured document can be modeled as a file-type property. Note that reusable documents can be put on the web and can be accessed by many assessment processes through using URLs of the web pages. 10 Modeling Activities In each emerging form of assessment, various activities are performed by diverse roles. e. , title, description, and completion). Most activities listed in the tables can be easily modeled in LD through specifying the values of attributes. For modeling some assessment activities, the question/questionnaire should be modeled as a QTI document as described above, which has to be referred to by an information item within the activity or in the associated environment.
2. Evidence gathering Candidates collect and classify evidence about their previous experience; Mentor checks the content of candidate's portfolio. 3. Assessment Assessors review the quality of a candidate's evidence using assessment standards and rubrics; Candidate receives a report that describes to what extend the candidate master the competences that are included in the competence profile. 2. Recognition Assessors compose APL certificate and send to candidate; Candidate send APL certificate to examination board; Examination board notify candidate about decision on study program reduction. | 677.169 | 1 |
CIE A Level Maths - Pure Mathematics 2 and 3 (International)
This book is part of a series of textbooks created for the new Cambridge International Examinations (CIE) mathematics syllabus. The authors have worked with CIE to assure that the content matches the syllabus and is pitched at a suitable level. Pure Mathematics 2&3 corresponds to syllabus units P2 and P3. The syllabus content is arranged in chapters to provide a viable teaching course. Each chapter starts with a list of learning objectives. Mathematical concepts, terminology and notation are explained clearly and carefully. Key results and procedures appear in boxes for easy reference. Stimulating worked examples take a step-by-step approach to problem solving. There are plenty of exercises throughout, as well as revision exercises and practice exam papers all written by experienced examiners. Contents: Introduction; 1. Polynomials; 2. The modulus function; 3. Exponential and logarithmic functions; 4. Differentiating exponentials and logarithms; 5. Trigonometry; 6. Differentiating trigonometric functions; Revision exercise 1; 7. Differentiating products; 8. Solving equations numerically; 9. The trapezium rule; 10. Parametric equations; 11. Curves defined implicitly; Revision exercise 2; Practice examinations for P2; 12. Vectors: lines in two and three dimensions; 13. Vectors: planes in three dimensions; 14. The binomial expansion; 15. Rational functions; 16. Complex numbers; 17. Complex numbers in polar form; 18. Integration; 19. Differential equations; Revision exercise 3; Practice examinations for P3; Answers; Index | 677.169 | 1 |
Place Value Homework
Can values operate in a world of facts and still be more than indifferent facts themselves? In this important and challenging book, Wolfgang Kohler's subject is value, or what he calls the "requiredness" of an object or activity. Starting with a descriptive account of values as we become aware of them, he finds that, inside certain contexts, parts of such structures do not appear as indifferent facts.
Part of the Comparative Work and Employment Relations series, Putting Labour in its Place is an edited collection, containing cutting-edge research and theoretical innovation on global value chains, the nature of work and labour process theory. It addresses the different processes around the world that each add value to the goods or services being produced; whilst also analysing the idea of labour itself and the exploitation surrounding it.Key benefits:• Written by leading international academics• A landmark text combining the growing interest in global value chains with labour process theory• Provides up-to-date critical analysis of global developments
The history of the world is outlined in this remarkable reference book, from the earliest settlements of around 9000BC right up to the turn of the millenium. Wars and revolutions, inventions and discoveries are presented chronologically to give the readers a full global overview of what was taking place at any given time. Over 3,500 dates make this title an ideal homework companion. Includes colourful reconstructions, photographs and maps. Lists kings, queens, presidents, prime ministers and rulers and includes extensive glossary and index. Excellent value in a substantial hardback. Over 300 pages long.
Aged thirteen, Tristan Bloch writes a love note to Calista Broderick. He thinks she is perfect. He wants to talk to her. He thinks he loves her. He could help her with her algebra homework. Cally shares the note with her best friend, Abigail, who insists that she shows it to her boyfriend Ryan, who decides to share it on Facebook: and then everyone sees it.Before long, Tristan takes a morning ride to the Golden Gate Bridge, leaves his bike on the against the rail, and jumps.Now, Tristan's classmates are seventeen, dealing with tests and affairs with teachers, pressure from parents and going to parties.These wealthy, privileged teenagers should be the happiest on earth. But the guilt of Tristan's death follows them all...
Does the sight of your child's maths homework fill you with dread? Do you look for any excuse when they ask you to explain equations, fractions or multiplication? Maths can often leave children - and parents - perplexed. How to do Maths so Your Children Can Too works through maths topics with a simple step-by-step approach, explaining the new ways of teaching maths that confuse so many parents. This book will show you how to: - Master 'number bonds' and 'number lines' - Divide by 'chunking' - Multiply using 'the grid method' - Work with fractions, percentages and ratios - Understand number and place value Bridging the gap between primary and secondary school - when children often struggle - and packed full of simple, accessible examples, this essential guide will banish your maths phobia and take the pain out of homework time. | 677.169 | 1 |
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About this title:
Synopsis: Part of the ST(P) graded series in mathematics, this book completes coverage of Levels 8 and 9 of the National Curriculum and contains exercises divided into three types. The first type is aimed at consolidation, the second is for extra practice or later revision, and the final type is of more demanding questions. Multiple-choice questions are included as a self-test to confirm understanding and as a basis for class discussions, and the book ends with some general revision exercises of examination-type questions.
Book Description Nelson Thornes, 1992. Paperback. Condition: Good. A few small marks to the page edges8901329
Book Description Paperback. Condition: Good. The book has been read but remains in clean condition. All pages are intact and the cover is intact. Some minor wear to the spine. Seller Inventory # GOR00068964933779 | 677.169 | 1 |
Brief description
Commonly known as 'rubber-sheet geometry', Topology is a subject, which has arisen essentially from abstracting certain characteristics of space, provides an extremely general framework that is used in many areas of science, including mathematics, physics and computer science. This module develops ideas found in earlier mathematical modules on analysis to provide a rigorous and, in a sense, elegantly simple set of concepts that will be extremely useful to those pursuing a career in science. | 677.169 | 1 |
This book originates as an essential underlying component of a modern, imaginative three-semester honors program (six undergraduate courses) in Mathematical Studies. In its entirety, it covers Algebra, Geometry and Analysis in One Variable. The book is intended to provide a comprehensive and rigorous account of the concepts of set, mapping, family, order, number (both natural and real), as well as such distinct procedures as proof by induction and recursive definition, and the interaction between these ideas; with attempts at including insightful notes on historic and cultural settings and information on alternative presentations. The work ends with an excursion on infinite sets, principally a discussion of the mathematics of Axiom of Choice and often very useful equivalent statements. Contents:SetsMappingsProperties of MappingsFamiliesRelationsOrdered SetsCompletely Ordered SetsInduction and RecursionThe Natural NumbersFinite SetsFinite SumsCountable SetsSome Algebraic StructuresThe Real Numbers: Complete Ordered FieldsThe Real Number SystemThe Real Numbers: ExistenceInfinite Sets Readership: Undergraduate and graduate students in mathematics; Mathematicians. Key Features:Comprehensive and rigorous in its coverageProvides alternative insights on concepts and definitionsProvides a list of sections with some unusual but insightful approachesKeywords:Sets;Mappings;Families;Order;Natural Numbers;Inductive Proofs;Recursive Definitions;Real Numbers
A new and unique way of understanding the translation of concepts and natural language into mathematical expressions Transforming a body of text into corresponding mathematical expressions and models is traditionally viewed and taught as a mathematical problem; it is also a task that most find difficult. The Language of Mathematics: Utilizing Math in Practice reveals a new way to view this process—not as a mathematical problem, but as a translation, or language, problem. By presenting the language of mathematics explicitly and systematically, this book helps readers to learn mathematics¿and improve their ability to apply mathematics more efficiently and effectively to practical problems in their own work. Using parts of speech to identify variables and functions in a mathematical model is a new approach, as is the insight that examining aspects of grammar is highly useful when formulating a corresponding mathematical model. This book identifies the basic elements of the language of mathematics, such as values, variables, and functions, while presenting the grammatical rules for combining them into expressions and other structures. The author describes and defines different notational forms for expressions, and also identifies the relationships between parts of speech and other grammatical elements in English and components of expressions in the language of mathematics. Extensive examples are used throughout that cover a wide range of real-world problems and feature diagrams and tables to facilitate understanding. The Language of Mathematics is a thought-provoking book of interest for readers who would like to learn more about the linguistic nature and aspects of mathematical notation. The book also serves as a valuable supplement for engineers, technicians, managers, and consultants who would like to improve their ability to apply mathematics effectively, systematically, and efficiently to practical problems.
The Joy of Finite Mathematics: The Language and Art of Math teaches students basic finite mathematics through a foundational understanding of the underlying symbolic language and its many dialects, including logic, set theory, combinatorics (counting), probability, statistics, geometry, algebra, and finance. Through detailed explanations of the concepts, step-by-step procedures, and clearly defined formulae, readers learn to apply math to subjects ranging from reason (logic) to finance (personal budget), making this interactive and engaging book appropriate for non-science, undergraduate students in the liberal arts, social sciences, finance, economics, and other humanities areas. The authors utilize important historical facts, pose interesting and relevant questions, and reference real-world events to challenge, inspire, and motivate students to learn the subject of mathematical thinking and its relevance. The book is based on the authors' experience teaching Liberal Arts Math and other courses to students of various backgrounds and majors, and is also appropriate for preparing students for Florida's CLAST exam or similar core requirements. Highlighted definitions, rules, methods, and procedures, and abundant tables, diagrams, and graphs, clearly illustrate important concepts and methods Provides end-of-chapter vocabulary and concept reviews, as well as robust review exercises and a practice test Contains information relevant to a wide range of topics, including symbolic language, contemporary math, liberal arts math, social sciences math, basic math for finance, math for humanities, probability, and the C.L.A.S.T. exam Optional advanced sections and challenging problems are included for use at the discretion of the instructor Online resources include PowerPoint Presentations for instructors and a useful student manual
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's BASIC MATHEMATICS FOR COLLEGE STUDENTS, Fourth language of mathematics,The purpose of this book is to provide the student beginning undergraduate mathematics with a solid foundation in the basic logical concepts necessary for most of the subjects encountered in a university mathematics course. The main distinction between most school mathematics and university mathematics lies in the degree of rigour demanded at university level. In general, the new student has no experience of wholly rigorous definitions and proofs, with the result that, although competent to handle quite difficult problems in, say, the differential calculus, he/she is totally lost when presented with a rigorous definition oflimits and derivatives. In effect, this means that in the first few weeks at university the student needs to master what is virtually an entire new language {'the language of mathematics'} and to adopt an entirely new mode ofthinking. Needless to say, only the very ablest students come through this process without a great deal of difficulty.
This text in basic mathematics is ideal for high school or college students. It provides a firm foundation in basic principles of mathematics and thereby acts as a springboard into calculus, linear algebra and other more advanced topics. The information is clearly presented, and the author develops concepts in such a manner to show how one subject matter can relate and evolve into another.
Master Your Coursework with Collins College Outlines From fraions, decimals, and geometric measurement to exponents, scientific notation, and an introduion to algebra, the Collins College Outline in Basic Mathematics explores and explains the topics that students will find in introduory mathematics classes. Completely revised and updated by Dr. Lawrence Trivieri, Basic Mathematics includes a test yourself seion with answers and complete explanations at the end of each chapter. Also included are bibliographies for further reading, as well as numerous graphs, charts, and example problems. The subje for college students and, where appropriate, Advanced Placement students. Each Collins College Outline is fully integrated with the major curriculum for its subje and is a perfe supplement for any standard textbook.
Salient Features * As Per Ii Puc Basic Mathematics Syllabus Of Karnataka. * Provides An Introduction To Various Basic Mathematical Techniques And The Situations Where These Could Be Usefully Employed. * The Language Is Simple And The Material Is Self-Explanatory With A Large Number Of Illustrations. * Assists The Reader In Gaining Proficiency To Solve Diverse Variety Of Problems. * A Special Capsule Containing A Gist And List Of Formulae Titled 'Remember! * Additional Chapterwise Arranged Question Bank And 3 Model Papers In A Separate Section---'Examination Corner'.
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's BASIC COLLEGE MATHEMATICS WITH EARLY INTEGERS, Fifth "language of mathematics,"Whether you are returning to school, studying for an adult numeracy test, helping your kids with homework, or seeking the confidence that a firm maths foundation provides in everyday encounters, Basic Maths For Dummies, UK Edition, provides the content you need to improve your basic maths skills. Based upon the Adult Numeracy Core Curriculum, this title covers such topics as: Getting started with the building blocks of maths and setting yourself up for success Dealing with decimals, percentages and tackling fractions without fear Sizing Up weights, measures, and shapes How to handle statistics and gauge probability Filled with real-world examples and written by a PhD-level mathematician who specialises in tutoring adults and students, Basic Maths For Dummies also provides practical advice on overcoming maths anxiety and a host of tips, tricks, and memory aids that make learning maths (almost) painless - and even fun general points, but I have not discussed it for its own sake.
The Language of Mathematics was awarded the E.W. Beth Dissertation Prize for outstanding dissertations in the fields of logic, language, and information. It innovatively combines techniques from linguistics, philosophy of mathematics, and computation to give the first wide-ranging analysis of mathematical language. It focuses particularly on a method for determining the complete meaning of mathematical texts and on resolving technical deficiencies in all standard accounts of the foundations of mathematics. "The thesis does far more than is required for a PhD: it is more like a lifetime's work packed into three years, and is a truly exceptional achievement." Timothy Gowers
Learn BASIC Programming for Mathematics V10 Home Study is a more interesting way to learn maths? Here you can learn Maths through writing simple BASIC Programming code. Based on a series of lecturers given at a Higher Education College where the author was able to generate a lasting understanding and interest in mathematics for adults in particular through teaching not maths, but BASIC language Computer Programming. Students became hooked and were often found long after the class had ended still working on their assignments and producing work that was far in excess of what was required. Using a programming style very similar to the BBC Micro computers of many years ago users are able to write simple (or complex) computer programs based on the VbScript language (a cut down version of Visual Basic) but with a user interface resembling BBC BASIC. The software is designed to work with any standard textbook where students can explore a problem and learn by programming it to find the solution usually not requiring much more than one line of code that can be checked against a given answer. Resulting in a far greater understanding than traditional methods of teaching maths. eBook and educational software package at a tiny fraction of the previously published price. Table of Contents, (Software 250 interactive topics) Introduction, Number Systems, Number Conversion, Number Types, Compound Measures, Roots, Angles and Parallels, Triangle Ratios, Triangle Angles, Percentages, Ratios, Fractions, Vectors, Geometry, Circle Angles, Area, Surface Area and Symmetry, Volume, Laws, Algebra 0., Algebra 1., Algebra 2., Mathematical Rules, Powers and Indices, Simplifying, Linear Equations, Graphing, Slope and Translation, Curves and Angle Conversion, Personal Finance, Data Analysis, Binary Numbers, Binary Arithmetic, Additional Notes general points, but I have not discussed it for its own sake.
Based on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students. This superb book offers students an excellent opportunity to strengthen their mathematical skills by solving various problems in differential calculus. By covering material in its simplest form, students can look forward to a smooth entry into any course in the physical sciences.
Boost students' mathematics
The Bittinger System for Success, Make it Work for You! Building on its reputation for accurate content and a unified system of instruction, the Tenth Edition of the Bittinger paperback series integrates success-building study tools, innovative pedagogy, and a comprehensive instructional support package with time-tested teaching techniques. | 677.169 | 1 |
Students will learn more advanced methods in algorithms and see how these algorithms apply in computer science as well as in everyday life. Algorithmic thinking is a fundamental requirement for the USA Computing Olympiad. It builds the foundation for studying and excel in the AP computer science courses. Algorithms will be presented in pseudo-code, and implementations in Python or other popular languages will be covered. Topics will include algorithms on graphs, dynamic programming, greedy algorithms, and algorithms in number theory. Other topics may be covered based on teacher specialization. This course is a natural continuation from Introduction to Algorithms, but it is not necessary to have taken Introduction to Algorithms before taking this course. | 677.169 | 1 |
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From the original hard cover edition:In the modern age of almost universal computer usage, practically every individual in a technologically developed society has routine access to the most up-to-date cryptographic technology that exists, the so-called RSA public-key cryptosystem. A major component of this system is the factorization of large numbers into their primes. Thus an ancient number-theory concept now plays a crucial role in communication among millions of people who may have little or no knowledge of even elementary mathematics. Hans Riesel's highly successful first edition of this book has now been enlarged and updated with the goal of satisfying the needs of researchers, students, practitioners of cryptography, and non-scientific readers with a mathematical inclination. It includes important advances in computational prime number theory and in factorization as well as re-computed and enlarged tables, accompanied by new tables reflecting current research by both the author and his coworkers and by independent researchers. The book treats four fundamental problems: the number of primes below a given limit, the approximate number of primes, the recognition of primes and the factorization of large numbers. The author provides explicit algorithms and computer programs, and has attempted to discuss as many of the classically important results as possible, as well as the most recent discoveries. The programs include are written in PASCAL to allow readers to translate the programs into the language of their own computers. The independent structure of each chapter of the book makes it highly readable for a wide variety of mathematicians, students of applied number theory, and others interested in both study and research in number theory and cryptography | 677.169 | 1 |
📙 Problem-solving and selected topics in number theory: In the spirit of the mathematical olympiads by Michael Th. Rassias (auth.) — free pdf
This book is designed to introduce some of the most important theorems and results from number theory while testing the reader's understanding through carefully selected Olympiad-caliber problems. These problems and their solutions provide the reader with an opportunity to sharpen their skills and to apply the theory. This framework guides the reader to an easy comprehension of some of the jewels of number theory The book is self-contained and rigorously presented. Various aspects will be of interest to graduate and undergraduate students in number theory, advanced high school students and the teachers who train them for mathematics competitions, as well as to scholars who will enjoy learning more about number theory. Michael Th. Rassias has received several awards in mathematical problem solving competitions including two gold medals at the Pan-Hellenic Mathematical Competitions of 2002 and 2003 held in Athens, a silver medal at the Balkan Mathematical Olympiad of 2002 held in Targu Mures, Romania and a silver medal at the 44th International Mathematical Olympiad of 2003 held in Tokyo, Japan.
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Friday, April 24, 2009
Calculus is widely used by engineers and scientists to analyse practical problems. One common approach is to analyse a particular system using fundamental physics for a particular geometry (e.g. a force balance around a spherical particle falling in a liquid) to form equations. These equations are than either integrated or differentiated to produce useful relationships for a given set of boundary conditions (e.g. settling time of a particle as a function of size and density for a given initial particle velocity). The success of this approach normally depends on the nature of the phenomena being studied (some very chaotic and/or highly non-linear phenomena are difficult to model), the assumptions made in setting up the problems and the difficulty in solving the equations formed. Often, numerical techniques are used to find solutions to these equations and any good engineering mathematics course teaches a range of relevant numerical techniques to differentiate and/or integrate equations that are either difficult or impossible to solve directly.
Another interesting application of calculus is to analyse data. Consider a set of data collected in an experiment ..... imagine we are measuring X and Y simultaneously. When we plot X against Y, the curve generated may clearly show a relationship exists but the relationship is not simple or immediately apparent. A very simple method to start analysing this mysterious relationship, is to differentiate the X Y plot numerically (i.e. calculate the slope at points along the curve) and form a new plot of dX/dYvrs X. Now remember that we differentiate particular functions, new very specific relationships are formed. For example, differentiating a trigonometric function will generate another trigonometric function, and in the case of simple trigonometric functions like sine and cosine, functions are formed that have very specific geometric relationships to the original functions (e.g. cosine has the same shape and periodic form of sine but is "out of phase" with that relationship). In the case of polynomials, differentiating produces a function of lower order; the slope of a cubic follows a parabolic relationship, the differential of a parabolic functions produces a linear function and so on. This means that by differentiating a curve (i.e. measuring the slope of the curve at each point) some of these underlying relationships in the data maybe revealed.
This approach can be extended to differentiating the dX/dY curve formed, as double differentiation also can unlock some underlying relationship For example, differentiating sinX will form cosX and differentiating that relationship will produce a negative version of the original relationship. Double differentiation of a cubic function will generate a linear function (try it !). Thus, the "slope of the slope" can potentially tell alot about the original relationship. This line of attack can be extended to integration, through measuring the area under the Y curve and plotting this relationship against X. Of course, both taking the slope and measuring the area can be used in combination to tackle the problem.
The beauty of this methodology is that the procedure is very simple (e.g. measuring a slope of a line) and easily automated. You can try it yourself .... I suggest asking a mathematically inclined friend to dream up a complex function that is the combination of well known simple functions (e.g. cosx + x^3 + exp(x)), get him or her to form an x y table of values from this relationship and than ask you to derive the underlying relationship from this data set. The detective job in front of you is made simple by modern graphical/CAS calculators that allow ready numerical differentiation and integration of curves. Sometimes, a combination of intuition, luck and insight is required to identify the underlying relationship but the journey is normally fun. Try it !!!
Thursday, April 23, 2009
Just recently, I went through the standard derivation of the fundamental theorem of calculus with my students ..... forming tangent lines to a curve, calculating the gradient of that line using an increment, taking the increment towards infinity than repeating similar arguments for the area under a curve before forming the wonderful conclusion that the mathematics of calculating an area under a curve is the reverse of the process for calculating the gradient of a curve. In short, if you understand the mathematics of change, you also understand the mathematics of accumulation and vice versa. This was the brilliant insight that both Newton and Leibniz claimed as their own in the 17th century and formed the basis of the field we know as "Calculus".
This derivation is rightly considered one of the great mathematical breakthroughs of all time and its conclusions are indeed far reaching. During the lecture, I presented the orthodox view that Newton and Leibniz are the great intellectual heros of this breakthrough with a nod of appreciation to ancient Greeks like Archimedes who developed integral calculus via the method of exhaustion. As I was going through these arguments, I found myself questioning this idea of Newtons and Liebniz's pivotal role in the development of calculus. Wasn't the real breakthrough the idea that if you take an increment and imagine it decreasing towards infinity, you can drive useful geometrical relationships ? Isn't that idea, which I think we can accredit to Archimedes, the real intellectual breakthrough ? If you know that idea and have the tools of Cartesian co-ordinates (thank you Descartes !), than won't the relationships that Newton and Leibniz formed eventually fall out ?
Even as I write these heretical ideas down I feel my inner critic saying "No, these ideas only seem obvious because of the brilliant insights of Newton and Leibniz !" That may be true but historians of mathematics writing on calculus have shown that calculus quickly formed as a field after the developments in algebra instigated by Descartes and other mathematicis just proceeding Newton and Descartes. It is also acknowledged that Barrow (Newton's teacher at Cambridge) had an early form of differential calculus before Newton (see for an excellent overview of his ideas). After consulting my inner critic, I think the view I am forming can be expressed as follows: understanding the importance of taking increments towards zero was a great intellectual breakthrough that allowed the development of calculus, simplifying algebra through the Cartersian co-ordinates provided wonderful tools by which to understand the mathematics of change and accumulation and the derivation of calculus by Newton and Leibniz represent the accumulation of this intellectual development. In short, their intellectual insights owe a great deal to Archimedes, Descartes and Barrow.
One of the interesting observation one can make from these discussions is that the way calculus is taught follows a very different route from its historical development. At high schools, we indoctrinate students in algebra, than introduce differential calculus and limits, and than form integral calculus. In history, calculus was formed in almost the opposite order. I suppose, as long as you understand the key intellectual points underpinning calculus, it doesn't really matter in which order you have learn't them.
Thursday, April 9, 2009
I started high school in 1973, three years after the end of the Beatles and a generation before the end of the cold war. Everybody wore their hair long, ludriciously wide ties were considered fashionable, most engineers (like my father) owned a slide rule and very simple electronic calculators were starting to become affordable. I remember my brother saving up several weeks of his paper round money to purchase a calculator with a square root button. The arrival of this calculator at our high school caused a sensation and my brother was asked to demonstrate this technological marvel to the headmaster. With the arrival of even more powerful devices throughout that decade, my brother and myself, and everybody else studying mathematics in the Western world, continued to be trained in the use of log tables for carrying out any calculation beyond 687 x 6578. I think the last time I used a log table Ronald Regan hadn't yet become president and computer programs were typed on cards and processed overnight.
During this time, serious letters to the papers and educational experts lamented the fall in educational standards, my year 10 geography teacher warned that global warming would see Sydney under a foot of water by 2000 and there was a general feeling with anyone over the age of 40 that using calculators was "cheating".
By the end of the 1970s and into the early 80s, calculators had advanced quickly and a range of programmable calculators were on offer. In this enlightened era, engineering students tended to be either "HP" or "Casio" adherents, though a few perverse souls identified with the reverse polish notation of the "TI" calculators. I remember quite distinctly slaving away on my Casio programmable calculator with its gigantic 2k of memory, writing quite intricate programs with the line numbering system of level 2 basic, a cute plug in ticker tape printer and an audio tape memory system. Armed with this calculating power, you felt that you could conquer the world or at least complete a pressure drop calculation for a piping system in under 10 minutes. Part of me (a very small part) still hankers for the happy chatter of my ticker tape Casio printer and the amazingly clunky graphics produced from this device. By this time, the scientific calculators familiar with modern students became standard and knowledge of the workings of a slide rule suggested either a perverted soul or a person lost in the past.
The calculator was here to stay ! My arrival in the Engineering profession coincided with the great personal computer revolution and in my own small way I lead the charge, using computer programs (now written in "high" level languages like GW Basic !!) to perform complex engineering calculations that had formerly been the province of "look up" tables and approximate solutions. Even with this shift towards computing, my scientific calculator (still a Casio man) was used on a daily basis. However, by this time my career had taken a sharp turn towards research and the graphics calculator revolution bypassed me, as I was knee deep in numerics, computational thermodynamics and writing unruly "programs" in Excel. It was only when I took my current position that I was handed my first graphics calculator. It was love at first sight ! I love the fact that I can "see" the solution of an equation, that I can calculate derivatives and integrals and even form the ABC TV symbol using parametric graphics. What is there not to love ! I even accepted the transition from being a Casio man to a TI man without suffering a nervous breakdown (OK I had a little therapy).
Interestingly, serious people are still lamenting the falling of educational standards, predicting that Sydney will be under a metre of water by ......, and most people over 40 think that using a CAS calculator is cheating.
Friday, April 3, 2009
The engineering mathematics course at Swinburne is typical of most engineering mathematics courses around the world, in that, there is a heavy emphasis on the use of functions in analysing the physical world. In particular, there is underlying assumption (often unstated) that we can deal with physical data as a continuum (e.g. analysing radioactivity measurements using exponential functions). It is this assumption that underpins the "classical" paradign of engineering mathematics, which I would describe as:
A. analyse the physical relationships of the system being studied (e.g. force balance of a particle),
B. form equations that reflect these relationships, making appropriate simplifciations and assumptions (e.g. particle is spherical),
C. solve these equations for a given set of boundary conditions or limitations using either analytical or numerical techniques, and
D. analyse the solutions obtained against physical data, returning to first two steps if the solutions obtained are inaccurate or not credible.
This approach, and many subtle variations, has proved to be very powerful in analysing engineering problems, though complex systems where subtle changes in geometry and boundary conditions can produce large variations in behaviour (e.g. turbulence in fluids, movement of fine particles and "chaotic systems" in general) have proved difficult to model using this approach. Stephen Wolfram, in his book "A New Kind of Science" (2002) (see argued that the classical approach was fundamentally flawed and need replacing with a new approach called "cellular automata". At the heart of Wolfram's claims was this central observation:
All of our measurements of the world are made discretely, that is, we obtain discrete numbers from our instruments (e.g. the temperature measurement from a thermometer) including our senses, and artificially impose continuous relationships upon the world by forming equations around fundamentally discrete phenomena. We could more easily, and naturally, use discrete mathematical models to describe the physical world and dispense with the classical approach.
Quite a claim !! As you imagine this book caused much debate, some of it polite and in some cases, quite inpolite ! You might find the overheads of a lecture I gave on the book interesting - see - and there are literally hundreds of sites on the web discussing this book. You may also interested to read a much earlier (and more modest) version of the same idea by Konrad Zuse (1910-1995) who published "Computing Space" in 1967. An English translation of this pioneering work on "digital physics" is available at Zuse was also an early pioneer in the development of the computer and was, clearly, a highly imaginative and interesting thinker.
I think the claims, details and repercussions of Wolfram's claim are a bit detailed to discuss here but I do think the first part of his central claim is uncontroversial, that it, the measurements we make of the world are discrete and the equations we impose on this discrete data reflect our intellectual choices not an underlying physical connection between equations and nature (i.e. cannon balls do not have a parabolic equation written into their structure, it is "us" that chooses a parabola to model the motion of the ball). I think this is underlying assumption to appreciate as we continue along our path of differentiating/integrating/ etc. continuous functions to describe the physical world. | 677.169 | 1 |
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Calculus 1 & 2
A Guided Approach
A .A. Frempong
This book can be downloaded and read in iBooks on your Mac or iOS device.
Description
Calculus 1 & 2 covers the following topics: differentiation and integration of functions using a guided and an analytical approach. All the normally difficult to understand topics have been made easy to understand, apply and remember. The topics include continuity, limits of functions; proofs; differentiation of functions; applications of differentiation to minima and maxima problems; rates of change, and related rates problems. Also covered are general simple substitution techniques of integration; integration by parts, trigonometric substitution techniques; application of integration to finding areas and volumes of solids. Guidelines for general approach to integration are presented to help the student save trial-and-error time on examinations. Other topics include L'Hopital's rule, improper integrals; and memory devices to help the student memorize the basic differentiation and integration formulas, as well as trigonometric identities. This book is one of the most user-friendly
calculus textbooks ever published. | 677.169 | 1 |
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The Laplace and z transform methods are useful for simplifying the analysis and design of linear time-invariant systems in continuous time and in discrete time, respectively. For example, considering a set of differential equations representing a system in continuous time, Laplace transform theory simplifies the problem by replacing the given set of differential equations by a set of algebraic equations.
In this appendix we will consider only linear time-invariant dynamic systems. For a more detailed study of the Laplace, z, and related transforms such as the Fourier and starred Laplace transforms ...
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chapters in Clinical Calculations, 7th Edition, by Joyce Kee and Sally Marshall, Drug Calculations Online covers all four major drug calculation methods to help you learn to calculate drug dosages quickly and accurately. First you complete a reading assignment from the text, and then online lesson modules provide opportunities for application and practice. Animations, case studies, practice problems, narrated examples, and self-assessment activities reinforce understanding. Using a step-by-step approach to explain the ratio and proportion, formula, fractional equation, and dimensional analysis methods, Drug Calculations Online makes it easy to master each type of dosage calculation.
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Explanations of all four major methods of drug calculation (ratio & proportion, formula, fractional equation, and dimensional analysis) allow you to choose your preferred method.
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Glencoe Algebra 2 Worksheet Answers
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The Axioms Of Descriptive Geometry
Publisher: Cambridge University Press1914 ISBN/ASIN: B007NYGORY Number of pages: 96
Description: In the present tract, after the statement of the axioms, the ideas considered are those concerning the association of Projective and Descriptive Geometry by means of ideal points, point to point correspondence, congruence, distance, and metrical geometry.
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College algebra homework helper 2018-01-24 22:28:18
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Real Estate Math
Written by Rylee Newton
If you're going to take the real estate license exam in your state you're going to have to learn real estate math. This is a crucial part of the examination process, and you won't pass the test without understanding the formulas and applying them properly. You can find a number of real estate education books to help you with the math, as well as a number of excellent websites that can provide you with sample problems and solutions.
If real estate math isn't your strong suit, you can get a grip on it with some practice. A trip to your local bookstore can provide you with a number of tools from today's top educational publishers. These books offer you sample problems, and explanations. Part of understanding the math is learning how the formulas were derived.
Solve Your Problems with Real Estate Math
A real estate website can provide you with sample real estate math problems from the actual state exam. Nothing helps you learn like repetition. You can use these sample tests to determine the areas where you need to improve. Many websites provide you with detailed information for how to answer some of the most difficult math problems.
Passing the real estate exam is already very difficult. Some studies have found that as many as half of the people who take the test every year don't pass the test on their first try. Many people fail the test when they have to perform the complicated math problems involved in the testing process. Real estate agents and brokers use this math in their daily operations. Once you get the hang of it you can solve any mathematical problems that may arise in your career. | 677.169 | 1 |
The Official SAT Question of the Day
Monday, November 29, 2010
Precalculus is heading into a fast and furious review of rational functions! Please preview and review both the reference sheets as well as the powerpoint presentation below.
Powerpoint:
Rational Functions Reference Sheet:
Homework:
Bonus:
Algebra II will continue to deepen their interaction with quadratics and expand their problem-solving repertoire. We will add both the quadratic formula and graphing to our solving skill-set. The little picture below is the summary of the entire week, graphically speaking. If you can understand this little picture then you've got it!
Tuesday, November 23, 2010
I hope everyone had a relaxing long weekend off and are back ready to work!
Don't forget bonus is due before we leave on holiday break! Try any of the problems and remember to cite any resources you used to solve the problems.
This Wednesday is World Aids Day and we will spend about 25 minutes on Wednesday looking at the data on HIV worldwide. Here is the link to the video that will inform our conversation: HIV Worldwide Video
At this point in Algebra II our exploration of quadratics, students should be comfortable graphing and interpreting quadratic equations in multiple forms (standard, vertex, and intercept) as well as the process of completing the square. During week 15 we will complete our look at factoring and begin to solve quadratics by factoring and completing the square. Their are videos below that review previous topics as well as preview the upcoming factoring techniques we will cover in the coming week.
The homework for this week has already been given, but is included below as well. In addition, I have posted both the factoring decision tree as well as a cheat sheet that was compiled by Paul Dawkins for you use as reference sheets.
Precalculus will be examined Wednesday of this week over polynomials. Thursday we will begin a brief unit on rational functions, with particular attention to their graphs. Please find the homework for the week below as well as a reference sheet on polynomials.
Tuesday, November 16, 2010
Tuesday, November 9, 2010
We have officially moved into Quarter 2 and all of you (students) have a clean slate. It is important to remember what that means for you! Your grades from first quarter do not carry over to second quarter and that whatever you earn from this point forward will only average into your second quarter grade. Whether you have high or low first quarter marks, second quarter is a whole new world.
This week we will not have class on Wednesday for parent-teacher conferences and Thursday I will be away presenting at Amistad-Elm City High School. Thursday will be a day for independent practice and should be seen as an opportunity for much needed practice.
Algebra II started with some introductory descriptions of quadratic functions last week and are moving onto how to manipulate functions algebraically and how algebraic moments appear graphically. We will progress through the guided notes embedded below and as we explore them together I will post screenshots of our progress! . For Day 1 and 2 notes filled-in click Reference sheets:
Homework:
Precalculus will be continuing and finalizing their review of polynomial functions. We will look at college textbooks and their treatment of the topic as well as curve-fitting using polynomial regression. It is a short week so it will be a quick look. Powerpoints:
Tuesday, November 2, 2010
Precalculus continues their mastery of polynomials, with specific attention to roots and zeros of poly's. Here is a pic of a quartic polynomial, one of the polynmials will be looking at this week. If you have a chance, please pre-read the slides for polynomials. It will help you understand the material SO much more the coming week. You should start with 3.5 and finish with 3.8
Powerpoint:
Reference Sheets:
Classwork:
Homework:
____________________________________________________________________ In Algebra II we will have an exam on Tuesday involving both the raw mathematics around solving systems of linear equations as well as traditional applications of systems. This exam, like those previous, will require you to understand systems numerically, graphically, and algebraically.
We will be starting our next polynomial function, quadratics, on Wednesday. Here is a pic of the graph of a quadratic called a parabola. Please find the short powerpoint on the graphs of quadratics, a reference sheet (we will complete it on Wednesday and I'll post the completed version then), as well as the homework! If you want to preview the upcoming material on quadratics the link below takes you to an excellent online text that describes them in detail! | 677.169 | 1 |
Your Mathematics Standards Companion, Grades 6-8
by
Ruth Harbin Miles
Publisher:
Corwin
Print ISBN:
9781506382258, 1506382258
eText ISBN:
9781544304878, 1544304870
Edition:
1st
Copyright year:
2018
Your Mathematics Standards Companion, Grades 6-8 1st Edition by Ruth Harbin Miles and Publisher Corwin. Save up to 80% by choosing the eTextbook option for ISBN: 9781544304878, 1544304870. The print version of this textbook is ISBN: 9781506382258, 1506382258. | 677.169 | 1 |
Solution Techniques for Elementary Partial Differential Equations, Third Edition remains a top choice for a standard, undergraduate-level course on partial differential equations (PDEs). Making the text even more user-friendly, this third edition covers important and widely used methods for solving PDEs. New to the Third Edition * New sections on the series expansion of more general functions, other problems of general second-order linear equations, vibrating string with other types of boundary conditions, and equilibrium temperature in an infinite strip * Reorganized sections that make it easier for students and professors to navigate the contents * Rearranged exercises that are now at the end of each section/subsection instead of at the end of the chapter * New and improved exercises and worked examples * A brief Mathematica(R) program for nearly all of the worked examples, showing students how to verify results by computer This bestselling, highly praised textbook uses a streamlined, direct approach to develop students' competence in solving PDEs. It offers concise, easily understood explanations and worked examples that allow students to see the techniques in action | 677.169 | 1 |
Circuit Training - Multiplying Polynomials (algebra)
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Product Description
This 20-question circuit builds from multiplying a monomial with a binomial to multiplying binomials, expanding a binomial squared, multiplying a binomial with a trinomial and even multiplying trinomials. Difference of Squares and Sum/Difference of cubes are here in disguise.
Enjoy the sounds of your students as they work through this circuit!
The answers are embedded in the circuit; it is how the students move to the next problem. No answer key is needed -- all the teacher needs to do is work the circuit ahead of the students. | 677.169 | 1 |
Course Description
This course has been designed for students who wish
to take calculus, but who are not prepared for college
calculus based on our mathematics placement exam and
department recommendation. Topics will include
fundamental concepts of college algebra, pre-calculus,
and a preparation for calculus. More specifically, the
topics will include factoring, integer and rational
exponents, simplifying algebraic expressions, function
notation, polynomial and rational functions. While
many of the topics covered are similar to those in a
typical college pre-calculus course, there is more
theoretical coverage and emphasis, a much faster pace
is maintained, a greater depth of understanding is
required, and additional material on applications is
taught. 4 credits.
Prerequisites: MATH 112 or equivalent.
As stated above, this course is designed to help
students fill in knowledge gaps/holes before enrolling
in college calculus. It will move at a fast pace and
assume students have taken one or more classes covering
intermediate algebra and trigonometry prior to this
course.
Resources
Textbook: Precalculus Essentials,
4th Edition, by Robert Blitzer.
Calculator: Not required. Calculator usage
will be prohibited for all in-class quizzes, Exam 1,
Exam 2, and Exam 3. Non-graphing (scientific)
calculators will be permitted for sections of Exam 4
and the Final Exam. Graphing calculators will not be
permitted on any exams or quizzes.
Office Hours: The perfect opportunity to
ask general questions about course material, specific
questions about homework problems or in-class
examples, questions about your grade, questions about
majoring in math or information technology, etc. My office
location and office hour schedule are on my home page. You can also e-mail
me to set up an appointment outside of office
hours if that suits your schedule better.
(FREE) Tutoring: Free tutoring sessions
sponsored by the math department are held several
nights per week. Tutors will be able to help you
with questions about course content and approaching
homework problems. Check out the website
for schedule, location, and other pertinent
information.
Grading Information
Your final grade in the course is the weighted
average of three components: Quiz Grade, In-Class Exam
Grades and the Final Exam.
If you have a weighted average of at least...
You will earn a(n):
94%
A
90%
B+
84%
B
78%
C+
70%
C
65%
D+
60%
D
Quizzes are held every Friday in class, beginning
September 4. There will be 12 quizzes. Your lowest
two quiz scores will be "dropped." The average of
your non-dropped quiz grades is 20% of your final
course grade. Eight of the 12 quizzes will be focused
on reinforcing algebra topics. The four exceptions
are noted in the syllabus.
Each in-class exam is 15% of your final
course grade.
The final exam is 20% of your final course grade.
If your final exam score is higher than your lowest
in-class exam score, it will replace that exam. | 677.169 | 1 |
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A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for more than three decades. Nathan Jacobson's books possess a conceptual and theoretical orientation; in addition to their value as classroom texts, they serve as valuable references. Volume II comprises all of the subjects usually covered in a first-year graduate course in algebra. Topics include categories, universal algebra, modules, basic structure theory of rings, classical representation theory of finite groups, elements of homological algebra with applications, commutative ideal theory, and formally real fields. In addition to the immediate introduction and constant use of categories and functors, it revisits many topics from Volume I with greater depth and sophistication. Exercises appear throughout the text, along with insightful, carefully explained proofs. | 677.169 | 1 |
"Elementary Number Theory," Sixth Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton's engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.
This second edition updates the well-regarded 2001 publication with new short sections on topics like Catalan numbers and their relationship to Pascal's triangle and Mersenne numbers, Pollard rho factorization method, Hoggatt-Hensell identity. Koshy has added a new chapter on continued fractions. The unique features of the first edition like news of recent discoveries, biographical sketches of mathematicians, and applications--like the use of congruence in scheduling of a round-robin tournament--are being refreshed with current information. More challenging exercises are included both in the textbook and in the instructor's manual. Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and in-service math teachers at the high school and middle school levels. * Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises * Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes * Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle EastThis is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.
An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.
This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject. | 677.169 | 1 |
Standards For Quick Solutions Of Essay Help
Algebra 2 is a step further than Algebra 1. Before starting out this stage in the useful process, students need to be thoroughly grounded with the foundations. Algebra 2 homework help may well answer some of the basic doubts about Algebra. But do you find it advisable that students need tried to do their personalized Algebra homework before seeing this source. In Algebra 2, students are launched to logarithms and exponents, graphic functions, ways of solving inequalities and equations with complex numbers.
Algebra 2 homework help trains scholars to first understand the matter and then locate the most efficient way to solve it. For instance, when looking to solve an equation, the first step is to take notice of the number of terms in the given equation. The next step, then, is to decide which type of invoice discounting to opt for to solve this equation. Algebra 2 assignments help also works for a student's critical thinking ability.
Benefits associated with the Algebra 2 Assignments help platform. This platform's motto is to make math concepts easier for students. As a result of availing themselves of this service, students can get rid of that complexities of the subject. Algebra 2 homework help gives you tips for the effective analyze of mathematics. The young people have to possess sharp memory skills in order to excel in this section of math.
Career prospects made attractive with Algebra 2.Those that excel in Algebra need great career prospects. They often not be aware of this, nonetheless they can get really high salaried jobs. They can even work with any educational institute and spread their knowledge.
The help book provides certain rather interesting ways of making remembrance skills sharper to assist the learning process. This is also helpful for students who do not don't forget the elementary lessons with Algebra. It is not possible to provide a detailed version here, but a basic reference is offered. This reference helps scholars brush up their memory to remind them in the lessons that are learned with Algebra 1.
Help meant for competitive examinations Algebra some homework help is a really effective platform when preparing meant for competitive examinations or a university entrance examination. A cosmetic foundation in this category of mathematics can be a must when a student is actually applying for the General Educational Development examination.
This process helps in case young people want to take up mathematics for further higher studies. That students gather a sound knowledge required in order to deal with complex mathematical problems at better levels. It is as convenient and friendly as troubleshooting.
The GED exam earns the taker the equivalent of a high school diploma, which is necessary for students who can not complete their high school courses. The university entrance examinations are the SAT and ACT. Students need not bother to find a personal teacher. All the problems are easily resolved with the solutions found at the Algebra some homework help site.
The path structure also includes polynomial arithmetic, rational expressions, radicals and complex numbers, quadratic system and cone areas. As new terms, these words might sound a little complicated. However, Algebra 2 studying help becomes a personal lead to make complex mathematics easier for students. | 677.169 | 1 |
These 19 lessons provide initial instruction or intervention on linear equations and inequalities of two variables and functions. The first 4 lessons define those equations and their solutions, provide instruction on graphing those equations and ...
3 ...
SQLUtilities extends the T-SQL scripting environment by adding 64 new general-purpose functions for text, date/time and numeric processing that can be used directly within queries and stored procedures at the database layer.
SQLFinancials extends the T-SQL scripting environment by adding financial analysis functions similar to those found in the MS Excel analysis toolpack, but which can be used directly within the database.
This app is highly recommended for any of us who deal with SQL. Do you know there are so many SQL functions that are available out of the box, and we need to write them again? Many times, because of our lack of knowledge we end up writing ...
Polynomials Even and Odd Functions is a powerful learning aid to help students prove algebraically whether a polynomial function is even, odd, both or neither. An interactive Coaching Calculator and Guide combine to speed the learning process.The ... unauthor
Many developers needlessly loop through data in an effort to present raw, normalized, data into a more readable, tabular format, not realizing; that, "characteristic functions" can accomplish the same goal, elegantly, and more efficiently.
Style Extension Functions project implements the "OpenGIS XSLT Map Style Sheet Specification", which defines the constraints on Map Style Sheets written in XSLT, and used for the transformation of GML to SVG.
Learn math anywhere on your iPhone, iPod or iPad. Once the application is installed, you do not need internet access. This application teaches you linear functions in an easy to understand fashion. Each point is described by a highly | 677.169 | 1 |
The Complete How to Figure It: Using Math in Everyday Life
how much home insurance is enough; whether it makes more sense to buy or lease a new car; the most efficient way to save for future needs, from vacations to college tuition; air-conditioning and heating requirements for a new house; how many rolls of wallpaper you will need for a particular room; and much more. Here are tips for getting the most out of a modest pocket calculator or home computer to make tedious calcuations easy, a handy chapter on "Math in a Hurry," and even tips on improving your chances in tennis, horse racing, and blackjack....more
Community Reviews
this book tells mathematically-challenged folks like myself how to figure out problems that come up in everyday life. it is useful. i learned from it, and it now sits on my shelf awaiting a crisis of the math flavor. | 677.169 | 1 |
Professional Development book
4.11 - 1251 ratings - Source
McDougal Littell Incorporated, Ron Larson, McDougal, Littell ... in a group
Number of groups 27 = 3 a€c x After solving the equation, have students check the
answer in the context of the equation. ... A function works better. INCLUSION To
help your visual learners, give students graph paper and have them create
geometric patterns of their own, similar to the ... Pattern Number (x) Number of
Boxes (y) 1 1 2 4 3 7 4 10 Lesson Title Lesson Goals Key Applications 12.1:
Writing Expressions and.
Title
:
Professional Development book
Author
:
McDougal Littell Incorporated, Ron Larson, McDougal, Littell
Publisher
:
- | 677.169 | 1 |
Topology
ISBN
9780131816299
Edition
2 ed
Publication Date
28 Dec 1999
Publisher
Pearson Education (us)
Author(s)
James R. Munkres
Overview
For a senior undergraduate or first year graduate-level course in Introduction to Topology. Appropriate for a one-semester course on both general and algebraic topology or separate courses treating each topic separately. This text is designed to provide instructors with a convenient single text resource for bridging between general and algebraic topology courses. Two separate, distinct sections (one on general, point set topology, the other on algebraic topology) are each suitable for a one-semester course and are based around the same set of basic, core topics. Optional, independent topics and applications can be studied and developed in depth depending on course needs and preferences. *Follows the present-day trend of exploring the subject much more extensively with one semester devoted to general topology and a second to algebraic topology. *Exercises - Varied in difficulty from the routine to the challenging. Supplementary exercises at the end of several chapters explore additional topics. *Deepen students' understanding of concepts and theorems just presented rather than simply test comprehension. | 677.169 | 1 |
Form 4 – Functions
Just-Math Center is organizing Learning Additional Mathematics on an On-Line basis. We are giving a 50% discount on our monthly fees for new students. Our normal fees is RM40 a month. So if you are new, take advantage of the 50% discount. The benefits of on-line learning :
No traffic congestion
No need for transportation
Save cost on transportation
Save time on journey to and from tuition center
Save cost of materials as materials will be delivered digitally via WhatsApp
Students can watch teaching videos over and over to fully understand the concept behind the topics covered.
Every Saturday, a Skype session will be conducted to address any areas that students are still unsure or requires additional help.
Homework submission will be hassle free. Just snap shot your homework and send it to me for review. Feedback and comments will be given wherever necessary.
Good for students who are reluctant to ask questions in class. You can ask questions in writing form or voice form via WhatsApp. Your concerns will be attended to either via picture or video.
Students get small volume homework assignment on a daily basis. Video will be accompanied for new areas of learning.
WhatsApp me if you are interested to give the On-Line learning of Mathematics a try. I will WhatsApp you the details of payment method.
Functions is the first chapter of the Form 4 additional mathematics. In this chapter students are introduced to some elementary concept of functions. Students are basically required to have an understanding of the following areas:
Relations between sets of data
The types of relations between the data sets
What constitute a function
The composite function (meaning a combination of more than one function)
The inverse of a function.
We shall look at the above areas briefly via some video below. Students are required to practice on the questions of the subjects to achieve mastery.
Now lets look at the first video. This video shows the components of relations between sets.
The second video looks at the types of relations between the sets of data. Relations can be categorized into four main category, i.e.
One to One relations
One to Many relations
Many to One relations
Many to Many relations
Only the first and second category above i.e. One to One Relations and the One to Many Relations are classified as fulfilling the requirement of a function. Item 3 and 4 are not considered functions.
Now lets take a look at the video.
The third video looks at the relations that fulfill the requirement of a functions, i.e. each object has only one image.
Lets review the video.
The subsequent video examines the area of Composite Functions. Composite functions are a combination of more than one function. Lets now examine the video to look at what a composite function looks like
The next video looks at Inverse Function. This video will explain fully what it means by inverse function. Take a look to understand what inverse function means. | 677.169 | 1 |
This is a PowerPoint presentation focused on "Writing Expressions and Equations" in Algebra 1. This lesson is best used with a clicker because each example is set to show up upon click. This way, the students can't get ahead of you and you only | 677.169 | 1 |
AMathematical Preliminaries and Notation
This appendix gives some definitions of fundamental mathematical concepts that are used in AI, but are traditionally taught in other courses. It also introduces some notation and data structures that are used in various parts of the book.
A.1Discrete Mathematics
The mathematical concepts we build on include:
sets A set has elements (members). We write sS if s is an element of set S. The elements in a set define the set, so that two sets are equal if they have the same elements.
tuples An n-tuple is an ordered grouping of n elements, written x1,…, xn. A 2-tuple is a pair, and a 3-tuple is a triple ...
With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, interactive tutorials, and more. | 677.169 | 1 |
MAS6320 Algebra II
This module will develop both the algebraic and geometric theories of commutative rings and modules. The most basic form of interaction between these two subjects can be seen as the relationship between polynomials (seen as elements in a ring) and their graphs. This relationship can then be extended to the relationship between certain kinds of ideals in a ring and the geometric object ("graph") such an ideal describes. At a basic level, this module can be seen as the study of turning algebra into pictures and describing pictures using algebra. To do so, we will study many important properties of commutative rings and their modules, and then explore the geometric objects that arise from various algebraic properties. Interpreted in the context of complex numbers, this analogy between algebra and geometry reflects many of the basic intuitions one has about graphs of polynomial equations, but we will also consider the geometry that comes about in more exotic situations, such as over finite fields.
There are no prerequisites for this module.
No other modules have this module as a prerequisite.
Outline syllabus
Part A
Review: rings and abelian groups
Tensor products
Modules
Direct sums and generators; Chinese Remainder Theorem
Prime ideals and radicals
Local rings
Algebras
Noetherian rings and modules
Integral extensions and Noether Normalization
Localization
Part B
Affine algebraic varieties
Hilbert's Nullstellensatz
Regular functions
Maps between varieties
Dimension theory
Derivations and "calculus" for algebraic varieties
Aims
To establish a basic groundwork of knowledge in commutative algebra.
To apply that knowledge to study problems of a geometric nature.
To develop an appropriate perspective on the techniques discussed.
To understand the connections between algebra and geometry.
40 lectures, no tutorials
Assessment
Assessment will be via 20 assigned problems, each with an equal weighting of 5%. These problems will assess the student's knowledge of the key concepts, their ability to synthesize and generalize these concepts, and their ability to present proofs logically and coherently. Individual problems will be assessed via descriptor, with clear standards and sample solutions provided to students at the beginning of the module. | 677.169 | 1 |
This curriculum for mathematics is designed as a teachable curriculum directed toward workplace needs. It is intended to be workplace specific and to move the learner toward a full 12th-grade level attainment. The two modules in this curriculum are further divided into modules dealing with specific topics. The first module, Basic Mathematics, consists of these seven parts: whole numbers, decimals, fractions, percents, measurement, ratio, and graphs and charts. The second module, Algebra 1, Geometry 1, & Statistics, has these seven parts: introduction to algebra, introduction to geometry, further development of algebra, ratio and proportion, further development of geometry, statistics, and the mathematics of linear equations. Each module is broken down into concepts, which are explained with examples and applications. Sources of drill and practice exercises are provided for each concept. (YLB) | 677.169 | 1 |
I have only taken mathematics up to trig and I was curious if I would be able to start reading books on more advanced topics like Abstract Alegebra and topology??
If you are mathematically mature enough and have no problem with proofs and abstract arguments, then there should be no problem in starting with abstract algebra and topology. Most students are not however.
At some point you will need calculus, but it is entirely possible to pick it up later and from a more advanced viewpoint. One of the first places where you will need calculus is in differential geometry because that subject pretty much studies objects on which one can do calculus locally, so the point of the subject (at least at a basic level) is to take what you learned in calculus and see how much you can extend to spaces that locally looks like the one you have seen.
So it can be done, but it probably shouldn't.
@micromass: I disagree. A good topology book (such as Munkres') will provide motivation enough to understand the concepts. You don't need to have studied calculus and analysis to know what an open and closed interval is, or what it means for a function to be continuous in an intuitive sense. I personally did topology without having taken calculus, and today I think it would have been much more efficient to do it the other way around, but it was by no means impossible.
The standard one-year course in calculus is probably 20% review, 20% cool stuff, and 60% techniques that are fundamentally unimportant (e.g., trig substitutions). For that reason, I can see why you might want to do other stuff without first slogging through a year of calc.
If you are asking this to ultimately understand physical theories such as GR or QFT (guessing because of your other posts) then you will absolutely need calculus to understand differentiable manifolds as was stated above.
Well, I am equally interested in physics but I am not looking to study these math subjects for physics reasons, but rather personal gratification. I will be taking my calculus sequence starting in january and I thought it would be nice to learn some other more interesting math topics. I am not saying Calculus is not interesting, but so far from my text book it just seems like a bunch of simple calculations and applying different methods to get a derivative ( I have not gone looking at integrals yet)
Yes Calc 1 - 3 can be extremely dry in general. If you find it unbearably boring then you could always go through Spivak "Calculus". It is the holy grail of proof based single - variable calculus \ introduction to analysis. Also try to enroll in honors calculus if your school offers it.
I am not saying Calculus is not interesting, but so far from my text book it just seems like a bunch of simple calculations and applying different methods to get a derivative ( I have not gone looking at integrals yet)
You're just studying from the wrong calculus book. Why don't you go through a book like Spivak. That book is much better than simple calculations and finding derivatives.
I just want to add in another recommendation for Spivak. I've already taken the standard Calc 1-3 sequence and I agree it can be quite boring. On the other hand, I'm working through Spivak right now and it's an absolute blast. It's very difficult at times, but it's extremely fun and rewarding.
Since you guys are telling me abstract alegebra doesn't need calculus but is very proof based, along with spivak calculus is very proof based, would it be a good idea to get an intro to proof book? As I have no experience with proofs. If you could recomend a good starter one that would be amazing.
How To Prove It by Daniel Velleman is what I used to get acquainted with proofs. It's pretty good, but working through the whole book might be overkill. Definitely work through the first few chapters and the chapter on induction, though.
In my opinion, calculus was when math started to be beautiful and elegant.
That's not fair to the Greeks and geometry. Ruler and compass constructions are awesome. The discovery of the platonic solids was such a beautiful fact that they attributed a mystical significance to it.
You can't do topology at all without calculus. The motivations for topological terms all come from calculus and analysis.
Well, you can't do formal point-set topology very easily, although the subject has no formal prerequisites. I think if someone wrote a good exposition of it, though, you could do SOME topology. For example, I could probably teach high school students to compute the Jones polynomial (though, truth be told, I don't really *understand* the Jones polynomial myself half as well as I would like). Practically speaking, though, it might be hard to find suitable books. I'm not familiar enough with them to say. Intuitive topology by Prasolov might fit the bill, for example.
I think it might be a good idea for a lot of people to skip ahead in math, just to find out that math can actually be interesting because the normal curriculum is mostly kind of boring, until you get to upper division stuff (multi-variable calc and linear algebra can be okay, and other subjects might be okay if they were taught differently). However, you can't get too far without running into the need for calculus.
If you want to get comfortable with proofs, maybe naive set theory would be a good subject to get some practice in, so you can separate out the skill of writing formal proofs from most of the other more conceptual difficulties.
It's good to get a lot of practice with calculus computations. I think you need to do some of those, too. I never read Spivak, so I don't know if he neglects that.
It's good to get a lot of practice with calculus computations. I think you need to do some of those, too. I never read Spivak, so I don't know if he neglects that.
He has some loosely computational things like induction proofs where you have to show some derivative formula or such. There are almost none but that is, in my opinion, for the better because honestly who likes computations haha.
He has some loosely computational things like induction proofs where you have to show some derivative formula or such. There are almost none but that is, in my opinion, for the better because honestly who likes computations haha.
Funny, I was just having this argument with a friend, who has a pretty similar outlook on math to mine, with a few very fundamental differences. I was on the other side of the argument, then, saying I don't like computations. He objected that he didn't like how mathematicians these days don't want to get their hands dirty. He solved a problem his big shot adviser couldn't solve because he computed some examples. He also said if you look at the notebooks of the great mathematicians, they have reams of computations.
I like avoiding computations by being clever, and I think computations can sometimes be unenlightening as far as understanding things goes (when trying to understand something deeply, I try to ban myself from doing any calculations where possible). But, you shouldn't be afraid to get your hands dirty, either.
Would an intro to proof book be the best thing to start with then??? I would love to be ahead of the game in terms of my mathematics degree ( I plan to get a minor or *maybe* a double major) If I have a good basis in writting proofs and I start learning abstract alegebra and topology then when I get to these classes it should be breeze???
Another question; if I were to study these topics extensivly enough would it be possible to skip the class and move on to the next level of lets say topology??? | 677.169 | 1 |
Calculus I (Math 1411). This is entirely a mathematical maturity requirement,
as we will use no calculus in this course.
Course Objectives:
Upon successful completion of the course, you will know and be able
to use the basic algebra of sets and of logic. You will be able to
identify and use common classes of relations. You will know basic
properties of arbitrary functions. You will be able to solve counting
problems involving combinations and permutations, including counting
problems with restrictions. You will know the basic definitions and
theorems of graph theory, and be able to apply them to specific
graphs. You will know the basic algorithms for traversing trees, and
be able to apply them to specific trees.
Note that this class will probably be quite different from other math
classes you have taken, in at least two important ways. First, in
contrast to calculus and related courses, the objects under
consideration are (as the course title suggests) discrete, not
continuous. This has the advantage that you can often explicitly list
all the pieces (try listing all the function values of a continuous
function!), but the disadvantage of not having continuity to "tie"
things together nicely. Second, although there is still a lot of
problem-solving, the problems and their answers have a very different
flavor: the problems are not equations to be solved, and the answers
often aren't even numbers. We also may spend more time explaining why
a particular solution works than in finding the solution.
Textbook:
Discrete Mathematics, 5th ed., Dossey,
et. al., Chs. 2, 4, 5, 8, Appendix A.
We will skip some sections, as announced in class.
The textbook is required at all class meetings.
Required Reading:Grades:
Quizzes (15%):
Suggested homework problems will be assigned most class days and will
generally be discussed at the next class.
There will be approximately weekly quizzes, with problems taken from
the homework. Quizzes are closed-book, closed-notes. Missed quizzes
cannot be made up, but your two lowest quiz scores will be
dropped.
It is very important that you do your homework before it is discussed
in class. You will only learn the material by doing it yourself, not
by watching others do it for you.
Projects (10%):
There will be 3-4 projects throughout the semester, where you consider slightly more in-depth problems, and write up their solutions more carefully. As part of a pilot project, all the projects will be drawn from real applications in biology (though you will not need to know any biology in order to complete them), where math and computer science are playing an increasing role.
Exams (15% each):
There will be three in-class exams on the following days:
Ch. 2: Thu. 22 Feb.
Ch. 8: Thu. 22 Mar.
Chs. 4,5: Thu. 26 Apr.
Final (30%)
comprehensive (including Appendix A)
The final will be on
Thu. 10 May, 10:00 a.m.-12:45 p.m.
Makeup exams can be given only in extraordinary and unavoidable
circumstancesI strongly encourage you to attend every class, though there is no particular grade penalty for absences. You are responsible to find out any assignment that must be made up if you are absent
1. We should contact Ruby Lynch Arroyo, who is in College of Ed, professor of practice, math teacher (math methods courses), but has presented at a literacy workshop
2. A good phrase "Literacy across the curriculum"
3. A good practice for reading is to give an "essential question" about something deeper in the reading material (students are good at who what when, but less good at how and why). He uses it to start class -- present question, students write it, then pair then share (write - pair - share)
4. He will send stuff
5. Strategies for reading include trying to "read like a writer" and "write like a writer". In English, they may treat a literary work as a model. We may ask: What are the big ideas of mathematics? How to think like a mathematician.
(For instance, here are 3 things English people all think: 1. Language arts is across the curriculum; 2. Everyone is a writer and reporter (e.g., even writing what's going on via phone); 3. Fiction is based on real-life events.)
5. A good method is "understanding by design", one piece of which is backwards design. This was promoted by [something like] Wiggins and McTigh [see one of the articles he sent]. Roughly: Design a unit with big ideas:
1. What will students know and be able to do?
2. What activities will they do to demonstrate that understanding
3. How will I know they have learned it? [assessment]
6. Some program called "National Writing Project"
Teachers as writers and students as writers. Teachers keep writing as the students are writing. The students see the teacher writing. | 677.169 | 1 |
Solution of Nonlinear Equations
Abstract
Solving linear and nonlinear equations is a major preoccupation of applied mathematics and statistics. For nonlinear equations, closed-form solutions are the exception rather than the rule. Here we will concentrate on three simple techniques—bisection, functional iteration, and Newton's method— for solving equations in one variable. Insight into how these methods operate can be gained by a combination of theory and examples. Since functional iteration and Newton's method generalize to higher-dimensional problems, it is particularly important to develop intuition about their strengths and weaknesses. Equipped with this intuition, we can tackle harder problems with more confidence and understanding. | 677.169 | 1 |
Included • Video Warm-Up: Students preview the lesson by watching a short video on YouTube and then come to class with some prior knowledge. Warm-up handout included to review what students learned from the video. • Notes & Practice: Three pages of guided notes and practice. • Homework: One page containing 10 homework problems. • Answer Keys: Completely worked out answer keys are included.
Please look at the preview to find out more information about this resource. | 677.169 | 1 |
Getting started with M208
I've been a bit lax with the studying over the last couple of weeks, but there's only so many days I can spend on comic books and videogames, so this week I've been trying to get a head start on M208 using the OpenLearn materials.
I'm about halfway through Unit I1: Real functions and graphs at the moment, and I'm really enjoying it so far. It starts with a fairly gentle refresher about the graph-sketching strategy covered in MS221, and then goes on to expand the strategy so that we can sketch composite and hybrid functions. This leads to some weird and wonderful graphs, like the graph of xcosx and (sinx)/x.
I was a bit worried at first that the introductory block of M208 would just be composed of material we'd already covered in MS221, to make sure that all the incoming students were up to speed on the same topics, but there's definitely enough new stuff to keep me interested, at least so far. I'm particularly intrigued by the idea of 'jumps' and 'corners' in graphs, like the 'hole' at x = 0 in sinx/x. Apparently we're going to look at these jumps and corners in more detail in the Analysis blocks, so I'm looking forward to that very much!
But that's way in the future, I reckon we probably won't be doing the Analysis units until spring/summer next year – I've got four units of group theory, and five units of linear algebra to get through first. This winter is going to be mathtastic! | 677.169 | 1 |
Millennium Math
Mathematics in a technological world
Technology in Service of Mathematics
We live in a time when technology is rendering much of our ability
to do routine, mechanical tasks unnecessary. Just as few of us need to
have skill in washing dishes or walking long distances, the time has
come when we do not need much skill mundane mathematical tasks
such as solving simultaneous linear equations or adding four-digit
numbers. To carry the metaphor further, just as the ability to
wash some pots and pans will always be needed, we certainly must
understand the ideas behind solving those equations, and perhaps
it can be quicker at times to do it in our heads than to pull the
computer out of our pockets. Nonetheless, modern instruction in
mathematics should be predicated on the assumption that the students
will always have computers at their fingertips.
This new situation is abhorrent to many mathematicians. We have
spent so much of our lives building courses around those mechanical
skills that it is difficult to change our tactics to
adapt to the times. On the other hand, I believe
strongly that we should embrace the new technology and build
our courses around it. The point is that we never cared much
whether our students could manipulate the formal expression
\(2x+1=2\) to evaluate \(x\); instead we wanted them to
come to this equation from a question such as: "a salad
dressing recipe calls for one cup of oil and equal parts
ordinary and balsamic vinegar. The recipe makes two cups.
How much of each kind of vinegar is required?" We wanted
them to understand that the solution to this problem is
unique. The details of finding that unique solution we should
be happy to leave to a machine.
This site is dedicated to discussing mathematical technology, and
developing approaches to its incorporation in 21st century
mathematics courses. | 677.169 | 1 |
IP Mathematics Book 2 Answers Booklet
4.11 - 1251 ratings - Source
Mathematics in schools offering the Integrated Programme is usually taught as an integrated subject, so that students will be able to better relate learnt knowledge to new knowledge and transfer conceptual understanding to application, as many mathematical concepts are interconnected. One driving force to write the series is to provide a guidebook especially for students in the Integrated Programme. The other is to share teaching ideas with other Mathematics teachers who love the subject as much as I do. Features: ii½ Each topic begins with a recap of key mathematical concepts to help students consolidate learning. ii½ Worked examples are included to enhance understanding and application of key concepts, with side notes explaining some of the working. ii½ Practice questions are tiered into three levels of difficulty. Level 1 aims to provide students with the necessary practice; Level 2 to further build the confidence and test studentsii½ understanding; Level 3 to challenge students with higher order thinking questions. ii½ ii½Math Wonderlandii½ is one highlight of the book. Activities include extension of the topic, suggested alternative assessment and questions to stretch mathematical thinking. The primary purpose of the Wonderland is to allow students to think deeply about what they have learnt and to appreciate the learning of Mathematics beyond classroom. ii½ Step-by-step solutions to all questions are provided as an additional resource to studentsii½ problem solving process. I hope this book will benefit students studying Integrated Mathematics, as well as those with aptitude for the subject who are preparing for the GCE O Level Mathematics and Additional Mathematics examinations.2340 a€" = 36 = 6.4 0111 E EœEd55 nf Cl 5 3 2340 _ _ E _ (Nˆ) EœEd55 of C2 8 36 5 3 =
65 Mass of C1 _ 150 Ey E Surface area : 65 cm2 I 36 6 g (to 3 S f) (ii) Number of
scale models : 36 (i) In MBE and ABCEAr (iii) Cost of painting one scale model:anbsp;...
Title
:
IP Mathematics Book 2 Answers Booklet
Author
:
Wong-Ng Siew Hiong
Publisher
:
Shing Lee Publishers Pte | 677.169 | 1 |
Foundation Courses
This programme is a step forward in laying a strong foundation from the beginning stage and to make students understand the concept of science and mathematics rather than mere memorizing and cramming. It will also help a student to secure good marks. It will include comprehensive coverage of CBSE syllabus in a synchronized way and all those additional and relevant topics of higher level, analytical skills, problem solving ability required for various competitive examinations (such as NTSE, Olympiad, KVPY, JSTSE etc.). | 677.169 | 1 |
3What is E7?E7 is a programming course for engineers. In the first half of the course, you will learn the elements of programming that you need to build your own programs and solve engineering problems. Although this course focuses exclusively on programming in MATLAB, many of the programming concepts are generalizable to other programming languages. The second half of the course is focused on mathematical models and numerical methods for solving them.Prerequisites:- Math 53It will help (but not a prerequisite) if you have - Some familiarity with calculus, linear algebra- Some notion of programmingSet of notesWe have a set of notes for you.–Everything you need to know to do the labs and the exams is in the notes. –Notes have examples, sample code, explanations of the concepts.–Notes have been written specifically for you, for this class, so there is a complete sync between the lectures, the labs, the exams and the material. –Notes will be posted on bSpace–First half of notes will be available at the Northside Copy Central for about $12. Second half will be available later on.Notes are not perfect (they are written for the course, and this is the first edition). There might be bugs:–If things seem wrong, you need to tell us (more on this later)–We will have a TerminaBug contest (more on this later).
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4Set of notesNotes are organized as follows:Text to explain Sample codeSet of notesNotes are organized as follows:Sample codeExamplesTips
5Set of notesNotes are organized as follows:Try it! (simple things to try)Warnings (common pitfalls)Set of notesNotes are organized as follows:Explanations about what is happeningConstruction: about the syntax
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6Set of notesNotes are organized as follows–Problems to be worked on paper–Problems to think about–Problem that you should try in the command prompt–Problem for which you should write a program–Problems which will require you to generate an error.
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Solve math problems algebra
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. 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 instantly solves your most difficult. Solve Your Math Problems Such as Basic Math, Pre-Algebra, Algebra, Trigonometry, Pre-calculus, Calculus, Statistics and much more by mathway. Algebra Calculator shows you the step-by-step solutions! Solves algebra problems and walks you through them.
Algebra. Quick! I need help with:. Help typing in your math problems. Simplifying Expressions Powers, Products, Like terms, Anything else: Polynomials Add/Sub., Multiply, Div., Expon. Algebra. Quick! I need help with:. Help typing in your math problems. Simplifying Expressions Powers, Products, Like terms, Anything else: Polynomials Add/Sub., Multiply, Div., Expon. Algebra Calculator shows you the step-by-step solutions! Solves algebra problems and walks you through them. Solving Algebra word problems is useful in helping you to solve earthly problems. While the 5 steps of Algebra problem solving are listed below, this article will focus on the first step. WebMath is designed to help you solve your math problems. Composed of forms to fill-in and then returns analysis of a problem and, when possible, provides a step-by-step solution. Covers.
Solve math problems algebra
Free math problem solver answers your algebra homework questions with step-by-step explanations. QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices. PTC Mathcad is the engineering algebra calculator that can help you quickly and easily solve algebra problems. 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 instantly solves your most difficult.
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. Algebra, math homework solvers, lessons and free tutors online.Pre-algebra, Algebra I, Algebra II, Geometry, Physics. Created by our FREE tutors. Solvers with work shown, write algebra. WebMath is designed to help you solve your math problems. Composed of forms to fill-in and then returns analysis of a problem and, when possible, provides a step-by-step solution. Covers. Need urgent help with math problems? No panic! We are here to cope with your complex math assignment. Exceptional quality is guaranteed.
Our pre-algebra calculator will not only help you check your homework but will also help give you extra practice to help you prepare for tests and quizzes.
Welcome to IXL's algebra page. We offer fun, unlimited practice in more than 200 different algebra skills.
Solving Algebra word problems is useful in helping you to solve earthly problems. While the 5 steps of Algebra problem solving are listed below, this article will focus on the first step.
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. Welcome to IXL's algebra page. We offer fun, unlimited practice in more than 200 different algebra skills. Find practice math problems with answers in algebra & calculus from the Cymath online math solver. The Cymath equation solver makes solving math problems easy. Solve Your Math Problems Such as Basic Math, Pre-Algebra, Algebra, Trigonometry, Pre-calculus, Calculus, Statistics and much more by mathway. | 677.169 | 1 |
Eleven contributions are selected from the eight working groups in the areas of elliptic surfaces and the Mahler measure, analytic number theory, number theory in functions fields and algebraic geometry over finite fields, arithmetic algebraic geometry, K-theory and algebraic number theory, arithmetic geometry, modular forms, and arithmetic intersection theory.
Focusing on the many approaches to analytic number theory, including arithmetic functions that illustrate the multiplicative structure of the integers (hence the term anatomy of integers, an expression coined by the authors and a third mathematician, Andrew Granville), this text provides as well 263 problems, with answeres included for all the even-numbered ones.
summability, integral transforms of hypergeometric functions, the constructive theory of approximation, orthogonal polynomials and Sobolev inner products, orthogonal and other polynomials on inverse images of polynomial mappings, and analytic number theory and approximation.
In a clever and logical system that builds from previous knowledge, this covers such core topics as divisibility and primes, congruences, cryptography, and quadratic residues, then addresses arithmetic functions, large primes, continued fractions, and diophantine equations, closing with advanced topics such as analytic number theory, elliptic curves, and the relationship between logic and number | 677.169 | 1 |
Physics / Algebra 2
This course is designed to cover the basic concepts of Algebra II and show how they are applied in the science of Physics. Topics in Algebra II include Functions and Equations, Graphing, Systems of Equations, Probability and Statistics, Polynomials, Rational Expressions and Trigonometry. Topics in Physics include Vectors, Kinematics, Dynamics, 2-dimensional Motion, Work and Energy, Simple Harmonic Motion, Momentum, Heat, Electricity and Magnetism, Optics and Waves. Labs and other activities will be incorporated to show real world applications of Physics and Algebraic concepts. | 677.169 | 1 |
Methods Techniques De Gruyter Textbook
Applied mathematics books at e books directory files with free access on the internet these books are made freely available by their respective authors and publishers. Radiology books at e books directory files with free access on the internet these books are made freely available by their respective authors and publishers | 677.169 | 1 |
S5, S6 Maths
The National 4 Mathematics Course enables learners to select and apply straightforward mathematical skills in a variety of mathematical and real-life situations. Learners interpret, communicate and manage information in mathematical form.
Assessment
Unit Assessments:Expressions and Formulae, Relationships, Numeracy
Course Assessment: Added Value Unit
Recommended entry:
Learners would normally be expected to have attained the skills and knowledge required by the National 3 Lifeskills Mathematics Course
Progression:
This Course or its Units may provide progression to other qualifications in Mathematics such as National 5 or to further study, employment or training
The Higher Mathematics Course is suitable for learners who are secure in their attainment of the National 5 Mathematics Course.
Learners will develop and extend their skills in selecting and applying mathematical techniques in a variety of contexts and situations. Learners will be given the opportunity to become skilled in interpreting, analysing, communicating and managing information in mathematical form.
The Course develops and expands a range of mathematical skills. It allows the learner to develop further skills in calculus and algebra. Areas such as number theory (which helps keep the internet secure), complex numbers (the uses of which are ubiquitous, ranging from the solution of equations to the description of electronic circuits) and matrices (used in game theory and economics) are introduced.
Assessment
Unit Assessments:Methods in Algebra and Calculus,Applications of Algebra and Calculus, Geometry, Proof and Systems of Equations | 677.169 | 1 |
A level mathematics d2 revision notes
Game Theory
Networks
Critical Path Analysis
If you have certain tasks that need to be completed, say planning for a D2 exam. Certain tasks have to be completed before you go on to start another, for example you need to buy the revision quide before you can revise from it. Once you have your list of activities, you need to fill in a presedence table, which contains the activities and which other activity it depends on. If there is an activity that does not depend on any thing else then it can start at any time. After you have completed your presedence table you are ready to draw an activity network (including nodes representing events and arcs representing activities). Start with a node that represents the start of the project (called the source node), add arces for each activity that can start at any time (don't worry about the exact position of each arc, as you might have to redraw a neat copy of the whole network at the end). Then add nodes for the end of any of the arcs which have another activity depending on it, then add another arc starting at that node. To make your life easier, add lables with the duration of each activity next to each arc, s you go along drawing the arcs.
Linear Programming
Matching and allocation
1) Bipartite Graphs and Matching Augmentation Algorithm
They are the graphs that have a set of right nodes and a set of left nodes. The nodes on one side of the the graph (e.g. left) can't join up to each other. This means that the nodes on the left have arcs going only to the right and the nodes on the right have arcs only going to the left. Google the image of a bipartite graph to reinforce what I've said.
For a bipartite graph, initially you are given a certain matching, where the nodes on the left match up to nodes on the right. Some times this matching can be improved (a better combination can be found). Most of the times you are given a question a bit like this one below:
Alan, Brad, Ciara and Dave have to pick one colour of out Pink, Green, Yellow and Red. Alan would prefer Green or Red. Brad would prefer Yellow or Green. Ciara prefers Pink, Green or Yellow. Dave only wants red. Finally, a teacher decides that Alan gets Red, Brad gets Yellow and Ciara gets Green. This leaves Dave with Pink which he didn't want initially. Can this matching be improved?
In this case, you put the A,B,C and D nodes on the left and the P,G,Y and R nodes on the right and then you draw arcs according to what they initially wanted in a bipartite graph - this shows the possible matchings. Once you've drawn the arcs to what they initially wanted, highlight those arcs that signify what the teacher gave them. In the case above, Dave didn't get Red so leave don't draw a node from D to P. This is known as an incomplete initial matching.
To improve the matching follow the algorithm below -
1) Let all the arcs from the possible matchings go from left to right and all the arcs showing the incomplete matching from right to left. 2) Start from the unmatched node on the left and try to follow the arcs to reach the unmatched node on the right where arcs in the possible matchings go from left to right and arcs in the initial incomplete matching go from right to left. 3) Pick the shortest path following Step 2. This is the alternating path. 4) If there is an arc in the alternating path that was in the initial incomplete matching, pick the other arc in the possible matchings that goes out of that particular node. If there isn't a node going common to the arc in the alternating path covered by the initial incomplete matching, leave it like that. 5) Repeat step 4 for any other nodes that might be like the above.
And you should have a final complete matching.
2) Hungarian Algorithm
This is a MINIMISING problem for ALLOCATION problems. So take away the numbers accordingly. But at the end if you have to find the final cost etc. remember to add the number you took away intially, back again at the end. This only works for square arrays (matrices) so add a dummy row accordingly which is the largest element (number) in the array (matrix).
Follow the algorithm below-
1) Reduce rows and then reduce columns (most of the times the question will which one to reduce first but if they don't say it, reduce the rows first and then the columns) Here take away the lowest value in the row or column away from each element in that row or column. If 0 is the lowest value, leave it like that because taking 0 away from each element is not going to change it. 2) You will get 0s so cover all the 0s with the least number of lines. 3) Add the least uncovered element (number in the array) to the zeros. 4) Take away the least uncovered element (number in the array) from EVERY element in the array. 5) Repeat this till you get enough 0s in each row and column. 6) Pick the 0s in a way that it's the only 0 in it's row or column. | 677.169 | 1 |
They take a little more strategy to solve problems than a calculator does, but they're faster than a calculator once you get the hang of them. You have to break that idea that all problems have to be in a format of (73)(45) = x before solving them. Most times, it's faster to solve your problem when the variable is buried somewhere in the middle of the equation. I know I want my answer to be 45, so I'll just scan down the scale for all the division problems that equal 45 until I get to 73 and then see what I had to divide by.
You have to buy a good graphing calculator before you reach a point where a calculator is capable of solving more types of problems, as well. I can solve problems with complex numbers, solve quadratic equations, do coordinate transformations from Cartesian to polar (or even spherical) - all things the cheaper calculators can't do. Of course, my TI-86 can solve cubic equations, or even higher powers. It can solve matrix problems, do cross products, etc, all a lot faster than you can on a slide rule. Plus, the graphing functions can make it easier to find roots for equations, area, etc than actually solving the problem.
Of course, you run into the same problems with slide rules that you do with calculators. As they get more elaborate with 20 some scales that make it quicker to solve different types of problems, you can forget the basic scales worked just as well at getting you an answer. Calculating something like the doping concentration of a semi-conductor can throw you for a momentary loop when you're working with numbers higher than your beloved log-log scales go.
You get a basic simplex slide rule with an A, B, C, D scales, a base 10 log scale, and your trig functions, Tan, Sin, and small angle Tan/Sin scale, you're really good to go for just about any problem. I still like my N4-ES vector hyperbolic dual base speed rule with it's 34 scales.
That slide rule actually works, by the way click on the slide and move it around; click on the cursor and move it around. Mine is slightly different - mine was the initial version, so it doesn't have the copyright, and has only 33 scales (I don't have the Ln scale above the L scale). Mine is in a lot better shape with no rust, as well.
(My avatar is a close up of the pocket Pickett N600-T my dad gave me).
Slide rules were status symbols before advent of calculators in 70-ies. The more mathematically advanced section in school you attended, the more advanced and
expensive slide rule you must have. Some were careful letting their slide rule protrude out from pocket or briefcase, showing their status and expected successful career.
In mid 70-ies I changed to circular slide rule, that was more handy sized for pockets.
Although some drawbacks, they had longer scales compared to size and not the least you never needed shifting because you never ran out of scale.
But at that time the electronic calculator rapidly replaced slide rules and earlier expensive slide rules were sold out cheaply - I bought a lot of such bargain slide rules
of different kind.
Regarding calculators I am astonished the several-button system has survived. I thought
one- or few-button solutions should replace. There are a lot of possibilities - but for some reason only the traditional several button system appear still exist for calculators
I was in high-school when the first 4-banger (+-x/) calculators came out. No memory, and they cost over $500. If you were caught with one on school property (even in the glove box of your car), you were expelled on the spot. We did use slide-rules, though. I kind of miss mine, even though I avoid math like the plague.
I think that I might have been the first to realize that if I greased up the rails and hooked up a good rubber band, I could nail someone in the forehead at over 50 feet with that slide.
edit: Oooh, Cyrus. That do bring back a memory or two, although I had only the basic Cessna flight computer. I have the mathematical skills of a tree, but I could work wind vector triangles like a demon.
You will never regret it. Just remember that you don't fly or drive an aeroplane; you wear it. Once it becomes a part of your body, there's no going back. Best of luck to you in your training and future piloting. | 677.169 | 1 |
Larson edwards calculus 10th edition pdf
Our interactive tutorials are designed to take you step-by-step larson edwards calculus 10th edition pdf the process of creating your own questions. Find out how easy it is to get started. Discover our wide selection of textbook content and advanced teaching tools.
Indeterminate forms and improper integrals, this course is designed to provide the basic computer skills required in typical college work. Learning and thinking, previous service learning experience, find out how easy it is to get started. Students will learn about such topics as leadership, independent research in biology under direction of a faculty member. Ruth Anna Abigail, students will begin to define and research a topic for their Capstone. Students will learn about and regularly practice basic listening skills, advanced level jazz technique course.
Various information systems; counts as one credit activity and one credit theory. Related aspect of physical fitness studied, survey of Western Civilization from A. Tests of significance, and context within which it has evolved from Renaissance to modern era. The course develops understanding of how to use Red Cross course materials, elementary score study, seeks to further develop student's leadership skills and encourage them to see themselves as "change agents" in their communities through exploration of civic engagement and social justic principles and activism. And encourages students to apply feminist theories to their lives and work.
Distribution of income, the course fexamines the responsibilities of business corporations to their many constituencies. Uses class activities, students will increase leadership and teaching skills by leading a service learning program. Major topics include variations of democracy, counts as one credit activity and two credits theory. Focuses on cultural and regional differences, and social justice. Linear and quadratic equations, student identifies "unknown" organic compounds. | 677.169 | 1 |
Don Cohen was born in Jersey City, N.J. He has a B.A. and M.A. from SUNY-Albany, NY and M.S.from RPI,in Troy, NY. He has taught all ages of students for 44 years, the last 22 of these as co-founder and teacher of The Math Program. After 7 years of teaching in a junior high school, he realized there must be a more enjoyable and effective way to teach math. He searched for alternatives. This led to designing new curriculum for N.Y. State; learning about mathematics and creativity from Bob Davis with The Madison Project; learning what real teaching is about by observing great teachers such as Sue Monell; teaching teachers; working on Plato (a computer-based education system at the U of I); all before Don and his partner invented The Math Program. These are the good old days! Don has three fine sons and six terrific grandchildren. He is a watercolor artist (see cover).
Review:
This book has been translated into Japanese and a gentleman there is planning to put the English and Japanese version on a CD-ROM, for sale in Japan and the rest of the world.
.. Much lip-service is given to education as it ought to be. Your materials guide us to education at its best. It's always exciting and rewarding to work with your materials. -- From the publisher
This is a book of problems to work on and think about. Archimedes, Newton, Euler, my students and I have worked on them also -- they are important. Important mathematically and important for understanding problems in your life. Get out some paper, a pencil and a calculator. Expect chaos and confusion, then you'll be on your way. This is not a linear book, but do get started on the ideas in chapters 1 and 2 first, then you could jump to chapter 6 or 8 or 13. If you are a parent or teacher please allow your child to look at the answer sheets and to work with other people on the problems! This book is not a 1 year course! One could spend two months on one problem. -- From the Publisher | 677.169 | 1 |
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Easy Mathematics Step-by-Step Author: Sandra Luna McCune
ISBN: 9780071767651 Format: Paperback Number Of Pages: 288 Published: 28 February 2012 Country of Publication: US Dimensions (cm): 23.0 x 18.0 x 1.0 Description: Take it step-by-step for math success!
The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find in Easy Mathematics Step-by-Step is a lot of endless drills. Instead, you get a clear explanation that breaks down complex concepts into easy-to-understand steps, followed by highly focused exercises that are linked to core skills--enabling learners to grasp when and how to apply those techniques.This book features: Large step-by-step charts breaking down each step within a process and showing clear connections between topics and annotations to clarify difficulties Stay-in-step panels show how to cope with variations to the core steps Step-it-up exercises link practice to the core steps already presented Missteps and stumbles highlight common errors to avoid
You can master math as long as you take it Step-by-Step!
About the Author
Sandra Luna McCune, Ph.D. is Regents Professor currently teaching as a mathematics specialist in t | 677.169 | 1 |
Pages
Sunday, May 1, 2016
IIT JEE Mathematics Study Plan 1. Sets
R.D. Sharma, Objective Mathematics, Chapter 1
Video Lectures
1.1 Sets
1.2 Description of a set
1.3 Types of sets
1.4 Subsets
1.5 Universal set
1.6 Power set
1.7 Venn diagrams
1.8 Operations on sets
1.9 Laws of algebra of sets
1.10 More results on operations on sets
1.11 Some important results on number of elements in sets
Sets Chapter - Study Plan
(1 May to 7 May)
Day 1 ( 1 May)
1.1 Sets
1.2 Description of a set
1.3 Types of sets
1.4 Subsets
1.5 Universal set
1.6 Power set
1.7 Venn diagrams | 677.169 | 1 |
Not a repository, but an individual free/open math text under development -- comments and feedback desired: It starts with elementary probability and then combines probability and stats with linear algebra, multivariable calculus, and differential equations. Aimed at folks who have seen the math before but need a refresher and a viewpoint that unifies seemingly disparate topics. Note that it uses Softcover, a great way to publish technical texts to several formats at once. | 677.169 | 1 |
Product Features
Student tested. Teacher approved.
Replace your print book with engaging videos that students love. Professor Edward Burger presents the fundamentals of College Algebra in an exciting, dynamic format that prepares your students for success in the classroom.
A Textbook Your Students Actually Use
Thinkwell provides the familiarity of a textbook that any instructor can easily adopt and adapt. But that's just the start.
Complete Video Instruction
For students who prefer video, Thinkwell provides instruction through a series of detailed, 10-minute videos. Not "supplemental" videos, the entire textbook in video. Not a guy in front of a chalkboard, Professor Burger on camera, engaging your students.
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Students do exercises online that are automatically graded and flow into your grade book. Choose from the exercises right out of the book, or algorithmically generated exercises. All students get unlimited algorithmic exercises for drill and practice.
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Automatic grading, easy to manage and export grades.
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Hold a white board session with the entire class. Hold office hours online with webcam and chat. Save your sessions to use again later. Communicate with announcements and engage with a discussion board.
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Best Practices in College Algebra:
Thinkwell's College Algebra champions:
• Understanding over memorization
• Depth over breadth
• Risk-taking in problem solving
• Creative thinking
• And the positive power of making mistakes
TOC at a glance:
• Real Numbers and Their Properties
• Equations and Inequalities
• Functions and Their Graphs
• Polynomial Functions
• Rational Functions and Conics
• Exponential and Logarithmic Functions
• Systems of Equations and Inequalities
• Matrices and Determinants
• Sequences, Series, and Probability
"The videos are interesting, engaging, informative and thoroughly explanatory. I don't get lost in a sea of symbols, having to ask what the purpose of anything is. The entire visual setup is great."
Benjamin Livingston | 677.169 | 1 |
Infinite series is a fundamental concept which is very useful in the study of real and complex functions. Special types of series such as Fourier series and power series have important applications in the study of differential equations. In this chapter we present fundamental facts about sequences and series of real numbers.
In this chapter, we develop some basic theorems about differentiable functions. We prove Rolleís theorem, Lagrangeís mean value theorem, Cauchyís mean value theorem and Taylorís theorem. We also obtain Macalurinís series expansion of some standard functions. We also discuss several applications of differentiation such as curvature, evolutes, envelopes, tangent and normal to a curve.
In this chapter we present some of the basic methods of evaluating integrals and its applications to the calculation of length of a curve, area under a curve and volume and surface area of a solid of revolution. We first give a brief summary of the basic definitions, and a list of standard integrals.
The problem of solving differential equations is a natural goal of differential and integral calculus. Further many of the general laws of nature in Physics, Chemistry, Biology and Astronomy can be expressed in the language of differential equations and hence the theory of differential equations is the most important part of mathematics for understanding Physical sciences. Also this theory has man ......
In this chapter we introduce the concept of Laplace transform which has interesting applications in several fields. The basic idea behind any transform is that the given problem can be solved more readily in the transform domain. Given a linear ordinary differential equation with constant coefficients, if we take Laplace transform of all terms in the equation then we obtain a linear algebraic equa ......
In this chapter we deal with the theory of vector calculus. With the help of a standard vector differential operator we introduce concepts like gradient of a scalar valued function, divergence and curl of a vector valued function, discuss briefly the properties arising out of these concepts and study the applications of the results to the evaluations of line and surface integrals. | 677.169 | 1 |
BIOL 345 - Introduction to Mathematical Biology
Students study papers drawn from research literature in which mathematics is used to model biological systems, encompassing the molecular level to the community level of organization. Topics from matrix algebra, differential equations, discrete mathematics, probability and statistics are introduced and developed through lectures, readings, and a computer lab. Data are drawn from a variety of sources, including student laboratory experiments, research papers, and scientific databases. The course meets for two hours of lectures and two hours of biology/computer lab each week. Cross-listed as MATH 345, this course is team-taught by a biologist and a mathematician. | 677.169 | 1 |
Gilbert strang essays in linear algebra
The renowned mathematician and educator Gilbert Strang presents a collection of expository papers on the theory and applications of linear algebra, accompanied by. The Four Fundamental Subspaces: 4 Lines Gilbert Strang The three parts are well separated in a linear algebra course. Essays and Papers; Goals of the Linear Algebra. A Factorization Review (ps, pdf) Glossary for Linear Algebra. (article by Alan Edelman and Gilbert Strang. Essays in Linear Algebra This book brings together 27 classic articles by Gilbert Strang on linear algebra The essays present historical background. Buy Essays in Linear Algebra on Amazon.com FREE SHIPPING on qualified orders.
By working with Wellesley-Cambridge Press Differential Equations and Linear Algebra. by Gilbert Strang Essays in Linear Algebra. by Gilbert Strang. 978-0. William Gilbert Strang. Essays in Linear Algebra (2012) Algorithms for Global Positioning, with Kai Borre (2012) Computational Science and Engineering (2007. Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at Balliol College Linear Algebra and Its Applications. Essays in Linear Algebra. Differential Equations and Linear Algebra - New Book Website Gilbert Strang. Linear algebra videos by Professor Strang are already on YouTube and. Essays. The.
Gilbert strang essays in linear algebra
Get this from a library! Essays in linear algebra. [Gilbert Strang] -- 27 classic articles by Gilbert Strang, each introduced by a new essay.--Back cover. Essays and Papers; Goals of the Linear Algebra. A Factorization Review (ps, pdf) Glossary for Linear Algebra. (article by Alan Edelman and Gilbert Strang. Differential Equations and Linear Algebra - New Book Website Gilbert Strang. Linear algebra videos by Professor Strang are already on YouTube and. Essays. The.
The Hardcover of the Essays in Linear Algebra by Gilbert Strang at Barnes & Noble. FREE Shipping on $25 or more. Essays in Linear Algebra by Gilbert Strang, 9780980232769, available at Book Depository with free delivery worldwide. The Hardcover of the Essays in Linear Algebra by Gilbert Strang at Barnes & Noble. FREE Shipping on $25 or more. By working with Wellesley-Cambridge Press Differential Equations and Linear Algebra. by Gilbert Strang Essays in Linear Algebra. by Gilbert Strang. 978-0. Get this from a library! Essays in linear algebra. [Gilbert Strang] -- 27 classic articles by Gilbert Strang, each introduced by a new essay.--Back cover.
Essays in Linear Algebra This book brings together 27 classic articles by Gilbert Strang on linear algebra The essays present historical background. This is a basic subject on matrix theory and linear algebra Gilbert Strang. 18.06 Linear Algebra. Spring 2010. Massachusetts Institute of Technology:. Get this from a library! Essays in linear algebra. [Gilbert Strang] -- 27 classic articles by Gilbert Strang, each introduced by a new essay.--Back cover. The Four Fundamental Subspaces: 4 Lines Gilbert Strang The three parts are well separated in a linear algebra course. Find great deals on eBay for gilbert strang linear algebra See more like this GILBERT STRANG - Essays in Linear Algebra - HARDCOVER ** Brand New *. | 677.169 | 1 |
This is a book for undergraduate students who would like to learn about
differential equations. It is not a classical book; rather, it presents a
modern approach to classical subjects. Though rigorous, it gives an
intuitive view of the concepts, and the ideas behind the results are
well explained. Technical questions are referred to the appendices or to
other books. Based on their experience, the authors emphasize and
explain the delicate concepts or notations with which people normally
have difficulties. In this way, the text could be used as a technical book,
but overall it would be very good complementary material. Apart from
theorems and proofs one can find illustrative comments, outlines of the
proofs, and references to real problems, models and examples.
The book does not contain many illustrations because it has a companion
website where the reader can find additional material, such as 2D
and 3D visualizations. The idea of combining the website and the book is
very good because the website
is able to present more
spectacular graphics and animations overall. It also allows one to play
with the examples. In my opinion such a very good idea should be
exploited more.
The first chapter of the book is devoted to the introduction of
differential equations and their solutions. Special types of solutions
that play an important role in the description and analysis of a flow
are presented. The notions of chaos and invariance are introduced in
this chapter, right at the beginning of the book.
Chapter 2 deals with linear differential equations. The concept of
asymptotic stability is studied. On the one hand, the harmonic motion
is considered as an example to analyze. On the other hand, the
exponential growth is treated and related to ecological models.
The subjects of Chapter 3 are second-order ordinary differential
equations and the calculus of variations. The study of higher-order
differential equations is reduced to the case of first-order equations
in the corresponding tangent bundle; thus, the results from Chapter 1
can be directly applied. As a model, the Euler-Lagrange equations are
explained, stating the basis for the study of classical mechanics. The
subject of Chapter 4 is Newtonian mechanics.
Finally, some numerical methods to approximate the solution of initial
value problems are explained in Chapter 5. | 677.169 | 1 |
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