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This book is an introduction to Mathematica, aimed at students, with each chapter focusing on a specific application of Mathematica. As a study guide it supplies a comprehensive, step-by-step approach to understanding the computer program. Numerous solved problems and examples demonstrate and reinforce ideas discussed in each chapter. Mathematica commands with examples are clearly classified according to type. The index of commands and applications expedites referencing. | 677.169 | 1 |
School of Arts and Sciences
Mathematics and Computer Science
Course Descriptions
CSM 154: MATHEMATICAL TECHNOLOGY 4 credits/ Powers
This course focuses on the use of technology as a tool for solving problems in mathematics, learning mathematics and building mathematical conjectures; electronic spreadsheets, a Computer Algebra System (CAS), and a graphing calculator; the use of these tools, programming within all three environments, including spreadsheet macros, structured CAS programming, and calculator programming. A TI-89 graphing calculator is required.
MTH 101: INTERMEDIATE ALGEBRA (F) 3 credits
This course addresses algebraic operations; linear and quadratic equations; exponents and radicals; elementary functions; graphs; and systems of linear equations. Students who have other college credits in mathematics must obtain permission of the department chair to enroll in this course. NOTE: Not to be taken to fulfill major requirements.
MTH 113: ALGEBRA AND TRIGONOMETRY (F) 4 credits
This course provides a review of algebra; simultaneous equations; trigonometry; functions and graphs; properties of logarithmic, exponential, and trigonometric functions; problem-solving and modeling. A TI graphing calculator is required.
MTH 114: APPLIED BUSINESS CALCULUS (F, S) 4 credits/ Powers
This course is an introduction to functions and modeling and differentiation. There will be a particular focus on mathematical modeling and business applications. Applications include break-even analysis, compound interest, elasticity, inventory and lot size, income streams, and supply and demand curves. The course will include the frequent use of Microsoft Excel. A TI-84 or TI-83 graphing calculator is required. Prerequisite: MTH 101 or its equivalent.
MTH 120: CALCULUS AND ANALYTIC GEOMETRY I (F, S) 4 credits/ Powers
Topics in this course include functions of various types: rational, trigonometric, exponential, logarithmic; limits and continuity; the derivative of a function and its interpretation; applications of derivatives including maxima and minima and curve sketching; antiderivatives, the definite integral and approximations; the fundamental theorem of calculus; and integration using substitution. A TI graphing calculator is required. Prerequisite: MTH 113 or its equivalent
MTH 150: MATHEMATICS: MYTHS AND REALITIES (F, S) 3 credits/ Powers
This course offers an overview of mathematical concepts that are essential tools in navigating life as an informed and contributing citizen, including logical reasoning, uses and abuses of percentages, financial mathematics (compound interest, annuities), linear and exponential models, fundamentals of probability, and descriptive statistics. Applications include such topics as population growth models, opinion polling, voting and apportionment, health care statistics, and lotteries and games of chance.
This course is the first half of a two-semester course in discrete mathematics. The intended audience of the course consists of computer science majors (both B.A. and B.S.) and IT majors. Topics in the course include logic, sets, functions, relations and equivalence relations, graphs, and trees. There will be an emphasis on applications to computer science.
MTH 261: DISCRETE STRUCTURES II (S) 3 credits
This course is the second half of a two-semester course in discrete mathematics. The intended audience of the course consists of computer science majors (both B.A. and B.S.) and IT majors. Topics in the course include number theory, matrix arithmetic, induction, counting, discrete probability, recurrence relations, and Boolean algebra. There will be an emphasis on applications to computer science. Prerequisite: MTH 260.
MTH 302: FOUNDATIONS OF MATHEMATICS (S) 3 credits
Topics in this course include propositional logic, methods of proof, sets, fundamental properties of integers, elementary number theory, functions and relations, cardinality, and the structure of the real numbers. Prerequisite: MTH 221.
MTH 321: REAL ANALYSIS 3 credits
This is a course that emphasizes the theory behind calculus topics such as continuity, differentiation, integration, and sequences and series (both of numbers and of functions); basic topology, Fourier Series. Prerequisites: MTH 222 and 302
MTH 322: DIFFERENTIAL EQUATIONS 4 credits
This course focuses on analytical, graphical, and numerical techniques for first and higher order differential equations; Laplace transform methods; systems of coupled linear differential equations; phase portraits and stability; applications in the natural and social sciences. Prerequisite: MTH 221.
MTH 330: MODERN GEOMETRIES (F, Even Years) 3 credits
Topics from Euclidean geometry including: planar and spatial motions and similarities, collinearity and concurrence theorems for triangles, the nine-point circle and Euler line of a triangle, cyclic quadrilaterals, compass and straightedge constructions. In addition, finite geometries and the classical non-Euclidean geometries are introduced. Prerequisite: MTH 240.
This is an introducory course to specialized areas of mathematics. The subject matter will vary from term to term. Prerequisite: junior mathematics standing
MTH 405: HISTORY OF MATHEMATICS (F, Odd Years) 3 credits
This course is an in-depth historical study of the development of arithmetic, algebra, geometry, trigonometry, and calculus in Western mathematics (Europe and the Near East) from ancient times up through the 19th century, including highlights from the mathematical works of such figures as Euclid, Archimedes, Diophantus, Fibonacci, Cardano, Napier, Descartes, Fermat, Pascal, Newton, Leibniz, Euler, and Gauss. A term paper on some aspect of the history of mathematics is required. Prerequisite: MTH 302. | 677.169 | 1 |
Casio Takes New Approach to Graphing Calculator for Students
Casio's Prizm fx-CG10 plots graphs over full-color images to help students visualize concepts.
Casio Education has introduced the Prizm fx-CG10, a new concept in educational graphing calculators that aims to impart mathematical concepts in addition to providing standard graphing functions. Using a new tool known as Picture Plot, the Prizm enables users to plot graphs over full-color photographic images, such as an Egyptian pyramid or the jets of an outdoor fountain, as way of relating complex mathematical functions to real-world concepts such as design and engineering.
Casio also offers teachers online training using streaming video and downloadable supplemental activities, as well as a loaner program, which enables interested educators to try the Prizm for 30 days. An application for the program | 677.169 | 1 |
He is also a poor maths student. However that means that the bits that he does understand, he understands and can explain really simply so that anyone [really, anyone!] can understand it.
Also, unlike your high school maths teacher he's interested in finding the quick ways that make maths work for individuals rather than focussing relentlessly on the underlying concepts. Learn what you need when you need it as he says!
what to bring
a calculator of some description - on your phone or computer is fine, or an actual calculator if you prefer.
costKoha / donation
our cancellation policy If you RSVP for a class and fail to show up, you will get a "no show". Three no shows means you will be removed from chalkle°. You may request to rejoin after six months. If you cancel your RSVP: - less than 3 days before = no refund - after 7pm the night before = no refund and a "no show" will be recorded
about chalkle° chalkle° connects people interested in teaching and learning from each other face-to-face -- in a fun, social environment! Choose from many affordable lessons up for grabs (from intros to master classes) in agriculture, art and craft, business, construction, digital media, food, languages, music and performance, science, technology and zombie survival... and many of the lessons are free!
Chalkle° connects people who want to teach with people who want to learn and enables learning for communities in real life. Small affordable classes mean high value for participants, extracting the knowledge and skills otherwise hidden within the community. Chalkle° redefines what it means to be a teacher or a learner, helping everyone undertake training in practical, artistic or academic skills while fostering lifelong learning.
who is behind chalkle°? Silvia Zuur and Linc Gasking from Enspiral have teamed up to create chalkle°.
hundreds of classes Chalkle° classes are face-to-face, easy to join and we have many affordable classes in the pipeline or already up for grabs (from intros to master classes) including:
Agriculture & Environment: Composting
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where do the classes take place? Our classes take place all across Wellington -- from Community Halls to the Train Station! We organize classes based off the interest of chalklers.
when will a class happen? As soon as we have a minimum number of people interested in a class, we get in touch with the teacher and arrange when they can next teach. So make sure you let us know which classes you would like to see happen by filling in the survey we send you when you join, so we can get busy organizing a time and a place for them! | 677.169 | 1 |
Catalog Entries
Introduces the mathematical structures and methods that form the foundation of computer science. Studies structures such as sets, tuples, sequences, lists, trees, and graphs. Discusses functions, relations, ordering, and equivalence relations. Examines inductive and recursive definitions of structures and functions. Discusses principles of proof such as truth tables, inductive proof, and basic logic. Also covers the counting techniques and arguments needed to estimate the size of sets, the growth of functions, and the space-time complexity of algorithms.
4.000 Credit hours
4.000 Lecture hours | 677.169 | 1 |
We are happy with Parmanand and the help he is providing Paul in Math. Theresa is also doing a good job helping him with English." Tom & Cynthia H.
Woodbury, MNAlgebra Tutors Pre-algebra is a common name for a course in middle school mathematics. In the United States, it is generally taught between the seventh and ninth grades, although students have taken this course as early as fifth or sixth grade. The objective of pre-algebra is to prepare the student to the study of algebra. Pre-algebra includes several broad subjects: Review of natural- and whole-number arithmetic; introduction of new types of numbers such as integers, fractions, decimals and negative numbers; Factorization of natural numbers; Properties of operations (associative, distributive and so on); Simple roots and powers; Rules of evaluation of expressions, such as operator precedence and use of parentheses; Basics of equations, including rules for invariant manipulation of equations; Variables and exponentiation. Pre-algebra often includes some basic subjects from geometry, mostly the kinds that further understanding of algebra and show how it is used, such as area, volume, and perimeter. Wikipedia Pre-algebra.
Algebra I & II - Algebra is a branch of mathematics concerning the study of structure, relation, and quantity. Together with geometry, analysis, combinatory, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots. Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields. Wikipedia Algebra
Abstract Algebra - Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulas and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings. Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory. Wikipedia Abstract Algebra | 677.169 | 1 |
If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
First principle: Understand the problem
"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:
What are you asked to find or show?
Can you restate the problem in your own words?
Can you think of a picture or a diagram that might help you understand the problem?
Is there enough information to enable you to find a solution?
Do you understand all the words used in stating the problem?
Do you need to ask a question to get the answer?
The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.
Second principle: Devise a plan
Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
Guess and check
Make an orderly list
Eliminate possibilities
Use symmetry
Consider special cases
Use direct reasoning
Solve an equation
Also suggested:
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Work backward
Use a formula
Be creative
Use your head/noggin
Third principle: Carry out the plan
This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.
Fourth principle: Review/extend
Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.
From Yahoo Answers
Question:Two blocks of masses m1=1kg and m2=2kg are connected by a massless cord passing over a massless, frictionless pulley. The pulley is pulled up with a force of F=50 N. What are the accelerations of the two masses with respect to the ground?
Question:
The mass show is hanging from light, smooth pulleys. It weighs 12.5 kg. How much force must be exerted in the direction shown to keep the mass stationary?
I know the answer is 62.5, but I have no idea how to go about getting that answer.
Answers:if the rope and pulleys are massless, then the tension in the rope will be the same everywhere, so if you pull on the rope with a force T, all portions of the rope will exert a force T
consider now the pulley supporting the weight; the rope on the left pulls up with a force T, and the rope on the right also pulls up with a force T
the combination of forces must equal the weight of the mass, so we have
2T=W
W=mg=12.5 kg x 9.8m/s/s = 122.5N
T=61.25 N
(if you are using g=10m/s/s, then you will get the answer that T=62.5N)
Question:Two blocks, m1 = 1 kg and m2 = 2 kg, are connected by a light string as shown in the figure***. If the radius of the pulley is 1 m and its moment of inertia is 5 kg m^2, What is the acceleration of the system? The figure is a pulley with m1 on the left side and m2 on the right side of the pulley.
*** Figures cannot be pasted onto yahoo, so the link is
It is problem number 12 (if you scroll down a bit), and the figure is on the right hand side. The answer is suppose to be (1/8)g. I would like to know how it was solved, I am using these problems as practice for the exam. Thank you in advance.
Question:I have troubles calculating the mechanical adavantage and need help answering some lab questions.
1) what are the units of % efficiency? (work divided by work=?) 2)how to calculate the mechanical advantage of a pulley system -3 fixed n 3 movable wheels? 3)what are the units of MA? 4)what is the relationship between mechanical advantage and the # of strands of string that are lifting a weight? 5) Since my pulley system has 3 fixed n 3 movable wheels, what should the input work be? if the work output is 0.1962 J?
Answers:1) Efficiency is a dimensionless quantity. It has no unit.
3) This mechanical advantage is also just a number.
For the rest of the answers you should describe your pulley system better.
From Youtube
Physics - Acceleration on a Pulley :Solving for acceleration given two masses on a frictionless pulley.
Inclines Tension Pulleys :Watch the full video at: Introductory physics is full of problems involving two simple machines: inclines and pulleys. Inclines are often a student's first introduction to motion in two dimensions and serve as great examples of the utility of vectors. Pulleys are used to change the direction of a force, and require a new concept, tension. This lesson introduces tension and solves simple problems involving inclines and pulleys. An understanding of vectors and the laws of motion is required. Don't forget to subscribe, for new lessons uploaded on Power Learning 21- | 677.169 | 1 |
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Comment: No writing, underlining, or highlighting. Eligible for FREE Super Saving Shipping! Fast Amazon shipping plus a hassle free return policy mean your satisfaction is guaranteed! Tracking number provided with every order. HONESTY IS MY POLICY (Compare Feedback!) Please read the following details about this book! Has wear but not excessively. A book which has clearly been used but not abused. From a private collection unless otherwise noted. Thank you for your orderProblem topics can include improper numbers, grouping symbols, using proportions, rationalizing fractions, multiplying and factoring expressions, solving linear and quadratic equations, working with inequalities, and using formulas in story problems.
Editorial Reviews
From the Back Cover
From signed numbers to story problems — calculate equations with ease
Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear — this hands-on-guide focuses on helping you solve the many types of algebra problems you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with fractions, exponents, factoring, linear and quadratic equations, inequalities, graphs, and more!
100s of Problems!
Step-by-step answer sets clearly identify where you went wrong (or right) with a problem
The inside scoop on operating and factoring
Know where to begin and how to solve the most common equations
How to use algebra in practical applications with confidence
About the Author
Mary Jane Sterling is also the author of Algebra For Dummies, Trigonometry For Dummies, Algebra I CliffsStudySolver, and Algebra II CliffsStudySolver. She taught junior high and high school math for many years before beginning her current 25-years-and-counting tenure at Bradley University in Peoria, Illinois. Mary Jane especially enjoys working with future teachers and trying out new technology.
Top Customer Reviews
I'm 35 and studying for a COMPASS test (college entrance exam for those who never took the ACT/SAT). I failed Algebra in High School, skipped classes alot, and remembered absolutely nothing, but I have a good head for numbers and logic, so I've found myself able to understand most of the concepts in this book so far. If you have no head for numbers, or feel that you aren't good with logic problems/eyeballing math questions, then you may want to also purchase the "Algebra For Dummies", as this book only *briefly* goes into each concept.
Also, as a side note: please don't give an item a review of 1 star simply because another dealer fell through on delivery: it can mislead people who just look at the overall rating of a product. :-)
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I have gone back to college after 20 years and need to take a College Algebra course. This workbook has been very helpful in reviewing things I went over a very long time ago and making sense out of them. I highly recommend it.
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I purchased the Algebra for Dummies to help me in my Algebra class. Though it is a gret book and explains everything in a very easy to understand way, I felt I needed more practice with problems. This workbook did the trick!
It correlates with the first book and you can use them together to fully understand the rules and concepts. A+++
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Having had to retake Algebra in high school, I considered myself worthy of this book's title. I bought the book and workbook and put in about 3-4 hours daily for 10 weeks. I also watched an episode of the original Star Trek daily, as Spock is my muse for math and science. I went through the book and workbook once and then through the workbook again and I was able to pass the placement at the city college I wanted to attend. I knew exactly when those two trains that left Kansas City at different times and at different speeds would be 870 miles apart. I was no longer intimidated by x. The boost that this gave to my confidence is priceless but sadly, Star Trek is not quite the same.
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This book is great for not only the beginner in algebra but also for those who need a brush up. I purchased these books to help me review algebra since I'm heading back to college. I like the workbook; after you do the work you can check the answers at the end of the chapter and she even explains how the answers were worked out. Both books are worked together so you can read over a whole chapter before you work in the workbook or just work in the workbook then read through the book when needed. Over all this is a great set of books for learning algebra.
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This book is easy to understand and makes learning fun and easy. I used it to help me brush up on math skills to be able to take college entrance exam. I passed with flying colors! I highly recommend it to everyone of any age.
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After being away from college for many years i will soon be heading back to take some math classes. I thought I would get a head start before my classes started by reviewing Algebra with this workbook. I've found it a worthwhile reference and book to practice with. The key to really understanding Algebra is being able to practice with as many problems as you can until you get it right. It's alot like learning to play the piano, from my experience.
Anyway, with that in mind, here are some areas of improvement that I would like to see in this book:
(1)The graphs are so tiny. It's very hard to draw on them. Bigger graphs would sure be helpful.
(2)More problems to solve would be a big help. The book says that it comes with 100's of problems to solve which is true. The problem is though, that some concepts it will review and then give you only one or two problems to practice on. If you get those two wrong, you will feel like you really haven't learned the concept yet.
(3)More detail on the Working with Inequalities Chapter - (Chapter 16) I still don't feel like I have a grasp of that chapter after spending many hours trying to understand it.
(4)More details on Chapter 21- Getting to the Point With Graphing. The problems in this chapter give no clear explanation as to how the results came about like in the other chapters.
Other than that it's a good book and worth a purchase.
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Sending feedback... | 677.169 | 1 |
ALEX Lesson Plans
Title: Conic Sections: Discovering the Degenerates
Description:
Through a mixture of online exploration, and teacher instruction, students will discover how the degenerate forms of the conic sections are formed and will be able to identify the degenerate case of each conic section Discovering the Degenerates Description: Through a mixture of online exploration, and teacher instruction, students will discover how the degenerate forms of the conic sections are formed and will be able to identify the degenerate case of each conic section.
Title: Conic Sections: Playing With Parabolas
Description:
Through a mixture of online exploration, and teacher instruction, students will discover how parabolas are formed and will be able to use the key components from a graph (vertex, focus and directrix,) With Parabolas Description: Through a mixture of online exploration, and teacher instruction, students will discover how parabolas are formed and will be able to use the key components from a graph (vertex, focus and directrix,) to generate the equation of a graph.
Title: Conic Sections: Playing with Hyperbolas
Description:
Through with Hyperbolas Description: Through
Title: Graphing at all levels: It's a beautiful thing!
Description:
This Arts Education (7 - 12), or Mathematics (9 - 12) Title: Graphing at all levels: It's a beautiful thing! Description: This
Title: Ellipse
Description:
ThisStandard(s): [MA2015] PRE (9-12) 37: (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. [G-GPE3]
Subject: Mathematics (9 - 12), or Science (9 - 12) Title: Ellipse Description: This
Title: Going the Distance for Circles
Description:
ThisStandard(s): [MA2015] GEO (9-12) 29: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [G-GPE1]
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Going the Distance for Circles Description: This
Thinkfinity Lesson Plans
Title: Analyze the Data
Description:
In
Standard(s): [MA2015] PRE (9-12) 18: c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [F-IF7d]
Subject: Mathematics,Science Title: Analyze the Data Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 | 677.169 | 1 |
Mathematics
Mathematics is one of the oldest academic subjects and is still very much alive and growing very fast. The advent of computers and data acquisition facilities have stretched the limits of what is possible in Mathematics to all branches of human endeavor. New developments are taking place all the time; some as a result of fresh ideas or reviewing of old techniques and problems from a fresh standpoint and others stimulated by the applications to physical, biological and social sciences, Economics, computing and so on. This has given rise to new IT based techniques of Mathematical study and created new computational methods for the purpose. | 677.169 | 1 |
Algebra 2 focuses on advanced mathematical operations, such as those pertaining to complex numbers, factorization, linear systems, matrices and elementary functions, which comprise the essential knowledge base for trigonometry and precalculus. Often times, students struggle because they lack the | 677.169 | 1 |
Product Description
This Saxon Algebra 1/2 kit includes the hardcover student text, softcover answer key & softcover test booklet, and the solutions manual. Containing 123 lessons, this text is the culmination of pre-algebra mathematics, a full pre-algebra course and an introduction to geometry and discrete mathematics. Topics covered include prime and composite numbers; fractions & decimals; order of operations, coordinates, exponents, square roots, ratios, algebraic phrases, probability, the Pythagorean Theorem and more. Utilizing an incremental approach to math, your students will learn in small doses at their own pace, increasing retention of knowledge and satisfaction!
The Solutions Manual features solutions to all textbook practices and problem sets. Early solutions contain every step, while later solutions omit obvious steps. Final answers are in bold type for accurate, efficient grading.
I love Saxon. Have taught it to my kids since they were in 1st grade. But this set of books wasn't well organized. Part of the answers are in one book, part in another. I still haven't found the explanation for the practice sets. Usually Saxon is great! What happened with this level?
This saxon is confusing, too repetitive in some areas, has insufficient instruction in other areas, and is NOT A MASTERY program.
There are too many problems per lesson. An "average" student cannot complete the lesson in an hours time. Students are left feeling that they are "poor" math students.
BETTER CHOICE would be video text algebra (it also has a pre algebra component) - i would HIGHLY recommend Video Text, which is a MASTERY program that teaches students the "why" of what they are learning | 677.169 | 1 |
9780521358nequalities (Cambridge Mathematical Library)
This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians | 677.169 | 1 |
Your question is kind of confusing. Combinatorics and set theory are parts of mathematics, so what do you really mean by "introducing mathematics through combinatorics/set theory"?
– Mercy KingJul 9 '12 at 16:07
It's made to introduce mathematics, but it will take you to a specific field, in the case: Combinatorics and Set Theory. Mathematics introductions could leave you in a broad range of fields.
– VoyskaJul 9 '12 at 16:10
1 Answer
1
The books you mentioned are not really introductions to mathematics through that topic. Rather the topic itself is a branch of mathematics; for example, calculus. Physics uses much mathematics, and much mathematics is motivated by physical problems; for example, differential equations. Some universities have undergraduate math syllabuses that cover a lot of physics; for example, Cambridge. For a good overview of undergraduate physics with lots of mathematical material, you may see The Feynman Lectures on Physics by Nobel prize winner Richard Feynman. | 677.169 | 1 |
Mathematica Tutorial: Solving Algebraic Equations
By Debra Czarneski
Sep 17, 2013
Linear algebra got you down? Tired of solving messy equations? Wish you had a genie that could magically solve all your algebra questions? Mathematica can help!
Tuesday, September 18, the Simpson Computational Modeling Club will hold a workshop showing you how to solve algebraic equations quickly and easily in Mathematica. We'll meet in Carver 105 from 5-6p. Stop by at your leisure.
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Why Simpson?
Our innovative curriculum is designed to help students think critically, solve problems creatively and communicate effectively — exactly the kind of skills that employers and graduate schools want most. | 677.169 | 1 |
1001 Basic Math & Pre- Algebra Practice Problems For DummiesPractice makes perfect—and helps deepen your understanding of basic math and pre-algebra by solving problems1001 Basic Math & Pre-Algebra Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in Basic Math & Pre-Algebra For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in your math course. You begin with some basic arithmetic practice, move on to fractions, decimals, and percents, tackle story problems, and finish up with basic algebra. Every practice question includes not only a solution but a step-by-step explanation. From the book, go online and find:One year free subscription to all 1001 practice problemsOn-the-go access any way you want it—from your computer, smart phone, or tabletMultiple choice questions on all you math course topicsPersonalized reports that track your progress and help show you where you need to study the mostCustomized practice sets for self-directed studyPractice problems categorized as easy, medium, or hardThe practice problems in 1001 Basic Math & Pre-Algebra Practice Problems For Dummies give you a chance to practice and reinforce the skills you learn in class and help you refine your understanding of basic math & pre-algebra.Note to readers: 1,001 Basic Math & Pre-AlgebraPractice Problems For Dummies,which only includes problems to solve, is a great companion to Basic Math & Pre-Algebra I For Dummies,which offers complete instruction on all topics in a typical Basic Math& Pre-Algebracourse | 677.169 | 1 |
9780130328Introductory Algebra for College Students (3rd Edition)
The goal of this series is to provide readers with a strong foundation in Algebra. Each book is designed to develop readers' critical thinking and problem-solving capabilities and prepare readers for subsequent Algebra courses as well as service math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. The Real Number System. Linear Equations and Inequalities in One Variable. Problem Solving. Linear Equations and Inequalities in Two Variables. Systems of Linear Equations and Inequalities. Exponents and Polynomials. Factoring Polynomials. Rational Expressions. Roots and Radicals. Quadratic Equations and Functions. For anyone interested in introductory and intermediate algebra and for the combined introductory and intermediate algebra | 677.169 | 1 |
Details about Trigonometry Workbook For Dummies:
From angles to functions to identities - solve trig equations with ease
Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear - this hands-on-guide focuses on helping you solve the many types of trigonometry equations you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with angles, circles, triangles, graphs, functions, the laws of sines and cosines, and more!
100s of Problems! * Step-by-step answer sets clearly identify where you went wrong (or right) with a problem * Get the inside scoop on graphing trig functions * Know where to begin and how to solve the most common equations * Use trig in practical applications with confidence
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Rent Trigonometry Workbook For Dummies 1st edition today, or search our site for other textbooks by Mary Jane Sterling. Every textbook comes with a 21-day "Any Reason" guarantee. Published by For Dummies. | 677.169 | 1 |
Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus — how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.
This is a basic course on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positivedefinite matrices. | 677.169 | 1 |
This booklet includes 11 problems of increasing difficulty that explore the properties of black holes. Students learn about the relationship between black hole size and mass, time dilation, energy extraction, accretion disks, and what?s inside a black hole by working with a series of math problems that feature simple algebra, scientific notation, exponential functions, and the Pythagorean Theorem. (8.5 x11, 28 pages, 11 color images, PDF file) | 677.169 | 1 |
This book presents first-year calculus roughly in the order in which it was first discovered. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading. more...
The notion of "human rights" is widely used in political and moral debates. The core idea, that all human beings have some inalienable basic rights, is appealing and has an important practical function: It allows moral criticism of various wrongs and calls for action in order to prevent them. The articles in this collection take up a... more... | 677.169 | 1 |
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Resources for Teaching Discrete Mathematics
Brian Hopkins, Editor
A resource for discrete mathematics teachers at all levels.
Resources for Teaching Discrete Mathematics presents nineteen classroom tested projects complete with student handouts, solutions, and notes to the instructor. Topics range from a first day activity that motivates proofs to applications of discrete mathematics to chemistry, biology, and data storage. Other projects provide: supplementary material on classic topics such as the towers of Hanoi and the Josephus problem, how to use a calculator to explore various course topics, how to employ Cuisenaire rods to examine the Fibonacci numbers and other sequences, and how you can use plastic pipes to create a geodesic dome.
The book contains eleven history modules that allow students to explore topics in their original context. Sources range from eleventh century Chinese figures that prompted Leibniz to write on binary arithmetic, to a 1959 article on automata theory. Excerpts include: Pascal's "Treatise on the Arithmetical Triangle," Hamilton's "Account of the Icosian Game," and Cantor's (translated) "Contributions to the Founding of the Theory of Transfinite Numbers."
Five articles complete the book. Three address extensions of standard discrete mathematics content: an exploration of historical counting problems with attention to discovering formulas, a discussion of how computers store graphs, and a survey connecting the principle of inclusion-exclusion to Möbius inversion. Finally, there are two articles on pedagogy specifically related to discrete mathematics courses: a summary of adapting a group discovery method to larger classes, and a discussion of using logic in encouraging students to construct proofs. | 677.169 | 1 |
Features Technology
The urge to explore is a common feeling in a lot of people, regardless of their career. curiosity is a natural facet of human nature. Even a lot of animals are inherently curious of their surroundings, though they tend to have an instinct that prevents them from putting themselves in immediate danger. Humans have been known to bypass this instinct in favor of the pursuit of knowledge.
Excavating is one such activity that can actually be more dangerous than most people think about. Whenever you set out to dig a large expanse of ground, it's always important to be aware of your surroundings. Cave-ins can result in severe injury, and you never know what may be uncovered.
It may seem like a rather disappointing journey, if you set out to dig and find nothing. However, there are highly sophisticated tools available that will give you an edge and sense of understanding of the ground beneath your feet. The AIG resistivity method employs the use of electrical currents that travel through the ground and bounce back. It works on a basis similar to echo-location, and you'll soon find that it comes in handy for many different digging projects.
The data relays back in the form of graph that pinpoints what kind of substances are located and how far underground they are. It may not provide a photograph of what's down there, but it certainly paints a picture that's vivid enough for anyone on an excavation project. It's a machine that's well worth the investment.
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Do you have any idea about how troublesome algebra is? If you have not gotten anything about it then you should be ready for your chance to meet with this lesson as soon as possible. It is the part of math lesson and one of the hardest subjects which have made students turned pale when they get many algebra assignments. When students find that they cannot do the assignments by themselves then they will start to look for eduboard - help with algebra liner which can help them to deal with their algebra assignment. Math has become one of the essential subjects that you have to do magnificently so that you will be able to pass your grade safely. With that duty on your shoulder then you should try anything so that you can have magnificent score in math, especially with algebra. If you cannot deal with your algebra then you will not be able to pass your grade with good mark.
Even it might downgrade all of your hard works with other subjects too. When you deal with math then you will need to have serious though about it. They will not give the score easily and you will suffer a long period payment if you cannot get high rank in math. You will not be able to get over it that fast since you should take that subject next semester until you can pass it with satisfactory. With this condition that you have to face then it is not a problem whether you like math or not since you will still have to pass it. That can be a proper reason why you should get online tutor that will not only give you help with your algebra homework yet also you can get them whenever you need their help. When you have online tutors to help you then you can deal with your assignments effectively.
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home
blurb
We build on the theory and techniques developed in Calculus. After a recap of the material that will be assumed, we'll start on the "core" of the course, whose topics will be: an introduction to differential equations, hyperbolic functions, integration techniques, uses of integration, polar coordinates, complex numbers, sequences and series, and power series. We will cover at least one more topic, or look at one or more of these in more depth, the choice to be determined by the interests of the class.
The core will approximately follow chapters six through ten of Strang's Calculus with a few omissions to be announced as we go.
This website will grow as the course progresses, and house assignments, grades, etc. See the links on the left for more. | 677.169 | 1 |
"Give me a place to stand and I will move the world." -Archimedes
From classic antiquity to the modern age, mathematicians have been moving our understanding of the world forward. Iona's Mathematics programs will prepare you to make an impact in some of today's most challenging fields–and move your own career forward in the direction that interests you.
In addition to gaining a solid foundation in mathematical methods and techniques, you'll develop the logical and analytical thinking skills that will enable you to become a valued problem-solver within a broad range of industries.
The Mathematics Department has a proven track record when it comes to job placement.
An opportunity to study broadly and deeply
Iona's mathematics curriculum emphasizes the three major fields of study, algebra, analysis and applied mathematics, while giving students the chance to delve more deeply into an area of personal interest. All courses within the major are taught by full-time faculty, not teaching assistants, as is the case at many other schools. Iona's commitment to small class sizes ensures that you will benefit from personalized attention that is paramount to student success.
Mathematics alumni make a difference in the world
Edward A. Silver: In addition to the bachelor's degree in mathematics from Iona College, Dr. Silver went on to earn an Ed.D. at Columbia University. He has taught at the middle school and high school levels in New York State and at universities in Illinois, California, and Pennsylvania. He currently serves as Dean of the School of Education at the University of Michigan, Dearborn. | 677.169 | 1 |
This book assembles in a single volume a collection of original and review articles encompassing topics in theoretical developments in operator theory, exploring its diverse applications in applied mathematics, physics, engineering and other disciplines. more...
Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course. You'll get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step... more...
From nutritional labels and box office statistics to terabytes and megapixels, the 21st century world is awash in numbers. How can the average Joe or Jane make sense of all that data? The key, Theoni Pappas argues, is math. In Mathematical Snippets, she draws readers into the fascinating world of math without overwhelming them with mind-numbing... more... | 677.169 | 1 |
Survey Participation
Math Majors Needed for Research Study
The purpose of this research project, titled How Are Spatial and Numerical Skills Related investigates the specific connections between different types of spatial skills and numerical skills. In this study, your spatial and mathematical skills will be tested and the data will be used to test specific hypotheses. | 677.169 | 1 |
Find an Auburn, NJExponents, logarithms, roots, and radicals are introduced, along with the new concepts of infinite sequences and series. Equations and inequalities with absolute value-terms, imaginary numbers, and logarithms are often confusing for many students and require careful explanations and examples. The subject of calculus covers a lot of material and usually requires multiple semesters | 677.169 | 1 |
Surprise! Complex numbers are
not just a "math oddity" that only we crazy mathematicians enjoy.
Since complex numbers provide a system for finding the roots of
polynomials, and polynomials are used as theoretical models in various
fields, complex numbers enjoy prominence in several specialized areas.
Among these specialized areas are engineering, electrical engineering and quantum
mechanics. Topics utilizing complex numbers include the
investigation of electrical current, wavelength, liquid flow in relation
to obstacles, analysis of stress on beams, the movement of shock absorbers
in cars, the study of resonance of structures, the design of dynamos and
electric motors, and the manipulation of
large matrices used in modeling. While many of
these applications are beyond the scope of an Algebra2/Trig curriculum, an
introductory glimpse of the application of complex numbers to electrical
circuits can be easily understood and manipulated by students.
Application to Electrical Engineering:
First, set the stage for the discussion and clarify some vocabulary.
Information that expresses a single dimension, such as linear distance,
is called a scalar quantity in
mathematics. Scalar numbers are the kind of numbers students use
most often. In relation to science, the voltage produced by a
battery, the resistance of a piece of wire (ohms), and current through a
wire (amps) are scalar quantities.
When electrical engineers analyzed alternating current circuits,
they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar
one-dimensional scalar quantities that are used when measuring DC
circuits. These quantities which now alternate in direction and
amplitude possess other dimensions (frequency and phase shift) that must
be taken into account.
In order to analyze AC circuits, it became necessary to represent
multi-dimensional quantities. In order to accomplish this task,
scalar numbers were abandoned and complex
numbers were used to express the two dimensions of frequency
and phase shift at one time.
In mathematics, iis used
to represent imaginary numbers. In the study of electricity and
electronics,j is used to
represent imaginary numbers so that there is no confusion with i, which
in electronics represents current. It is also customary for
scientists to write the complex number in the form a + jb.
Introduce the formula E = I • Z
where E is voltage, I is current, and Z is impedance.
Possible Student Questions:
The impedance in one part of a series circuit is 2 +
j8 ohms, and the impedance in another part of the circuit is 4 -
j6 ohms. Find the total impedance in the circuit. Answer: 6 + j2 ohms | 677.169 | 1 |
Preparing for the AP* Calculus AB and BC Exams
This six week course will help students prepare for the AP* Calculus AB Exam.
* AP Calculus is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product.
Sessions
Course at a Glance
8-10 hours/week
English
Instructors
Categories
About the Course
This course reviews the topics for the AP Calculus AB Exam. Each week of the six week review will include review videos for an important block of material, complete with videos of worked example problems and exam tips. There will also be an extensive online quiz given each week, with multiple problem versions that students can use to practice, along with an practice tests to help students prepare for the AP* Calculus AB Exam.
* AP Calculus is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product.
Course Syllabus
Week 1: Limits, Continuity and Derivatives
Week 2: More on Derivatives, Curve Sketching and Applications of Derivatives
Week 3: Antiderivatives and the Fundamental Theorem of Calculus
Week 4: Integration Techniques
Week 5: Applications of Integration
Week 6: Exam Practice and Tips
Recommended Background
Concurrent enrollment in an AP* AB Calculus course, or prior exposure to the material in AP* AB Calculus.
Course Format
This
class will consist of several lecture videos over content taught in high school
AP* AB Calculus. There will be quizzes over the content reviewed each week
as well as optional stand alone homework assignments. We will also make
full use of the class discussion forum to check for understanding and prepare
you for the AP* AB Calculus Exam given in May. There will be a practice
exam given at the end of the course.
FAQ
Will I get a Statement of Accomplishment after completing this class?
Yes. Students who successfully complete the class will receive a Statement of Accomplishment signed by the instructors.
What resources will I need for this class?
For this course, all you need is an Internet connection, and the time to read, watch videos, work problems, discuss, and enjoy mathematics.
How will I benefit from taking this class?
This course will help you make a 5 on the AP* Calculus AB Exam.
* AP Calculus is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. | 677.169 | 1 |
How can we solve the national debt crisis? Should you or your child take on a student loan? Is it safe to talk on a cell phone while driving? Are there viable energy alternatives to fossil fuels? What could you do with a billion dollars? Could simple policy changes reduce political polarization? These questions may all seem very different, but they... more...
Written in an engaging style that renders complex concepts accessible to a wide readership, and with more than 450 exercises, this textbook introduces the basics of real analysis as well as more advanced topics such as measure theory and Lebesgue integration. more...
L.E.J. Brouwer: Collected Works, Volume 1: Philosophy and Foundations of Mathematics focuses on the principles, operations, and approaches promoted by Brouwer in studying the philosophy and foundations of mathematics. The publication first ponders on the construction of mathematics. Topics include arithmetic of integers, negative numbers, measurable... more...
In this monograph, which is an extensive study of Hilbertian approximation, the emphasis is placed on spline functions theory. The origin of the book was an effort to show that spline theory parallels Hilbertian Kernel theory, not only for splines derived from minimization of a quadratic functional but more generally for splines considered as piecewise... more...
Model Answers in Pure Mathematics for A-Level Students provides a set of solutions that indicate what is required and expected in an Advanced Level examination in Pure Mathematics. This book serves as a guide to the length of answer required, layout of the solution, and methods of selecting the best approach to any particular type of math problem.... more... | 677.169 | 1 |
Tagged Questions
For questions related to the teaching and learning of mathematics. Note that Mathematics Educators StackExchange may be a better home for narrowly scoped questions on specific issues in mathematics education.
I am writing my P exam for actuaries. I have the solution manual but I ran into this question, which confused me. I understood the solution but it did differently in how I wouldve tackled the problem.
...
So far, I've gotten that the line is parallel to the plane $x = 2 + t$, $y = -3 + 2t$, $z = 1 + 4t$
With the vector of that being $U$ is $(1,2,4)$ and the plane $2y-z = 1$ with the vector $V$ being ...
A long time ago (but I can't remember when), I was introduced to the (pedagogical) concept of writing a proof as giving a winning strategy for a game. Basically, given a statement $\forall x\exists y ...
At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
I'm going to teach very elementary combinatorics (limited to basic enumeration) during two weeks to middle school students. At the beginning, I want to demonstrate the importance of counting in real ...
Hey guys I need help showing if a function is a vector space or not. I believe we show is addition and multipication holds. but I don't know. Also how do I find out dimensions of such functions.
The ...
this is my differential equation course. I got back into school after a couple of years. This is just the start of this course and I am having difficulties in one of these practice problems.
This is ...
Proving: $A$ is closed iff $A = \bar{A}$.
"To the right": If $A$ is closed, $ A = \bar A$
If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
I'm going to be a teaching assistant and I'm currently looking for books/reviewed articles/journals written by mathematicians or people who taught mathematics (at a university level) about pedagogy ...
I am searching for math competitions for college students. Of course I am familiar with Putnam, but I am looking for a lower ranking competition. Either US-national or regional (South/Texas). I tried ...
Math people:
Until recently, at least, there existed at least one Web page containing complete Putnam competition problems and solutions from the past twenty years or so. In retrospect, I see that I ...
To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
Is there a faster way to solve math problems? I'm talking about proving theorems, proving certain properties of a function, etc. The way I do it is I write out the problem and all relevant definitions ...
My question is quite simple. I have been googling a lot lately trying to find a solution to this:
Given a sequence of n integers $[1,2,...,n]$. If we pick two numbers randomly from the set say, a and ...
I want to learn Mathematics but I don't know where to start. Sometimes I really get frustrated as I am a Software Engineering graduate (currently working) and I feel like I don't know anything about ...
I have some weird gaps in my learning, I can do lambda calculus and some basic category theory, but I do not know how to do some of the most basic of arithmetic.
(I am in my mid 20s)
Seeing this gap ...
I was reading a book on coding theory, there was a definition fot the Hamming's Distance and also one example. Understanding purely from the definition was hard but the example helped to give meaning ...
How would you respond to a middle school student that says: "How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. ...
I've heard that in the first class of my degree are teaching concepts like metalanguage. What does it mean. Could you give some examples? I've searched on google that metalanguage means representing ... | 677.169 | 1 |
Description
outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This "Schaum's Outline" gives you More than 2,400 formulas and tables. It covers elementary to advanced math topics. It is arranged by topics for easy reference. Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time - and get your best test scores!
About John Liu
Seymour Lipschutz is on the faculty of Temple University and formally taught at the Polytechnic Institute of Brooklyn. He received his Ph.D. in 1960 at Courant Institutes of Mathematics and Sciences of New York University. The late Murray R. Spiegel received the M.S. degree in physics and the Ph.D. in mathematics from Cornell University. He had positions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute, and served as a mathematical consultant at several large companies. His last position was professor and chairman of Mathematics at the Rensselaer Polytechnic Institute, Hartford Graduate Center. | 677.169 | 1 |
Calculus Know-It-All: Beginner to Advanced, and Everything in Between ...
Read More private tutor--without the expense! His clear, friendly guidance helps you tackle the concepts and problems that confuse you the most and work through them at your own pace. Train your brain with ease! Calculus Know-It-ALL features: Checkpoints to help you track your knowledge and skill levelProblem/solution pairs and chapter-ending quizzes to reinforce learning Fully explained answers to all practice exercises A multiple-choice exam to prepare you for standardized tests "Extra Credit" and "Challenge" problems to stretch your mind Stan's expert guidance gives you the know-how to: Understand mappings, relations, and functionsCalculate limits and determine continuityDifferentiate and integrate functionsAnalyze graphs using first and second derivativesDefine and evaluate inverse functionsUse specialized integration techniquesDetermine arc lengths, surface areas, and solid volumesWork with multivariable functionsTake college entrance examinations with confidence And | 677.169 | 1 |
Announcements and Homework Midterm date is set: Friday, Oct. 21. Please refer to the Study Guide for more information.
Please refer to the List of Mini-quizzes for the most updated list of definitions and statements of theorems that can be asked on a mini-quiz.
Please refer to the List of Homework Assignments for the written homework assignments and due dates.
Prerequisites Calculus II or equivalent, and some natural curiosity.
Text We are using the free online book by Jiri Lebl, Basic Analysis, with Pitt supplements by Frank Beatrous and Yibiao Pan. Please let me know if you are interested in purchasing a hard copy for $16-$18.
Overview The course covers the foundations of theoretical mathematics and analysis.
Topics include sets, functions, number systems, order completeness of the
real numbers and its consequences, and convergence of sequences and series
of real numbers.
Core topics
Logic, proofs and quantifiers. Basic set theory. Functions and relations.
Elementary properties of the natural numbers; mathematical induction.
Axiomatic introduction to the ordered fields of rational and real numbers.
Elementary inequalities.
The Completeness Axiom; Archimedean Property of the real numbers; density
of the rational and irrational numbers in the real numbers.
Countability of the rationals; decimal expansions of real numbers;
uncountability of the real numbers.
Sequences and an introduction to series; the geometric series; limits;
Limit Laws.
The Monotone Convergence Theorem.
The Bolzano-Weierstrass Theorem.
Cauchy sequences; Cauchy completeness of the real numbers.
Course Goals This course is a prerequisite for MATH 0420, and together they provide a rigorous foundation of the one-variable Calculus. Besides learning the topics listed above, you will get a better understanding of what constitutes a rigorous mathematical proof. This is a writing-intensive course, and during the semester you will be writing mathematical proofs of increasing degree of complexity. You will also learn to recognize and correct some mistakes in mathematical proofs.
Notes on the Structure of the Course 1. Recitations are an indispenable part of the course. Attendance and appropriate participation are required, part of the course grade comes from the recitation grade, which will be assigned by your recitation instructor.
2. In the beginning of every lecture you will be asked to answer 2-5 simple questions on the definitions and statements of theorems. The primary purpose of these mini-quizzes is to ensure that you can follow the topic of each lecture.
3. In addition to the daily mini-quizzes, you will have regular quizzes (during recitations, announced in advance), and homework. Their purpose is to help you achieve the mastery of the material required for solving problems on the midterm and the final, and to improve your mathematical writing skills. Some of these quizzes and homework may specifically address mathematical writing and proof comprehension/analysis.
Grading
Daily Mini-quizzes
20%
Regular Quizzes and Homework
20%
Recitation Grade
10%
Midterm (Date to be announced)
20%
Final (Date and place to be announced)
30%
Academic Integrity Cheating/plagiarism will not be tolerated.
Violations of the
School of Arts and Sciences Policy on Academic Integrity
will result in a minimum sanction of a zero score for
the quiz, exam or paper in question. Additional sanctions may be imposed,
depending on the severity of the infraction.
Homework Policy You may work with other students and/or use library/web resources, but each student
must write up his or her solutions independently. Copying solutions from other
students will be considered cheating, and handled accordingly.
Disability Resource Services If you have a disability for which you are or may be requesting an
accommodation, you are encouraged to contact both your instructor and the
Office of Disability Resources and Services,
216 William Pitt Union (412) 624-7890 as early as possible in the term. | 677.169 | 1 |
Plantersville, TX Calculus
David S
John W.
...For example in the study of a course like biology, the student learns how to approach the material, highlight key points, and check in the margin items that are not clear. He/she works on the items that require clarification and erases the pencil checks as he/she gains understanding. In that way, the student always identifies areas requiring extra work and focusses on those items | 677.169 | 1 |
Calc-Add
A calculator and adding machine with many features including: the ability to stay on top; commas in the entry box; Micro Window; toggle between two window sizes; Tape Display (history), saving, printing and annotating; and a Stationary Decimal.
Math Flight 2.2.1 Learning mathematics can be a challenge for anyone. Math Flight can help you master it with three fun activities to choose from! With lots of graphics and sound effects, your interest in learning math should never decline.
Visual Kalman Filter 2.8 "Visual Kalman Filter " is a visual math tool to simulate Kalman filter for linear or nonlinear system. Only three steps you need do,and you'll get the curve and the estimated results..
The Math Slate The Math Slate provides elementary school students with practice in the basic math skills of addition, subtraction, multiplication, and division, through a simple, friendly interface that resembles a chalkboard. Teacher control is emphasized.
QB - Math Do you think your kids may benefit from more math practice? If so, here's a Windows program that will help! QB - Math is an easy to use Windows program that lets you or your kids practice Multiplication, Addition, Subtraction and Division.
Functions_1 Functions_1 discusses different type of relationships, one-to-one, one-to-many and many-to-one, and how to differentiate between them and which relationship is a function or not, and how to graph it.
Machine Design Tools Evaluation (Pocket PC) A must have for every Machine Design Engineer! A specialty calculator for PocketPC. Forget having to remember complicated formulae, just enter the values and see the solution!
Univerter Univerter is a sophisticated units conversion utility designed with ease-of-use and functionality in mind. | 677.169 | 1 |
This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
more
The first part of this preface is for the student; the second for the instructor. But whoever you are, welcome to both parts. For the Student You have finished secondary school, and are about to begin at a university or technical college. You want to study computing. The course includes some mathematics { and that was not necessarily your favourite subject. But there is no escape: some finite mathematics is a required part of the first year curriculum. That is where this book comes in. Its purpose is to provide the basics { the essentials that you need to know to understand the mathematical language that is used in computer and information science. It does not contain all the mathematics that you will need to look at through the several years of your undergraduate career. There are other very good, massive volumes that do that. At some stage you will probably find it useful to get one and keep it on your shelf for reference. But experience has convinced this author that no matter how good the compendia are, beginning students tend to feel intimidated, lost, and unclear about what parts to focus on. This short book, on the other hand, offers just the basics which you need to know from the beginning, and on which you can build further when needed. less | 677.169 | 1 |
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures don't know, do you ever thing about things that are imaginary, like unicorns or leprechauns, or the Tooth Fairy, things that are really - let's face it, I'm thinking about things that are imaginary. That's right. This is nice. We're in a happy place now. But really, sometimes things that are imaginary are required, because they're just not real. Let me show you something that just ain't real. In fact, you may remember, when you're taking square roots, if I said to you, "What's the ?" what you've got to say there is you've got to say, "Okay, no problem. I've got to find some number that, when I multiply it by itself, it's going to equal -9." Now, wait a minute, if I take a number and multiply it by its self, whether it's positive or negative, when I multiply it by itself, it becomes positive, unless it's zero, in which case then it's zero. There's no way that I'm going to be able to take two numbers, multiply them together and get -9. So this is just not going to happen, this is just not real. So therefore, it must be imaginary. It's pretend time.
Now, if we were to pretend, let's just pretend for a second that this really was some sort of wacko number. Well, what would we say about it? So now, let's just have a little fantasy. I could write this, for example, as -1 times 9. And then I could use properties of exponents, because remember a square root is just an exponent of , and say that's just . Now, , I happen to know there's no imagine required for that. That's just 3, cut and dry. So let me write the 3 out in front. And then I've got . So it's that's sort of the imaginary part. It's not a real number. And, in fact, we have a name for that, we call it i, i for imaginary, just like my imaginary friend.
So if I call that i, and usually we use a fancy - we don't use a capital I because we don't want to make a big stink about it, we just use a lowercase i. And that stands for - it's not a real number, it's an imaginary number, . So that's what this symbol means. By the way, this is a little trivia fact you can impress your friends with if you want, that if you're an engineering person, do you know what they use? They use little j. They have the thing sort of dangle this way, but who cares? Anyway, we'll use i, because that's the way it's supposed to be. The engineering people don't know what they're doing.
Anyway, I'd write this as 3i, and that means it's 3 times this imaginary number. Okay, so when you're armed with that, now you can actually start writing all sorts of numbers, because what we can do is write what are called complex numbers. There it is. And what a complex number is, is a number that has i's in it, that's all. So, for example, something like this, if I took 2 + , I could write that in the following way: well, I'll just keep the 2 there, but we just saw is actually the same thing as 3i, because the = 3 and = i. So, in fact, an imaginary number is any number that looks like this, some real number plus a real number times i. So this is called a complex number. This number here is called the real part. This is the real part, because it's just real. Now, this part actually is the part that's the imaginary part, so this 3 is called the imaginary part. So the real part of this number is 2, the imaginary part of this number is 3. If I look at this number, how about 3 + , I could write that as 3 plus - and while the = 5, but then I have , which is i. So this is a complex number, 3 + 5i, and the real part is 3 and the imaginary part is 5.
So, now you can actually start dealing with imaginary numbers. And guess what? That means it must be imaginary arithmetic. Cool! We'll see what's up next.
Prerequisites
Complex Numbers
Introducing and Writing Complex Numbers Page [1 of 1 | 677.169 | 1 |
This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.
Reviews:
"...sure to become a standard as an introductory graduate text in combinatorics."
Bulletin of the AMS"
"As a researcher, Stanley has few peers in combinatorics...the trove of exercises with solutions will form a vital resource; indeed, exercise 6.19 on the Catalan numbers, in 66 (!) parts, justifies the investment by itself. Both volumes highly recommended for all libraries."
Choice
"Volume 2 not only lives up to the high standards set by Volume 1, but surpasses them... Stanley's book is a valuable contribution to enumerative combinatorics. Beginners will find it an accessible introduction to the subject, and experts will still find much to learn from it."
Mathematical Reviews
'… an authoritative account of enumerative combinatorics.'
George E. Andrews, Bulletin of the London Mathematical Society
'What else can be added to the comments upon this excellent book?'
EMS Newsletter | 677.169 | 1 |
Find a Redondo BeachNavigate through various word problems that will increase their math literacy. Start looking at the next unit to help them get ahead.
-Students at a medium level of mastery: Pose questions to guide their thinking. Build content mastery through increasing math fluency.
...As It is essential that students and t...
...Algebra is the basis for future math classes. This is the core of a student's success in math. The thing about algebra that is difficult to understand is that there are often several ways to approach a problem. | 677.169 | 1 |
Intermediate Algebra
This algebra course covers radicals, exponents, concepts of relations and functions, exponential and logarithmic functions, linear and quadratic functions, and the solutions of equations from these topics.
Subject:MATH Course Number:64 Section Number:1961 Units:4 Instructor:
Instructor information about this course
Learning Management System (LMS) for this course:Custom LMS link: Course start page: Course email:jharland@miracosta.edu Office: Office hours: Phone: Instructor notes:WE USE FREE OPEN SOURCE MATERIALS FOR THIS CLASS. Students are not required to purchase online access to any programs and are not required to purchase a printed textbook, but may do so. ON OR BEFORE FIRST DAY OF CLASS, STUDENTS MUST GO TO to go to the syllabus. STUDENTS MUST FOLLOW DIRECTIONS FOR ENROLLING AND SUBMITTING WORK IN THE ONLINE MATERIALS ON THE FIRST DAY OF CLASS OR MAY BE DROPPED. Students will need to have daily access to a computer with good internet access. Students should plan on dedicating a minimum of 12 hours a week to this course. This class requires students to take proctored tests in person. Appointments are required, and should be made at least a few weeks in advance of each test due date. | 677.169 | 1 |
Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understa
Formula MAX is a universal app with a collection of over 1150+ Physics, Chemistry and Maths formulas, more formulas to be added constantly through updates. Use your Formula MAX app across your iOS devi
Available in many languages, Maths Formulas is a perfect app on App Store that provides all basic and advanced formulas in mathematics. It's very convenient for all students in high school or universit
"over 100 math formulas at high school level" - "Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of t...
"see an answer to think about a math formula " - "Find an answer. However, you already see it.
See an answer to think about a math formula. It's English styled math drills.
[How to play]
1. Create 20 | 677.169 | 1 |
It requires real understanding and good work over the entire school year. Math concepts build on each other so that future topics depend on understanding previous material. Therefore, misunderstanding one topic can cause continuous problems down the road | 677.169 | 1 |
About the U of T Mathematics Network
It is designed to encourage high school students
to actively participate in doing mathematics,
providing cooperative, competitive, interesting,
and interactive projects, as well as
more traditional problems and other quality resource material.
The Network is also intended to promote communication
and mathematical discussion between high schools (in both curricular
and extra-curricular settings) and the university,
and to establish a connection
from the high-school mathematical experience to mathematics at the
university level and beyond.
Through this Network we aim to provide
Opportunities for students
to actively participate in doing mathematics:
a variety of events to which students can come,
interactive projects and activities,
featuring mathematical challenges as well as
interdisciplinary topics that integrate mathematics into other areas;
answers and
explanations of questions that often arise
as one begins to think about mathematics at a more advanced level,
along with expositions of a variety of topics;
resource material for classroom or individual use;
A forum for communication:
opportunities for questions and discussions about mathematics
at the high school level, university level, and beyond,
and the
use of mathematics in other fields;
personal contact between university and students through a variety
of events and activities, such as the annual Mathematical Sciences day
and the in-depth SOAR summer camp begun last year;
A link from the high-school mathematical
experience to mathematics
at the university level and beyond:
student involvement in more advanced mathematical projects such
as the SOAR summer camp;
presentations of current mathematical research activities and
descriptions of the use of mathematics in various fields of study here
at the University.
For more information on Network activities and events, please contact: | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
An introduction to the mathematics of money : saving and investing
This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction, recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts, certificates of deposit, student loans, credit cards, mortgages, buying and selling bonds, and buying and selling stocks. The book is self contained and accessible. The authors follow a systematic pattern for each chapter including a variety of examples and exercises ensuring that the student deals with realities, rather than theoretical idealizations. It is suitable for courses in mathematics, investing, banking, financial engineering, and related topics.Read more...
Abstract:
This introduction to the mathematics of finance is suitable for undergraduates in mathematics, economics and business programmes. Assuming no background or experience in investing, the book introduces the reader to principles of investing that will be of life-long practical use.Read more...
Reviews
Editorial reviews
Publisher Synopsis
From the reviews: "This book is written for students without assuming a background or any experience in investing. A basic knowledge in real analysis is necessary. The student is introduced to elements of saving and investing that are of lifelong practical use. These includes saving, checking accounts, certificates of deposit, student loan, credit cards, mortgages, buying and selling bonds of stocks. The authors follow a systematic pattern with a variety of examples and exercises. ... suitable for fundamental courses in mathematics, investing, banking, financial engineering, and related topics." (Klaus Ehemann, Zentralblatt MATH, Vol. 1114 (16), 2007) "This book is designed to serve as an undergraduate text on the fundamentals of personal savings and investing. ... The book includes an appendix that covers basic concepts and techniques in probability and mathematical statistics. ... follows a different philosophy; it allows the results and examples to speak for themselves. ... it serves as a valuable resource for attaining savings, investment, and retirement goals." (Joseph Cavanaugh, The American Statistician, Vol. 62 (2), May, 2008)Read more... | 677.169 | 1 |
MERLOT Search - materialType=Assignment&keywords=mathematics
A search of MERLOT materialsCopyright 1997-2016 MERLOT. All rights reserved.Tue, 3 May 2016 20:12:08 PDTTue, 3 May 2016 20:12:08 PDTMERLOT Search - materialType=Assignment&keywords=mathematics
4434Calculus of the Dinner Table: Mathematical Modeling
Calculus students are presented with a write-pair-share activity that initially involves the construction of a model based on direct variation and later involves the use of calculus as a means by which to analyze the model. Suitable for either Calculus I or Calculus II students. Note: This project has a sequel entitled Fundamental Theorem of Calculus: An Investigation (listed under Interactive Lectures) in which the Fundamental Theorem of Calculus is investigated via the constructed model.Tue, 29 Sep 2009 10:54:50 -0700Fundamental Theorem of Calculus: An Investigation
Calculus students are presented with a write-pair-share activity that leads them to a practical understanding of the Fundamental Theorem of Calculus. The activity involves analyzing a function that describes eating speed in a hypothetical dinner table experience. Suitable for either Calculus I or Calculus II students.Note: This project has a prequel entitled Calculus of the Dinner Table: Mathematical Modeling (listed under Interactive Lectures) in which students construct the mathematical model for the king's eating speed. This prequel provides an excellent and engaging prelude to this activity.Tue, 29 Sep 2009 10:56:49 -0700Earth-Moon-Sun Dynamics
This site has fully developed activities, handouts, and even a test covering sunrise and sunset direction, moon face, phases of the moon, eclipses, and seasonal changes.Tue, 7 Sep 2010 22:08:24 -0700Elementary Statistics
This course is the study of descriptive statistics; probability; discrete and continuous (including binomial, normal and T) distributions; sampling distributions; interval estimation; hypothesis testing; linear regression and correlation. It is recommended for majors in the fields of biology, mathematics, social sciences, education and business. Wed, 29 Apr 2015 20:33:13 -0700Applied Math Lesson Plan
This fall I will be teaching a new course entitled "Applied Mathematics" which is intended for students who demonstrate a need to reduce the Algebra II requirement in the Michigan Merit Curriculum due to academic difficulty in Algebra I and/or Geometry. The course features interwoven strands of algebra and functions, statistics, and probability, with a focus on applications of mathematics. Students will learn to recognize and describe important patterns that relate quantitative variables and develop strategies to make sense of real-world data. The course will develop students' abilities to solve problems involving chance and to approximate solutions to more complex probability problems by using simulation. The goal that will be addressed in this lesson is to review Algebra I fundamentals, more specifically mathematical models (price-demand model, formulas as models, and operations with real numbers) to lay the foundation for the semester.Mon, 11 Jul 2011 09:03:19 -0700Balancing Equations
This is a technology rich lesson plan that can be used in a mathematics class to explore balancing equations.Sun, 17 Mar 2013 16:28:55 -0700Choose Your Best Way Lesson
Lesson focuses on how mathematic models help to solve real problems and are realized in computers. Students work in teams to build a graph model of their city map while learning how mathematic models work. Student should be encouraged to use this model to solve real problems.Fri, 9 May 2014 12:54:26 -0700Exploring Space Through Math Series
This site provides several activities involving space exploration that fit into the mathematics curriculum including geometry, algebra 2, and precalculus. Students are asked to read about the space exploration related application and then answer several questions that require these mathematics topics in order to successfully answer them. They study math while they also learn about the International Space Station, the future lunar lander, and other NASA projects.Sat, 9 Apr 2016 18:14:14 -0700Exploring Space Through Math: Algebra 2
This site provides several activities involving space exploration that fit into the Algebra 2 Curriculum. The mathematics that is used includes using algebraic equations to model space exploration work such as the lighting in the International Space Station and communication on a future lunar outpost. Students are asked to read about the space exploration related application and then answer several questions that require these algebraic topics in order to successfully answer them.Sat, 9 Apr 2016 17:54:53 -0700Exploring Space Through Math: Precalculus Series
This site provides several activities involving space exploration that fit into the Precalculus Curriculum. The mathematics that is used includes using trigonometry to manipulate a robotic arm in space, modeling projectile motion, using spherical coordinates to track the ISS, investigating the elliptical orbits of rondevous to the ISS, and several other applications to NASA activities. Students are asked to read about the space exploration related application and then answer several questions that require these pre-calculus topics in order to successfully answer them.Sat, 9 Apr 2016 17:54:53 -0700 | 677.169 | 1 |
Subject
Mathematics
World Population Activity II: Excelpart of Examples (Activity 2 of 2)In this intermediate Excel tutorial students import UNEP World population data/projections, graph this data, and then compare it to the mathematical model of logistic growth.
World Population Activity I: Excelpart of Examples (Activity 1 of 2) This activity is primarily intended as an introductory tutorial on using Excel. Students use Excel to explore population dynamics using the Logistic equation for (S-shaped) population growth.
Medicine in the Bodypart of Pedagogy in Action:Library:Teaching with Spreadsheets:Examples In this activity, we will explore how the dosage and frequency of a medicine taken affect the amount of medicine present in the blood.
Subject: Mathematics
Trends in Alkane Boiling Pointspart of MnSTEP Teaching Activity Collection:MnSTEP Activity Mini-collection This activity is an investigation into the relationship between alkane length and boiling points. Students develop a mathematical model of this relationship and use it to make predictions and error analysis. | 677.169 | 1 |
Details about Elementary Differential Equations:
The 10th edition of Elementary Differential Equations is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical and sometimes intensely practical. The authors have sought to combine a sound and accurate exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students.
Back to top
Rent Elementary Differential Equations 10th edition today, or search our site for other textbooks by William E. Boyce. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley. | 677.169 | 1 |
Mathematics
Philosophy
The ISD Mathematics curriculum aims to open the minds of students to the beauty and utility of mathematics by providing skills in mathematical modeling and problem solving. Students build a foundation of number and spatial sense upon which the concepts of geometry, algebra and number theory will be developed. The Mathematics curriculum focuses on problem solving and conceptual understanding within real and imagined situations and provides students with multiple methods of representation and communication of processes and results. Students practice using creativity responsibly, by finding multiple alternative solution methods, using available technologies and finding ways to apply these methods to authentic and relevant problems for the modern world.
Guiding Principles:
We believe that effective mathematics education is:
relevant, both personally and globally,
integrated and applicable across disciplines and in ordinary real life situations,
coherent, through spiraled instruction that is built on previously mastered concepts and skills,
engaging, using hands-on, developmentally appropriate content and methods to help learners build understanding,
rigorous, developing core mathematical strengths,
explorative, using open-ended tasks to encourage multiple strategies for problem-solving,
collaborative, allowing students to work together in creative teams,
inspirational, by empowering students to reflect on the beauty of the form and relationship of numbers and operations in the world around them. | 677.169 | 1 |
The MOOC is Part 1 of a two-part course in Numerical Methods. The course covers the mathematical procedures of differentiation, nonlinear equations and simultaneous linear equations. We had the MOOC on Udemy but we are migrating it to CANVAS in two stages. CANVAS has a broader appeal for free MOOCs, it has a user friendly interface, looks familiar for many students using CANVAS, and has the capability of online quizzes that are algorithmic.
Start your journey today whether you are learning numerical methods for the first time or just need a refresher. Unlike other MOOCs, you have a lifetime access to the course and you can pace yourself. Ask questions within the course and we will keep the conversation going!
About: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral). Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).
To make the fulcrum of a bascule bridge, a long hollow steel shaft called the trunnion is shrink fit into a steel hub. The resulting steel trunnion-hub assembly is then shrink fit into the girder of the bridge. This is done by first immersing the trunnion in a cold medium such as dry-ice/alcohol mixture. After the trunnion reaches the steady state temperature of the cold medium, the trunnion outer diameter contracts. The trunnion is taken out of the medium and slid though the hole of the hub. When the trunnion heats up, it expands and creates an interference fit with the hub.
In 1995, on one of the bridges in USA, this assembly procedure did not work as designed. Before the trunnion could be inserted fully into the hub, the trunnion got stuck. So a new trunnion and hub had to be ordered at a cost of $50,000. Coupled with construction delays, the total loss was more than hundred thousand dollars.
Why did the trunnion get stuck? This was because the trunnion had not contracted enough to slide through the hole.
Now the same designer is working on making the fulcrum for another bascule bridge. Can you help him so that he does not make the same mistake?
For this new bridge, he needs to fit a hollow trunnion of outside diameter in a hub of inner diameter . His plan is to put the trunnion in dry ice/alcohol mixture (temperature of the fluid – dry ice/alcohol mixture is -108 degrees F) to contract the trunnion so that it can be slided through the hole of the hub. To slide the trunnion without sticking, he has also specified a diametrical clearance of at least 0.01 inches between the trunnion and the hub. What temperature does he need to cool the trunnion to so that he gets the desired contraction?
In spite of learning management systems making assigning letter grades simpler, these systems still leave much to be desired when asked to incorporate extra credit or curving of an assessment grade. To overcome this drawback, I download the grades to a excel spreadsheet and calculate the overall score of each student. Based on the overall score, one needs to then assign a letter grade for the transcripts. To assign 13 different plus/minus grading letters manually can be a good candidate for making a mistake in a large size class. So I use a VBA code (how to access VBA editor) to assign letter grades.
PS. You can modify the above given VBA code as needed. If you want a less hard-coded version, you can modify by having two more inputs – 1) a vector of lowest limit of score for a particular letter grade, and 2) a corresponding vector of the same length of letter grades.
Problem: Through three data pairs (0,0), (3,9) and (4,12), an interpolating polynomial of order 2 or less is found to be y=3x. Prove that there is no other polynomial of order 2 or less that passes through these three points.
When solving a fixed-constant linear ordinary differential equation where the part of the homogeneous solution is same form as part of a possible particular solution, why do we get the next independent solution in the form of x^n* possible form of part of particular solution? Show this through an example.
QUESTION: What is the largest base-10 positive number that can be stored using 7 bits, where the 1st bit is used for the sign of the number; the 2nd bit for sign of the exponent; 3 bits for mantissa, and the rest of the bits for the exponent?
ANSWER: Remember the base is 2.
1st bit will need to be zero as the number is positive.
2nd bit will need to be zero as that will make the exponent positive as 2^positive. number will give higher number than 2^negative number.
The mantissa bits will need to be 111 as you are looking for largest number and that will give the number to be 1.111 (the 1 before radix point is automatic) in base of 2 or 1*2^0+1*2^(-1)+1*2^(-2)+1*2^(-3)=1.875 in base of 10.
Now the exponent: it uses 2 bits. This will need to be 11 in base 2 and that is 3 in base 10. So the exponent part is 2^(+3)=8. | 677.169 | 1 |
Mathematics A Discrete Introduction
9780534398989
ISBN:
0534398987
Edition: 2 Pub Date: 2005 Publisher: Thomson Learning
Summary: With a wealth of learning aids and a clear presentation, this book teaches students not only how to write proofs, but how to think clearly and present cases logically beyond this course. All the material is directly applicable to computer science and engineering, but it is presented from a mathematician's perspective. | 677.169 | 1 |
This document provides information about how to strengthen partnerships between secondary, tertiary, industry training and employment with better pathways through education at Level 3. Pathways and partnerships will engage and retain more learners by providing:
Abstract: What is the common thread in the following? There are points in the ocean where there is no tidal variation. There are antipodal points on the earth's surface where both the temperature and the barometric pressure are the same. If you place a sheet of paper on a table then pick it up and crumple and fold it (and even stretch it if you can) without tearing then replace it within the spot where it was then some point on the paper will be back where it was or directly above. You can build a fence through very rugged terrain and still stop your sheep wandering. The answer is topology. Amongst other things I shall explain the links.
Tuesday 3 May, 6:30pm in Room 303.G02, University of Auckland City Campus
The Casio Senior Mathematics Competition is again underway. This is an important competition as it not only allows us to recognise the top Mathematicians but also allows top students, from all regions, to meet and share ideas. The format of the competition is remaining the same as in previous years with a preliminary round, which is 90 minutes long and marked within the school. Results are then forward to the regional markers to allow us to select the top 200 competitors. The papers then forwarded to NZAMT are used to select the 15 finalists. Both rounds consist of a written paper prepared by Examiners appointed by NZAMT.
A new resource designed to empower secondary school students to use maths concepts in contexts within and beyond the subject has been published. The LEMMA series: Mathematics tasks that promote higher order thinking is a collection of booklets that contain sequences of metamathematical activities which encourage students to focus on problem solving and critical thinking.
The resource was developed by Associate Professor Caroline Yoon, head of the Mathematics Education Unit at the University of Auckland, through funding from the Beeby fellowship, a partnership between the New Zealand National Commission for UNESCO and the New Zealand Council for Educational Research (NZCER).
The innovative resource differs from a standard maths textbook as it addresses higher level skills and higher level thinking that are much harder to assess and often missed. The series is linked to the mathematics and statistics learning area of the national curriculum and encompasses communication and writing skills.
"Rather than being a text book trying to cover a wide area, the resource is designed to offer flexibility to teachers who can choose to use the resources that best fit with their own classroom and students," Dr Yoon explains.
NZQA National Assessment Moderators for Mathematics and Statistics can be contracted to provide workshops to teachers to increase assessor confidence in making assessor judgements consistent with the national standard for the internally assessed NCEA standards.
The 2016 International Mathematical Modeling Challenge IM2C - An Announcement and an Invitation March 16, 2016 – May 9, 2016 Rationale: The purpose of the IM2C is to promote the teaching of mathematical modeling and applications at all educational levels for all students. It is based on the firm belief that students and teachers need to experience the power of mathematics to help better understand, analyze and solve real world problems outside of mathematics itself – and to do so in realistic contexts. The Challenge has been established in the spirit of promoting educational change. More Information
Auckland University of Technology, New Zealand, Nov 26-27th 2015 Day 1: Using Pen-Enabled Tablets in STEM Education: Practice and Theory Day 2: Workshop on Designing Approaches for using Pen-Enabled Tablets in New Environments (optional) This Symposium will explore current developments and future directions in the use of new technologies in Tertiary STEM Education, focussing on the use of pen-enabled Tablet PCs. Registration There is no registration fee, with morning and afternoon teas and lunch provided. Numbers are limited, so registration is required by Monday 16th November. To register, please go to: Presentation Submissions Participants are invited to submit abstracts for presentations that specify their preferred presentation duration, ranging from 'quick fire' Pecha Kucha style (5-10 min) through to longer, more formal presentations (up to 20min). Abstracts should be up to 150 words, and be submitted before Monday 16th November. Authors of abstracts submitted earlier will receive early notice of acceptance. Symposium abstracts and other outputs will be reviewed and published online. Please email your abstract to: peter.maclaren@aut.ac.nz | 677.169 | 1 |
A Course in Real Analysis
2nd Edition
Description
The second edition of A Course in Real Analysis provides a solid foundation of real analysis concepts and principles, presenting a broad range of topics in a clear and concise manner. The book is excellent at balancing theory and applications with a wealth of examples and exercises. The authors take a progressive approach of skill building to help students learn to absorb the abstract. Real world applications, probability theory, harmonic analysis, and dynamical systems theory are included, offering considerable flexibility in the choice of material to cover in the classroom. The accessible exposition not only helps students master real analysis, but also makes the book useful as a reference.
Key Features
* New chapter on Hausdorff Measure and Fractals * Key concepts and learning objectives give students a deeper understanding of the material to enhance learning * More than 200 examples (not including parts) are used to illustrate definitions and results * Over 1300 exercises (not including parts) are provided to promote understanding * Each chapter begins with a brief biography of a famous mathematician
Readership
One- or two-semester course in real analysis for upper-level undergraduate and graduate students in mathematics, applied mathematics, computer science, engineering, economics, and physics
Table of Contents
Set Theory
The Real Number System and Calculus
Lebesgue Measure on the Real Line
The Lebesgue Integral on the Real Line
Elements of Measure Theory
Extensions to Measures and Product Measure
Elements of Probability
Differentiation and Absolute Continuity
Signed and Complex Measures
Topologies, Metrics, and Norms
Separability and Compactness
Complete and Compact Spaces
Hilbert Spaces and Banach Spaces
Normed Spaces and Locally Convex Spaces
Elements of Harmonic Analysis
Measurable Dynamical Systems
Hausdorff Measure and Fractals
Details
Reviews
"...truly marvelous...weaves an interesting, lively, and crystal clear sequence of ideas comprising the heart of modern analysis. The order of presentation is so carefully chosen and the exposition is so masterful as to possess the traits of a literary art form."--MAA Reviews, January 23, 2015
"The exposition is very clear and unhurried and the book would be useful both as a text and a book for self-study. The last chapters go beyond what is usually covered in analysis courses and this is all to the good." -- Sigurdur Helgason, MIT ** "There is a literary quality in the writing that is rare in mathematics texts. It is a pleasure to read this book. The exercises are a strong feature of the book and the examples are well chosen and plentiful." -- Peter Duren, University of Michigan ** "The outstanding features of the book are the wealth of examples and exercises, the interesting biographical data, and the introduction to wavelets and dynamical systems." -- Duong H. Phong, Columbia University ** "McDonald and Weiss have crafted a treasure chest of exercises in real analysis. Just an amazing and broad collection. Students and researchers will surely benefit from the enormous amount of superb exercises." -- Enno Lenzmann, University of Copenhagen ** "I was very impressed by the motivating discussions of a number of difficult concepts, along with their fresh approach to the details following. Their general philosophy of starting with concrete ideas, and slowly abstracting, worked well in communicating even the most difficult concepts in the course." -- Todd Kemp, University of California, San Diego | 677.169 | 1 |
books.google.co.uk - Geometric algebra (GA), also known as Clifford algebra, is a powerful unifying framework for geometric computations that extends the classical techniques of linear algebra and vector calculus in a structural manner. Its benefits include cleaner computer-program solutions for known geometric computation... to Geometric Algebra in Practice | 677.169 | 1 |
Fast Fourier Transform - Algorithms and Applications presents an introduction to the principles of the fast Fourier transform (FFT). It covers FFTs, frequency domain filtering, and applications to video and audio signal processing.
As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of the essential parts in digital signal processing has been widely used. Thus there is a pressing need from instructors and students for a book dealing with the latest FFT topics.
Fast Fourier Transform - Algorithms and Applications provides a thorough and detailed explanation of important or up-to-date FFTs. It also has adopted modern approaches like MATLAB examples and projects for better understanding of diverse FFTs.
Fast Fourier Transform - Algorithms and Applications is designed for senior undergraduate and graduate students, faculty, engineers, and scientists in the field, and self-learners to understand FFTs and directly apply them to their fields, efficiently. It is designed to be both a text and a reference. Thus examples, projects and problems all tied with MATLAB, are provided for grasping the concepts concretely. It also includes references to books and review papers and lists of applications, hardware/software, and useful websites. By including many figures, tables, bock diagrams and graphs, this book helps the reader understand the concepts of fast algorithms readily and intuitively. It provides new MATLAB functions and MATLAB source codes.
The material in Fast Fourier Transform - Algorithms and Applications is presented without assuming any prior knowledge of FFT. This book is for any professional who wants to have a basic understanding of the latest developments in and applications of FFT. It provides a good reference for any engineer planning to work in this field, either in basic implementation or in research and development | 677.169 | 1 |
This interactive maths resource will identify trouble areas and suggest further studies, and is full of video tutorials and real-life examples to help you revise and practise, progress and get top exam results.
With Collins Revision Algebra you can now hone your mathematics skills wherever you are.
• Be inspired by interactive animations, exciting video clips of students teaching a problem, real-life examples of maths at work, worked exam questions, tutorials and more
• Choose precisely which topics to revise and practise with material corresponding to the Collins New GCSE Maths scheme
• Test yourself with interactive assessment questions that identify trouble areas and suggest relevant further revision
Videos are downloaded separately by the app, directly to your device, upon installation.
Collect all four Apps covering all four GCSE maths strands, for a total of 900 practice questions, 300 assessment questions and 130 video clips | 677.169 | 1 |
Offering worked examples and solutions leading to practice questions, this book helps students to learn maths. It features sample past exam papers for exam preparation, and regular review sections. It includes a CD ROM which contains what students need to motivate and prepare themselves.
Synopsis:
Edexcel and A Level Modular Mathematics S2 features: *Student-friendly worked examples and solutions, leading up to a wealth of practice questions. *Sample exam papers for thorough exam preparation. *Regular review sections consolidate learning. *Opportunities for stretch and challenge presented throughout the course. *'Escalator section' to step up from GCSE. PLUS Free LiveText CD-ROM, containing Solutionbank and Exam Cafe to support, motivate and inspire students to reach their potential for exam success. *Solutionbank contains fully worked solutions with hints and tips for every question in the Student Books. *Exam Cafe includes a revision planner and checklist as well as a fully worked examination-style paper with examiner commentary | 677.169 | 1 |
Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on upcoming webinars we think may be of interest to the MaplePrimes community. For the complete list of upcoming webinars, visit our website.
Creating Questions in Maple T.A. – Part #1
This webinar will demonstrate how to create questions in Maple T.A., Maplesoft's testing and assessment solution for any course involving mathematics. The presentation will begin with an overview of the basic types of questions available, and then delve into how to create various types of questions in Maple T.A. Incorporating algorithms and feedback directly into questions will also be touched on. Finally, the session will wrap up with an explanation and several examples of how to create better questions using the question designer.
This first webinar in a two part series will cover true/false, multiple choice, numeric, mathematical formula, fill in the blank, sketch, and FBD questions. A second webinar that demonstrates more advanced question types will follow.
In this webinar, Dr. Robert Lopez will apply the techniques of "Clickable Calculus" to standard calculations in Linear Algebra.
Clickable Calculus, the idea of powerful mathematics delivered using very visual, interactive point-and-click methods, offers educators a new generation of teaching and learning techniques. Clickable Calculus introduces a better way of engaging students so that they fully understand the materials they are being taught. It responds to the most common complaint of faculty who integrate software into the classroom – time is spent teaching the tool, not the concepts.
It puzzles me that some of the choices for Clickable Math don't return a result. The image below shows four choices for clickable math. I would like it to "factor" exactly as shown in the upper left choice, but nothing happens after clicking the choice. (No matter how many times I try.) The choices "2D Plot" and "Isolate" return results ("Complete the square", in this case, also does not work).
My guess is that I'm asking it to operate on only the denominator of the rhs of the equation, but then why is the choice displayed at all. I've noticed this for some time in certain cases (like since version 17 was release).
Is there a way to make the choice return a result ? In this case, I don't find an equivalent command (factor, the denominator) using a right click. (In this case, I'll use another command "covert" "parfrac" . . .)
At some point in my calculations, I want to do some simple manipulations wiht the use of the «clickable» features in M17. For the moment, I can see directy how to do it by hand and I do it by copy-paste and use the mous to change it a little bit. But I just cannot find a simple way to do it by simple click.
You will find in this attach question, the maipulations I want to do. It is starting from equaiton (7). Am I at the edge where it is simple by hand than with the features?
Apart from the online description of this new Maple 16 feature here, there is also the help-page for subexpressionmenu.
I don't know of a complete listing of its current functionality, but the key thing is that it acts in context. By that I mean that the choice of displayed actions depends on the kind of subexpression that one has selected with the mouse cursor.
Apart from arithmetic operations, rearrangements and some normalizations of equations, and plot previews, one of the more interesting pieces of functionality is the various trigonometric substitutions. Some of the formulaic trig substitutions provide functionality that has otherwise been previously (I think) needed in Maple.
In Maple 16 it is now much easier to do some trigonometric identity solving, step by step.
Here is an example executed in a worksheet. (This was produced by merely selecting subexpressions of the output at each step, and waiting briefly for the new Smart Popup menus to appear automatically. I did not right-click and use the traditional context-sensitive menus. I did not have to type in any of the red input lines below: the GUI inserts them as a convenience, for reproduction. This is not a screen-grab movie, however, and doesn't visbily show my mouse cursor selections. See the 2D Math version further below for an alternate look and feel.)
The very first step above could also be done as a pair of simpler sin(x+y) reductions involving sin(2*a+a) and sin(a+a), depending on what one allows onself to use. There's room for improvement to this whole approach, but it looks like progress.
I am not quite sure what is the best way to try and get some of the trig handling in a more programmatic way, ie. by using the "names" of the various transformational formulas. But some experts here may discover such by examination of the code. Ie,
eval(SubexpressionMenu);
showstat(SubexpressionMenu::TrigHandler);
The above can leads to noticing the following (undocumented) difference, for example, | 677.169 | 1 |
Math & You: The Power & Use of Mathematics
Browse related Subjects ...
Read More latest educational research, "Math & YOU" helps students develop the quantitative skills needed to be successful in school and the workplace, using real data, problems based on everyday situations, and activities built around topics that are recognizable and relevant. With this approach, students become comfortable with quantitative ideas and proficient in applying them. In addition, to support the printed text, "Math & YOU" provides an online eBook accompanied by additional teaching aids, all part of a robust companion Web site. Hardcover edition available upon request. Ask your local W.H. Freeman representative. Math & YOU HallmarksConfidence with Mathematics. One of the goals of the ""Math & YOU"" program is to help students become comfortable with quantitative ideas and proficient in applying them. Students routinely quantify, interpret, and check information such as comparing the total compensation of two job offers, or comparing and analyzing a budget Cultural Appreciation. "Math & YOU" provides examples and exercises that help student to understand the nature of mathematics and its importance for comprehending issues in the public realm. Logical Thinking. "The Math & YOU" program develops habits of inquiry, prepares students to look for appropriate information, and exposes them to arguments so that they can analyze and reason to get at the real issues. Making Decisions. One of the main threads of the "Math & YOU" program is to help students develop the habit of using mathematics to make decisions in everyday life. One of the goals of the text is for students to see that mathematics is a powerful tool for living. Mathematics in Context. The "Math & YOU" program helps students to learn to use mathematical tools in specific settings where the context provides meaning. Number Sense. "The Math & YOU" program begins with a chapter that reviews the meaning of numbers, estimation and measuring. Throughout the rest of the program students develop intuition, confidence, and common sense for employing numbers. Practice Skills. Throughout the "Math & YOU" program students encounter quantitative problems that they are likely to encounter at home or work. This helps students become adept at using elementary mathematics in a wide variety of common situations.
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Good. The cover shows heavy wear. There is light highlighting or handwriting through out the book. Fast shipping and order satisfaction guaranteed. Your purchase benefits charities and literacy groups No Jacket. Very Good-Hardcover with no DJ. Light soiling and shelfwear to covers. Textblock has a small bit of soiling, which has very lightly bled onto a few pages. Otherwise, pages clean and tight in binding. Pictures available upon request. A locally owned, independent book shop | 677.169 | 1 |
Customer Reviews
Useful for anyoneMay 5, 2014 by pearl Gregory
"Precalculus with Limits, written by Ron Larson (of Pennsylvania State University and Behrend College), was developed on the premise of creating a textbook to help prepare students for a college level Calculus course. This text helps student learn concepts by incorporating real world utilization of topics. Learning is enhanced through an added web site subscription which includes more guidance and additional math problems to nail down many of the textbook's concepts. Topics include algebra, three dimensional analytic geometry,core precalculus, and an introduction to college level calculus.Overall, this is an excellent text preparing students for the upcoming challenges that Calculus can present."
Summary
Larson's PRECALCULUS WITH LIMITS is known for delivering the same sound, consistently structured explanations and exercises of mathematical concepts as the market-leading PRECALCULUS, Ninth Edition, with a laser focus on preparing students for calculus. In LIMITS, the author includes a brief algebra review to the core precalculus topics along with coverage of analytic geometry in three dimensions and an introduction to concepts covered in calculus. With the third edition, Larson continues to revolutionize the way students learn material by incorporating more real-world applications, ongoing review, and innovative technology. How Do You See It? exercises give students practice applying the concepts, and new Summarize features, Checkpoint problems, and a Companion Website reinforce understanding of the skill sets to help students better prepare for tests. | 677.169 | 1 |
Browse related Subjects ...
Read More and CAD operations in order to facilitate critical thinking, problem solving, and basic mathematics literacy. Filled with real-world applications and designed to cover a range of skills and levels of difficulty, the fourth edition includes updated figures, illustrations, problem sets, examples, and solutions in order to give you the skills you need to succeed in the field of drafting | 677.169 | 1 |
Flummoxed by formulas? Queasy about equations? Perturbed by pi? Now you can stop cursing over calculus and start cackling over Math, the newest volume in Bill Robertson's accurate but amusing Stop Faking It! best sellers. As Robertson sees it, too many people view mathematics as a set of rules to be followed, procedures to memorize, and theorems... more...
A Transition to Advanced Mathematics promotes the goals of a ``bridge'' course in mathematics, helping to lead students from courses in the calculus sequence to theoretical upper-level mathematics courses. The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse... more...
A no-nonsense practical guide to geometry, providing concise summaries, clear model examples, and plenty of practice, making this workbook the ideal complement to class study or self-study, preparation for exams or a brush-up on rusty skills. About the Book Established as a successful practical workbook series with more than 20 titles in the language... more...
Introducing sophisticated mathematical ideas like fractals and infinity, these hands-on activity books present concepts to children using interactive and comprehensible methods. With intriguing projects that cover a wide range of math content and skills, these are ideal resources for elementary school mathematics enrichment programs, regular classroom... more...
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... more...
Any way you slice it, fractions are foundational Many students struggle with fractions and must understand them before learning higher-level math. Veteran educator David B. Spangler describes powerful diagnostic methods for error analysis that pinpoint specific student misconceptions and supplies specific intervention strategies and activities... more...
This volume supports the belief that a revised and advanced science education can emerge from the convergence and synthesis of several current scientific and technological activities including examples of research from cognitive science, social science, and other discipline-based educational studies. The anticipated result: the formation of science... more... | 677.169 | 1 |
Quantitative AnalysisEasy to understand-even for learners with limited math backgrounds, this book uses a modeling approach to provide thorough coverage of the basic techniques in quantitative methods and focuses on the managerial applications of these techniques. An interesting and reader friendly writing style makes for a clear presentation, complete with all the necessary assumptions and mathematical details.Chapter topics include probability concepts and applications, decision models and decision trees, regression models, forecasting, inventory control models, linear programming modeling applications and computer analyses, network models, project management, simulation modeling, and more.For an introduction toquantitative analysis, quantitative management, operations research, or management science-especially for those individuals preparing for work in agricultural economics and health care fields. | 677.169 | 1 |
Perth Amboy AlgebraMoinuddin M.
...Realistically speaking almost all the problems, particularly in the multiple choice sections, are based on reasoning, and one can logically often narrow down the choices of possible answers to two (out of the given 5) and even to just one of the correct answers. Reasoning is often needed in many...Bryan W. | 677.169 | 1 |
Space Mathematics: Math Problems Based on Space Science
Created by NASA for high school students interested in space science, this collection of worked problems covers a broad range of subjects, including mathematical aspects of NASA missions, computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus. In addition to enhancing mathematical knowledge and skills, these problems promote an appreciation of aerospace technology and offer valuable insights into the practical uses of secondary school mathematics by professional scientists and engineers. Geared toward high school students and teachers, this volume also serves as a fine review for undergraduate science and engineering majors. Numerous figures illuminate the text, and an appendix explores the advanced topic of gravitational forces and the conic section trajectories. | 677.169 | 1 |
Details about Elementary and Intermediate Algebra:
Master algebraic fundamentals with Kaufmann/Schwitters ELEMENTARY AND INTERMEDIATE ALGEBRA 5e. Learn from clear and concise explanations, multiple examples and numerous problem sets in an easy-to-read format. The text's 'learn, use & apply' formula helps you learn a skill, use the skill to solve equations, then apply it to solve application problems. With this simple, straightforward approach, you will grasp and apply key problem-solving skills necessary for success in future mathematics courses.
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Rent Elementary and Intermediate Algebra 5th edition today, or search our site for other textbooks by Jerome E. Kaufmann. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cengage. | 677.169 | 1 |
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RELEVANT EQUATIONS: Eberly gives purpose to math problems
COLUMBIA CITY — For the students who say they'll never use superior mathematical equations after they graduate from school — teacher Brett Eberly concurs.Eberly, a math teacher at Eagle Tech Adademy, Columbia City, has taken a new approach to teaching his students — giving them skills to last a lifetime."It's not realistic to think they'll use this math for the rest of their lives," Eberly said. "My goal is to help them build problem-solving skills so when they're presented with a problem in the working world, they'll have the experience and can work their way through that problem."Eberly has been recognized nationally by Stanford University for work he has done at Eagle Tech.He is part of a small group of educators in the New Tech realm and is one of three math teachers in the U.S., that is working in conjunction with Stanford to help students be prepared for life after graduation.Stanford analyzed projects each teacher assigned to their math students, and Eberly's was chosen as the best in the nation and will be used as a pilot project by the university.To read the rest of this story, see the Sept. 13 | 677.169 | 1 |
Complex Analysis
Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.
This unusually lively textbook introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. The book offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. For the second edition, the authors have revised some of the existing material and have provided new exercises and solutions.
The third edition contains a new section on Schwarz–Christoffel mappings, an improved treatment of the Schwarz Reflection Principle, and an expanded section on applications to number theory.
Applications of The Residue Theorem to the Evaluation of Integrals Sums.
Further Contour Integral Techniques.
Introduction to Conformal Mapping.
The Riemann Mapping Theorem.
Maximum
Modulus Theorems for Unbounded Domains.
Harmonic Functions.
Different Forms of Analytic Functions.
Analytic Continuation; The Gamma and Zeta Functions.
Applications to Other Areas of Mathematics.
Appendices.
Answers | 677.169 | 1 |
New to this Edition:
Introduction to Algebra
Second Edition
Peter J. Cameron
Description
Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the theory of groups, rings and fields, coding theory, and Galois theory. With over 300 exercises, and web-based solutions, this is an ideal introductory text for Year 1 and 2 undergraduate students in mathematics.
Author Information
Introduction to Algebra
Peter J. Cameron
Reviews and Awards
Review from previous editionThis clearly written exposition is accompanied by well-chosen exercises. This book should be useful as a textbook for most undergraduates courses on algebra. -
This is an extremely engaging introduction to abstract algebra by one of this country's most prolific and creative algebraists. Recognising that although the axiomatic method is unavoidable it is intially uncomfortable for many students, he adopts a relatively informal style which is constantly encouraging without ever lapsing into imprecision. Aided by a relaxed, friendly expository style, his expertise, sureness of touch and contagious enthusiasm for algebra shine through on every page this is a book to study, savour and enjoy -
'Altogether this is a concise but solid introduction into algebra and linear algebra' Internationale mathematische Nachrichten - | 677.169 | 1 |
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Why do some children seem to learn mathematics easily and others slave away at it, learning it only with great effort and apparent pain? Why are some people good at algebra but terrible at geometry? How can people who successfully run a business as adults have been failures at math in school? How come some professional mathematicians suffer terribly when trying to balance a checkbook? And why do school children in the United States perform so dismally in international comparisons? These are the kinds of real questions the editors set out to answer, or at least address, in editing this book on mathematical thinking. Their goal was to seek a diversity of contributors representing multiple viewpoints whose expertise might converge on the answers to these and other pressing and interesting questions regarding this subject.
The chapter authors were asked to focus on their own approach to mathematical thinking, but also to address a common core of issues such as the nature of mathematical thinking, how it is similar to and different from other kinds of thinking, what makes some people or some groups better than others in this subject area, and how mathematical thinking can be assessed and taught. Their work is directed to a diverse audience -- psychologists interested in the nature of mathematical thinking and abilities, computer scientists who want to simulate mathematical thinking, educators involved in teaching and testing mathematical thinking, philosophers who need to understand the qualitative aspects of logical thinking, anthropologists and others interested in how and why mathematical thinking seems to differ in quality across cultures, and laypeople and others who have to think mathematically and want to understand how they are going to accomplish that feat. | 677.169 | 1 |
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Integral calculus
How do you find the area under a curve? What about the length of any curve? Is there a way to make sense out of the idea of adding infinitely many infinitely small things? Integral calculus gives us the tools to answer these questions and many more. Surprisingly, these questions are related to the derivative, and in some sense, the answer to each one is the opposite of the derivative.
AP Calculus practice questions
Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).
The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle! | 677.169 | 1 |
Hi all, I just began my presents gifts class. Boy! This thing is really difficult ! I just never seem to understand the point behind any concept. The result? My grades suffer. Is there any guru who can lend me a helping hand?
Don't fret my friend. It's just a matter of time before you'll have no problems in answering those problems in presents gifts. I have the exact solution for your math problems, it's called Algebrator. It's quite new but I assure you that it would be perfect in assisting you in your algebra problems. It's a piece of program where you can answer any kind of algebra problems with ease . It's also user friendly and displays a lot of useful data that makes you learn the subject matter fully.
Algebrator is the program that I have used through several algebra classes - Algebra 1, College Algebra and Remedial Algebra. It is a truly a great piece of math software. I remember of going through difficulties with least common denominator, greatest common factor and complex fractions. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I highly recommend the program. | 677.169 | 1 |
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MS Math 3.0 Rolls Out This Month
By Michelle Rutledge
05/17/07
##AUTHORSPLIT##<--->
Microsoft recently announced the launch of Math 3.0, a math and science educational tool for students in grade levels 6-12, as well as entry-level college students. The software is designed for use at home, to assist students with math and science concepts and homework, or for visual examples in the classroom.
"Microsoft Math provides a space for nurturing student learning in mathematics with dynamic visualizations. The program provides essential ingredients for classroom environments designed to challenge all students to engage in visual thinking," said Margaret L. Neiss, mathematics education professor at Oregon State University in a prepared statement.
According to Microsoft, the software includes study material for six different math and science subjects. Features in the program include:
Graphing calculator;
Step-by-step math solutions;
Formulas and equation library;
Unit conversion tool;
Triangle solver; and
Handwriting support.
Math 3.0 runs $19.95 for a single license. Volume licensing is available for educational institutions. | 677.169 | 1 |
Technical Support
MathMax Web Site
This Web site is a valuable online resource for both instructors and student users of the MathMax developmental
math paperback series. Teachers can find additional materials to enhance their courses, while students can strengthen
their understanding through extended chapter openers, interactive math tutorials, practice exercises, and chapter
reviews; all written specifically to accompany the book content.
The MathMax Web Site ssupports the MathMax developmental math paperback
series, which includes the following titles:
MathMax Web Site Features
This Web site currently offers a full array of resources to help students strengthen their understanding of
the mathematics they are studying: InterAct Math software tutorials, Extra Practice Worksheets, Chapter Reviews,
and extended chapter openers.
InterAct Math is an interactive tutorial that allows students to work through exercises with hints,
guided solutions, and immediate feedback. The InterAct plug-in (the Internet version of the AW InterAct Math Tutorial
Software) can be downloaded from this site and used to work the exercises in a Web browser. (Available for Windows
95 or 98 platform only.)
Extra Practice Worksheets are available for selected sections throughout the book. Each worksheet provides
additional exercises for a specific section (or sections)of the book. Using the Adobe Acrobat Reader, students
can download and print the worksheet, work the exercises, then return to the Web site to compare their answers
to those provided.
Chapter Reviews are available for selected sections throughout the book. Each worksheet provides additional
exercises for a specific section (or sections)of the book. Using the Adobe Acrobat Reader, students can download
and print the worksheet, work the exercises, then return to the Web site to compare their answers to those provided.
Extended Chapter Openers build on the real world applications provided as chapter openers in the textbook.
The Web site's chapter openers provides links to related Web sites for further exploration. These links provide
students and instructors with an excellent opportunity for devising additional application projects.
In addition to the main features described above, students and teachers can get in-depth information about
all of the supplements and learning resources available for MathMax: The Bittinger System of Instruction. Instructors
can also benefit from a special section--written by a member of the Bittinger author team--that offers tips for
presenting those topics that students have difficulty grasping.
Hardware Requirements
In order to properly view all of the material on this site, you must:
be using a Web browser
(preferably version 3.0 or above of Netscape Navigator or Internet Explorer)
download the InterAct plug-in
(The InterAct plug-in is available for Windows 95 or 98 platform only.) | 677.169 | 1 |
Department of Mathematics C.K.G.M. College, Perambra is conducting a state level seminar on 17th and 18th sep 2009. The aim of the seminar is to underline the importance of Python and other free software in Mathematics Education..."
Maths is a famously lonely discipline - I should know, having spent three years of my life grappling with a single equation (the equation won). Mathematicians meet, and collaborate, it's true; but what would a truly open source approach to the process of solving mathematical problems look like? Maybe something like this...
A former assistant professor from Harvard, now at UofW, Dr. William Stein, and several students, have created a new open-source complex math solving tool called Sage. It is an Internet-based graphical tool which allows the user to do basically anything mathematically, from "mapping a 12-dimensional object to calculating rainfall patterns under global warming."
"ScienceDaily (Dec. 7, 2007) — Until recently, a student solving a calculus problem, a physicist modeling a galaxy or a mathematician studying a complex equation had to use powerful computer programs that cost hundreds or thousands of dollars. But an open-source tool based at the University of Washington won first prize in the scientific software division of Les Trophées du Libre, an international competition for free software..."
GeoGebra, a GPL-licensed teaching and learning tool that integrates geometry, algebra, and calculus, benefits both teachers and students alike. Developed by Markus Hohenwarter at Florida Atlantic University, GeoGebra constructs geometrical figures and demonstrates the relationship between geometry and algebra. | 677.169 | 1 |
Extra Workbooks Available for Additional Students
You'll be pleased to know you can purchase extra workbooks for Teaching Textbooks Math 3 without having to buy a complete extra Teaching Textbooks set. If you have two students doing this grade together or another student coming up to this grade, and already own the CDs, you can purchase an extra workbook here.
This book comes with an answer key. Workbook only just means no CDs.
CDs Required to Use This Workbook
Please note that this workbook alone will not have adequate instruction to teach the course without the CDs. If you do not already own Teaching Textbooks Math 3 including the CDs, click here for the complete set.
About Teaching Textbooks
Math for Homeschoolers
A number of years ago we did the unthinkable and took a poke at a sacred cow when we said that we didn't like the conventional big-name home school math programs. Though some families were steadfast in their defense, there has since been an increasing number of families who have come to realize that math programs originally designed to be used in government schools are wholly inadequate to meet the needs of home educators.
You Don't Have to be a Math Teacher!
What is wrong with adapting a school-based text for home use? There are a couple of significant issues with doing that. First, because a typical school text assumes a knowledgeable math teacher, the explanations are often meager and brief. If you double as a math teacher, this will present no problem to you. However, many parents either did not take upper level math as a teen, or if they did, the process is just a distant, unpleasant memory. So sparse explanations in typical texts can be frustrating to both child and parent. Second, government schools' math texts often contain excessive use of terminology, making simple ideas seem complex.
Designed for Independent Learners
However, the Teaching Textbooks series is produced with home educators in mind, so it tackles those issues head-on. Firstly, since the program is designed specifically for independent learners, it offers far more explanation than any others on the market, and the tone is friendly and conversational. Ultimately what sold me on this program are their computer CDs with down-to-earth, step-by-step multimedia solutions for every problem in the book, plus complete step-by-step solutions for every test problem.
Friendly Conversational Tone
The lectures - one for each lesson in the textbook - provide hours of instruction. Students will appreciate listening to (and watching) an explanation for each lesson rather than reading it out of a book. Designed for homeschoolers studying independently, Teaching Textbooks uses far less irrelevant jargon than other textbooks, while still retaining all the terms that students need to know for those important standardized tests, and the tone is friendly and conversational. This is a brilliant idea, long overdue and skillfully done.
Workbook Includes Lesson Summaries
The Teaching Textbooks program also incorporates a workbook in which your child will do daily math assignments. Designed specifically for independent learners, the Teaching Textbooks workbook contains a summary of the lectures from the CD, which allows your child to review key points from the lesson as necessary while working the problems. The workbook also comes with an answer key and a test bank.
Review Method
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Using vectors to model
Free Course
This free course, Using vectors to model,After studying this course, you should be able to:
show an awareness of some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions
understand how vectors can be represented in three (or more) dimensions
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Using vectors to model
Introduction
This courseThis OpenLearn course provides a sample of level 2 1st April 2011 | 677.169 | 1 |
Series
Lessons: 152
Total Time: 20h 3m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 05/18/2010
Last Updated At: 11/30/2011
In this 150+ lesson series, you'll go through prerequisites for Trigonometry and then the ins and outs of Trig. The first lessons will walk you through all of the algebra-related prerequisites that you should be familiar with in advance of a trigonometry class. This will include graphing basics, relationships between two and three points, circles, equation graphing, function basics, working with and doing operations on functions, notions of domain and range, slope, lines, linear equations, graphing shits and stretches, symmetry, reflections, quadratic functions, composite functions, rational functions, and function inverses.
Armed with an understanding of these prerequisites, we'll dig into a study of Trigonometric functions. We'll start with angles, angle measure, radians, right angle trig, trig functions, and inverse trig functions. We'll also look at graphing basic trig functions.
Next, we'll move on to look at trigonometric identities, including sum and difference identities, double-angle identities, etc. We'll learn about basic trig identities and simiplifying trigonometric expressions. Then, we'll take these learnings to get a feel for how to prove different trig identities and solve different trigonometric equations.
After our study of identities, we'll move on to a section on trig applications. In this portion, we'll learn about the Law of Sines, the Law of Cosines, vector basics, etc.
Then, we'll move on to a few lessons on complex numbers and polar coordinates before wrapping up with a series of lessons on conic sections that cover parabolas, hyperbolas, ellipses, etcBelow are the descriptions for each of the lessons included in the
series:
College Algebra: Using the Cartesian System Thinking Visually Find distance & midpoint betwn points
Professor Burger proves the distance formula (distance = square root [(x1-x2)^2 + (y1-y2)^2] ) using the pythagorean theorum. Using this formula, you can find the distance between two points on a line. Professor Burger goes on to prove the midpoint formula ( [x1 + x2]/2, [y1 + y2]/2). The midpoint formula is in the form of a point on a line and is the average of the points on the x and y axes. trade 2nd Endpoint of a Segment Collinearity and Distance Triangles Circle - Center-Radius Form articles Circle Center and Radius Decoding the Circle Formula
In this lesson, you will learn how to find the (x, y) coordinates for the center of a circle graphed on the Cartesian coordinate system as well as the the radius of the circle from a formula or expression that doesn't easily lend itself to the standard circle formula (e.g. r^2 = (x-h)^2+(y-k)^2)). To do this, you'll generally use a math technique called, Completing the Square Word Problems Involving Circles Locate Points to Graph Equations Thinking Equation's x & y-Intercept Functions and the Vertical Line Test
A function is basically a machine that takes an input value (x) and processes it to produce an output value (y). With a function, if an x value is known, you can find the y value. When graphed, a curve is a function if it passes the vertical line test. In this lesson, Professor Burger will show you how the vertical line test means and how to recognize when a curve does not pass the vertical line test. The vertical line test looks to verify that, for every value of x, only one y value is produced. If something doesn't pass the vertical line test, it is called a relation and not a function Function Notation and Values
In this lesson, Professor Burger will show you how to correctly denote functions and values. By definition, a function has only one value of y for each value of x. A function can always be expressed using the term f(x) instead of y. This lesson will walk through when to use this notation and how to use it correctly to indicate what you want it to be. Additionally, Professor Burger will show you how to verbally say the new notation in addition to how to write it. Last, he'll walk you through a few examples involving functions and their notation and evaluation straight Increase Intervals Functions
In this lesson, we will learn to define, recognize, write, graph and evaluate piecewise math functions. Piecewise-defined functions are unique in that they are defined to be equal to different things for different ranges of variable values. Thus, more than one expression defines the function (e.g. for x<2, f(x)=x^2 but for x>=2, f(x)=10x).
This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth! Solving Function Word Problems Finding Domain and Range
While a function always satisfies the vertical line test (for any value of x there is only one value of y), there are functions in which the domain of the function does not include all values of x. In this lesson, we look at the domain of a function (all of the values of x for which we can evaluate the function and find a value of y) and the range of a function (all the values of y that may be generated by evaluating the function for some value of x). In addition to learning about evaluating a function to find the domain and range, Professor Burger will graphically show you how to identify the domain and range Domain and Range: An Example Satisfying the Domain of a Function
In this lesson, you will learn how to find all of the allowable x values for a particular function (the function's domain). An allowable x value is one in which you can evaluate the function. There are certain types of numbers which are not allowable, like square roots of negative numbers, numbers with 0 in the denominator, etc. If you evaluate a function and end up with one of these types of numbers, then the x value is deemed to be outside of the domain for the function. Professor Burger will also show you how to correctly denote the domain of a function once you determine what it is An Introduction to Slope the Slope Given Two Points
In this lesson, you will learn how to find the slope, or the relative increase (also known as a pitch), of a line if you are given two points on that line (x1, y1) and (x2, y2). The slope (denoted by the letter, m) of a line is defined by the change in y divided by the change in x. First, you must calculate the change in the two distances (or the change (x1 - x2) and the change (y1 - y2)). You will also learn the shorthand for writing the equation of a slope and the phrase 'rise over run.' After learning how to find the slope of a line, you will practice with several sets of points and lines with different slopes (including verticle lines and horizontal lines) and also practice graphing those lines to view the slope. Professor Burger also examines what it means when a slope is undefined (or the change in x = 0), and when a slope = 0 (or the change in y = 0). Using Intercepts Working with Specific Slope from equations Use Point & Slope Equations in Slope-Intercept Form
Professor Burger teaches the algebraic expression for lines, or the equation of a line. The standard form for a line is written Ax + By = C. More complex algebraic expression include the slope-intercept form, y = mx + b, where m=slope and b= the point where the line crosses (or intercepts) the y-axis. Professor Burger proves the validity of this expression, and shows you how to graph a line from the slope-intercept equation.
Learn how to determine the slope of a line here equations Given Two Points
Using the slope-intercept equation of a line, Professor Burger teaches you how to write the equation of a line if you are given two points on that line. Given two points, you can find the slope. Once you have found the slope of the line, you can input any point on that line into the equation with the slope to solve for b. Once you have found the slope and b, you have the slope-intercept equation (y = mx + b).
Learn how to find the slope of a line: expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
Beg Algebra: Writing Point-Slope Form Equations
This lesson introduces another format for the equation of a line called the point-slope form. The point-slope form for the equation of a line is y - y1 = m(x - x1), where m=slope and x1 and y1 are the coordinates of a point on the line (x1, y1). Professor Burger proves the validity of this equation, which is derived from the formula for the slope.
Learn how to find the slope of a line in Slope-Intercept Form, Parallel & Perpendicular Line Slopes
Professor Burger explains parallel and perpendicular lines, teaching you how to identify if two lines are parallel or perpendicular, by looking at the formulas. Two lines are parallel if they have the same slope. Perpendicular lines are slightly more complicated, as they have slopes that are negative reciprocals. After demonstrating these principles, Professor Burger walks you through some example problems.
To learn more about slopes, visit Important Effectiveing Equations with Shifting Curves along Shift or Translate Curves on specialty Stretching aetermining Symmetry Reflections Reflectinggebra: Quadratic Function Nice-Looking Quadratics Using Patterns Max Height in Real World Find Vertex by Completing the Square
In this lesson, you will learn how to find the vertex of a parabola given the formula for the parabola. To do this, you will complete the square. By completing the square of the parabola equation, you will be able to get the equation into a standard form that can be more easily evaluated. A parabola is a conic section in which the locus of points constructing it are equidistant from the focus and the directrix. Once we've identified the vertex of a parabola, we can get a good sense for how the parabola is positioned on the Cartesian coordinate plane Write Quadratic Equation w Vertex Quadratic Maximum or Minimum teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
College Algebra: Operations on Composite Functions
In this lesson, you will learn about a method that you can use to combine functions. The composition of two functions is the way to combine two functions. In this lesson, you will learn how to combine functions (for example, to find f(g(x)) ). There are specific ways to denote these types of composite functions, and you will also learn how to correctly write composite functions (f(g(x)) or (f o g)(x) ). To compose a function (find the composition of functions), you'll have to take the answer of one function and plug it into the other function (to find something like, 'g composed of f of 3'. Professor Burger will also highlight why g(f(3)) is not always equal to f(g(3)). Composite Function Components Functions to Form Composite videos Difference Quotient Basic Vertical journals Horizontal Rational Functions Ex Quadrant Where an Angle Lies Coterminal Angles
Coterminal angles are angles that have their terminating rays in the exact same location. By definition, any two angles whose difference is some multiple of 360 degrees. In this lesson, Professor Burger will show you what coterminal angles look like, how to determine if two angles are coterminal using simple math and then he will review several examples of coterminal angles. He will also demonstrate visually how coterminal angles behave and why the definition is appropriate for these types of angles Find Angle Complements & Supplements
This trigonometry lesson introduces and explains the terminology behind complementary angles (angles that add up to 90 degrees) and supplementary angles (that add up to 180 degrees). Professor Burger gives you the definition of these two types of angles, shows you what these angles will look like when combined and and shows you how to find a complement angle or a supplement angle to a provide angle with a known degree measure. Professor Burger will also explain whether supplementary angles and complementary angles could be negative.
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at: Convert between Degrees and Radians
In this lesson, Professor Burger teaches the basics of degrees and radians as they relate to the measurement of angles. He will cover how these terms are related (e.g. 360 degrees= 2*pi radians), how they are different from each other and why they are used in different situations . This lesson should answer and assortment of questions, including: Why do we use 360 degrees? Is there a better way to measure angles? Radian measurement is a different way to measure angles, and is the method of angle measurement used in trigonometric functions. You will learn how to measure angles using radians instead of degrees, how to convert from degrees to radians, and what you should memorize to simplify this conversion. Additionally, Professor Burger will explain the rationale behind using radians in place of degrees at times (mostly with trigonometric functions). Finally, you will review several examples of this conversion the Arc Length Mathematics Intro to the trig find a Right Triangle angle
This lesson teaches you to evaluate trigonometric functions to find one (non-right) angle of a right triangle. To do this, Professor Burger will walk through an example in which he presents a right triangle with two known sides and one known angle. From this information, all other angles and sides can be determined using trigonometric functions for the angle (sine, cosine, tangent, cosecant, secant, cotangent, etc). To determine the unknowns, he will apply a range of different formulas (involving the identification and use of opposite and adjacent sides of the various angles). Trigonometric functions are ratios of the sides of a triangle. You will cover an example finding all the trigonometric functions for a right triangle, beginning with finding the hypotenuse using the Pythagorean theorem. Then you will learn how to find the rest of the trig functions for a triangle if you given one function Angle Given Trig Function Find Right Triangle sides
Professor Burger explains how to use trigonometry to find the unknown sides of right Triangles. First, he explains how to evaluate and use trig functions with the help of a calculator. Then he will teach you how to find the hypotenuse of a right triangle using trigonometry, as well as how to find the adjacent side of a right triangle, given the measure of the hypotenuse. Once you have the measures of two sides of a right triangle, you will be able to deduce what the third side is equal to by applying the Pythagorean Theorem. Professor Burger will also highlight that trigonometry uses functions (e.g. you are not multplying by sine, but finding the function, sine, of a number). This is an especially important distinction to remember when manipulating trigonometric expressions and applying trigonometric properties Height of a Building Evaluate Trig Functions for an Reference Angles and Trig make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
Trigonometry: Find Trig trig Functions of Important Angles Intro to Sine and Cosine Graphs
In this lesson, you will examine the graphs of both the following trigonometric functions: sine and cosine. Professor Burger will show you how to graph sin and cos and teach you the acronym ASTC (All Students Take Calculus). Prof Burger also defines and shows you where to look for to evaluate the amplitude, period, and zeros of the sine and cosine graphs and shows you how to find and determine the maximums and mininimums for both sine and cosine functions. Finally, he will compare the graphs of the two functions, demonstrating that they have an identical shape with merely a shift between them to differentiate the two functions from each other. You'll also learn the importance of the π/2 interval in plotting and remembering the trigonometric function graphs of cosine and sine Coefficients
After learning how to graph the sine and cosine functions, now we will modify the graphs of these functions by adding in coefficients. Professor Burgers shows you a simple, 2-step process to determine the graphs. First, he will teach you about changes in the coefficient of the function. The introduction of a coefficient changes the amplitude of the graphed trigonometric function (sine or cosine). This is difference between AM (amplitude modulation) radio stations; changes in amplitude produce AM radio signals.The amplitude is equal to the absolute value of the coefficient of the trigonometric function. Prof Burger will also show you how changing the coefficient of the independent variable changes the period of the graphed sine or cosine function. This is the difference in FM radio stations (frequency modulation). The period =(2 Pi) / coefficient of X in Max, Min and Zeros of SIN & COS with Sine or Cosine
Professor Burger provides a real-world application of periodicity using tidal waves (waves that happen after an earthquake, also called tsunami waves) as an example. In the example, the waves are moving 540 feet/second (or 370 mph) and peaking at a hight of 50 feet every twenty minutes (period of the wave). The question posed is what is the length between each wave? Ocean waves, like sound waves, have a sine curve. You will use this knowledge and the distance formula (D = r*t) to solve a word problem about tsunami waves. These types of sinusoidal waves occur frequently in nature (and often in math word problems). Knowing how to approach and evaluate them is key to being able to solve all of them Phase Shifts
Now that you have learned how to graph the sine and cosine functions, Professor Burger asks the question ""How does changing the x-value affect the graph?"" He shows you how adding or subtracting to the x-value can actually change graphs of the sine and cosine functions, a process called translation. Professor Burger also warns you about classic mistake #8, reminding you that adding and subtracting to the x-value actually creates the opposite effect when graphed (adding to X moves the graph in the negative direction). Finally, Professor Burger shows you how to simplify the equation y = 3sin(x + Pi/2) using translation. The key lies in the fact that adding or subtracting pi/2 or 2*pi to a sine or cosine function means there are some shortcuts that you can take to determine what the graph of the function looks like (e.g. the graph of sine of (x+pi/2) is the same as the graph of cosine and the same as the graph of sine of (x+2*pi)). Period, Amplitude, Shifts
Professor Burger shows you how to use all of the tools at your disposal to effectively graph complicated trigonometric functions involving sine and cosine. He will show you how to recognize changes in period, amplitude, and vertical and phase shifts in the equation and how to correctly incorporate them into your trig function graph. He will also show you a three-step process of translating the equation, graphing the intermediate steps, and finailzing the graph. The examples you will use are y = -2sin(x- Pi/4)+1 and y = 2cos(Pi*x)-2. These equations both involve complications like those listed above (as indicated by their added constants and coefficients). tan, sec, csc, cot
This lesson introduces the graphs of all the other trigonometric functions (cosecant, secant, tangent, cotangent), using the sine and cosine graphs for points of comparison. Professor Burger shows you how to graph tanx using the identity tanx = sinx/cosx. This graph has asymptotes at all the multiples of Pi/2 and a period of Pi/absolute value of b. Next, you learn to graph secx, which is equal to 1/cosx. This means that secant has an asymptote anywhere cosx = 0. Next, Prof. Burger graphs cosecant, using the identity that cscx = 1/sinx. This graph is identical to secx, but shifted, like the relationship between sin and cos. Finally, you will learn to graph cotx, which is equal to 1/tanx. This means that there will be asymptote where tanx = 0, and zeros where tanx has asymptotes Fancy Graphing: TAN, SEC, CSC, COT Identify Trig Functions from Graph Inverse trig Function Intro Evaluating Inverse Trig Functions
Professor Burger shows you how to evaluate inverse trigonometric functions, which include arc sine, arc cosine, arc tangent, etc. He reminds you that inverse functions are asking "What is the angle whose function is X?" Thus, the output for any inverse trig function should be an angle for which, if you apply the indicated trig function, you get the indicated value as a result. Then he walks through finding the inverse of sine, the inverse of cosine, and the inverse of tangent. Finally, Prof Burger shows you how to interpret the presence of a negative sign and how to evaluate inverse trig functions using a calculator (indicated in radians). Inverse Trig Function Equations
An inverse function asks the question ""What is the angle whose function is X."" In this lesson, you will learn to solve equations that include an inverse function (arc sine, arc cosine, arc tangent, etc). Professor Burger first shows you how to untangle the equation, re-writing it so that you can understand for what you are solving. He will also show you examples when there may be an infinite numbers of solutions, and how you will need to correctly denote this answer. Finally, he suggests that you check your answers by graphing, and shows you how. This lesson will include several examples of evaluating problems involving arc sin, arc cos, etc. You will begin by seeing how to approach and solve a problem like 'inverse cosine of cosine x = pi/4' While it would seem that the cosine and inverse cosine here would cancel, you will learn in this lesson why this is not the case and how you can correctly solve for the answer Journal Trig Function Composition & Inverse Application: Is He Speeding? Fundamental Trigonometric Identities
In this lesson, Professor Burger will reveal and explain several basic trigonometric identity proofs. He will begin by reviewing the definitions of sine, cosine, and tangent. From these definititions, he will prove tanx = sinx/cosx. Then, he uses the Pythagorean Theorem to show you the proofs for 3 more trigonometric identities: cos^2 + sin^2 = 1, 1+ tan^2 = sec^2, and 1 + cot^2 = csc^2. Finally, Professor Burger will tell you which of these identities and proofs you need to memorize and which you can derive simply and don't need to fret about memorizing in advance of your test All Number Simplifying Using Trig Identities
Professor Burger demonstrates how to use the fundamental trigonometric identities to simplify complex triognometric expressions. Many seemingly complex expressions can be greatly simplified with simple application of several trigonometric identities. You will practice using expressions such as tanx * cosx and [1+(tan^2)x]/(csc^2)x. In these problems, you will see how substituting 1/cos^2 for sec^2 or 1/sin for csc or sin/cos for tan. By applying the definitions of the different trig functions, you'll often be able to substantially simplify a problem to the point where it will be very easy for you to evaluate and solve it Fractional Trig Expression Trig Function Binomial Prod Factoring Trigonometric Expressions
Just as you can simplify trigonometric expressions using the trig identities, you can often simplify the expressions by factoring, as you would with other types of expressions. Simplifying using the identities and factoring will save you a lot of effort in solving trig problems. In order to recognize opportunities to factor when working trigonometric problems, Professor Burger recommends that you use trigonometry identities to convert trig functions to sin and cos, whenever possible. Some examples you will learn how to simplify include (sin^2) x+ (sin^2)x(cos^2)x and sinx - (cos^2)x - 1 and sin^2x + (2/cscx)+x and Functions - Odd, Even, Neither?
In this lesson, Professor Burger teaches you how to determine if a function is even, odd, or neither. He begins by defining even and odd functions and graphing them. A function is even if the function of negative x is equal to the function of x. The graph of an even function is symetric across the y-axis. A function is odd if the function of negative x is equal to the negative function of x. The graph of an odd function is symetric around the origin. After defining these, Professor Burger identifies whether sin and cos are even or odd, and then shows several more examples, including tan x, sin (2x), (sin x)/x, and x cos x. Lastly, Professor Burger describes and illustrates what a function looks like that is neither odd nor even. In this case, it is not symmetric to the Y axis or the origin: Other Examplesigonometric Equations
This lesson will teach you how to solve equations involving trigonometric functions. Professor Burger shows you two ways to look at these equations, on a graph, and using reference angles. You will learn to rephrase the equation to determine what it is really asking, ""What value of X makes the function of X = n?"" You will learn how to write your answer to indicate infinitely many solutions and the step-by-step process of solving the equations. Examples of problems covered in this lesson involve trig functions, roots, fractions, variables and coefficients, including problems like cos x = 1/2 and sinx = -(2^(1/2)/2). You'll also learn when and why most trig problems like these have multiple (or infinite) solutions and how to correctly identify and denote these solution setsig Equations by Factoring
Professor Burger teaches how to solve more complicated equations (tanx * sin^2x = tan x) involving trigonometric functions in this lesson. Solving these types of problems involve use of trig identities, factoring, etc and how to find all of the viable solutions for these types of problems. In the problem listed above, Professor Burger will show you how to factor the equation in order to help simplify and then solve it. Professor Burger also gives a warning about cancelling out in equations that involve trig functions. By canceling, you risk missing valid solutions and solution setsAbout Professor Edward Burger: with Coefficients
Now that you have learned how to solve simple trigonometric equations, Professor Burger will show you how to solve trig equations that have a coefficient in the argument (e.g. sine of 2X versus just sine of X). These are also called multiple angle equations. In evaluating these trigonometry equations, you will generally only be asked to find solutions in one or two periods of the functions, so you will not have an infinite number of solutions (most often, you'll solve for solutions between 0 and 2*pi). Sample problems from this lesson include (2*sin 3*theta) = 1 and tan^2(2X) = 1 and sin2x*tan2x + sin2x = 0 and Quadratic Formula
Sometimes, trigonometric equations cannot be factored. To solve these equations, Professor Burger shows you how to apply the quadratic formula to find solutions to these equations. This is a multi-step process that starts with simplifying the equation. After the equation is simplified, you will be able to solve the quadratic formula and then enter that answer in to solve for X. In the lesson example, Professor Burger uses both a calculator and graphing to ensure he has the correct points. This lesson explains the covered material by walking through sample equations 3sin^2(2x) + sin (2x)-1 = 0. This equation has 2 solutions over the interval from zero to 2*pi. This lesson is loaded with warnings about easy mistakes to make and pitfalls to be wary of when evaluating problems like this.
Taught by Professor Edward Burger, this and Trig Equations
This lesson provides a more real-world application of trigonometric equations using a word problem. Professor Burger walks you through solving the trig word problem about spring motion. The motion of the particular spring in question is described by the function: sin (2T)+ 3^(1/2)*cost (2T), where T is the time in seconds. The problem asks us to solve for all times, T, when the object is located where it started the experiment. As he demonstrates how to solve the word problem, Prof Burger uses many of the trig information he taught in previous lessons, including identities, graphing, and angles. Finally, he reminds you to check your answer to make sure that the solutions are allowable. Additionally, he highlights that you should 'reality check' your answer as it's obviously not possible to have solutions for T that are negative given that this is a real-world example and time should never be negative Angle Sum and Difference Identities
Using the sum and difference identities, Professor Burger shows you how to solve a trig function for an unknown angle. For an example, he uses the sin(15 degrees). You can use known angle values for 45 degree and 30 degrees and the formula for the difference of two sines to find the solution. The formula for the difference of two sines is sin(x1 - x2) = sinx1cosx2 - cosx1sinx2. Hence, if you know what the sin and cos values of 30 and 45 degrees are, you should be able to plub them into this formula to arrive at the sine value of 15 degrees. The beauty of the sum and difference formulas for trig functions is that they allow us to decompose a problem we don't know the answer to into component parts to which we do know the answer, thus solving the original problem Mathematics Confirming a Double-Angle Double-Angle Identities
Double-angle identities allow you to simplify trigonometric equations with a 2 as the coefficient. (similar formulae exist for trig functions with 1/2 or 3 as the coefficient). In this lesson, Professor Burger uses the equation cos2x = sinx as an example. If this equation were simply cos x = sinx, we could divide to re-write the formula as sinx/cosx = tan x = 0, but in this case, we have a coefficient in advance of one of the arguments, which is why we need to use the double-angle formulas. After using the double-angle formulas in the provided example to simplify, you can further simplify these equations using trig identities (like the Pythagorean identity) and factoring. These tools will help you to solve many trig equations. The duble angle identities for sine, cosine, tangent and cotangent are: sin2x = 2sinxcosx, cos2x = cos^2x-sin^2x, tan 2x = 2tanx/(1-tan^2x), and cot2x = (cot^2x-1)/2cotx Multiple-Angle Identities Word Probs mathematics Cofunction Power-Reducing Half-Angle Identities,Trig Sines
Trigonometric functions (sin, cos, tan, etc) originally arose from the ratios of the sides of right triangles. But we can still use sine to evaluate the sines of angles that are in a triangle but not in a right triangle, using the Law of Sines. The Law of Sines states that [(sin a)/A] = [(sin b)/B] = [(sin c)/C], where a is the angle opposite side A (and so on for b/B and c/C). Sometimes the angles, a, b, and c, in this equation are denoted by the Greek symbols for alpha, beta, and gamma. Professor Burger shows you how to think about and use this this law by working through a number of different examples. This law lays the foundation for proving properties about triangles that don't have a right angle, including the calculation of the lengths of their sides and the measures of their angles radiansve Triangle - 2 Sides & 1ving a Triangle (SAS): Example Sines: Cosines
In this lesson, Professor Burger begins with a review of the Law of Sines. He then introduces the Law of Cosines, which extends the Pythagorean Theorem to triangles that are not right, allowing us to solve for any angle in the triangle. The Law of Cosines, also called the Al-Kashi Law and the Cosine Formula and the Cosine Rule, states that, for the angle you are solving for, opposite side ^2 = (sum of the squares of adjacent sides) - 2 * (product of the adjacent sides) (cos of the desired angle). The law of cosines is most useful when computing the third side of a triangle when two sides and their enclosed angle are known (SAS) or when computing the angles of a triangle in which all sides are known (SSS).
This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!
Taught by Professor Edward Burger, this Cosines (SSS) Law of Cosines (SAS): Heron's Vectors Colorado Vector Magnitude & Direction
In this lesson, Professor Burger will show you how to find the magnitude and direction angle of a vector provided in standard form (and how vector notation associated with standard form looks and should be interpreted). He will also review how to depict a vector graphically that we have described by standard form. The magnitude is the length of a vector (reminder: it must be positive), and we use the Pythagorean theorem to calculate the vector's magnitude. The direction angle is always measured counter-clockwise from the positive side of the x-axis. Once we know the vector's length, we can use trigonometric functions to calculate the direction angle of the vector. Last, Professor Burger solves for the magnitude and direction of some of the vectors using a calculator Adding Vectors & Multiplying Scalars
Professor Burger shows you how to add and subtract vectors and use scalar multiplication to elongate or shrink vectors while maintaining their direction angle. The magnitude of a vector can be altered with scalar multiplication. A scalar is simply a number (positive or negative or a fraction) used to multiply a vector by, with the vector keeping its same direction and changing magnitude. Vectors can also be added and subtracted by simply adding or subtracting the components. It is also simple to find the answer graphically by creating a parallelogram with the two vectors, which Professor Burger demonstrates Components of a theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
Trigonometry: Finding a Unit Word Problems - Velocity or Forces Writing Rewriting Powers of i Add and Subtract Complex Numbers
A complex number is a number of the form a+bi where a and b are real numbers and i is equal to the square root of 1. This video lesson will walk you through the basics of addition and subtraction when dealing with complex numbers Multiplyividing Complex Number-Graph, Absolute Complex Numbers - Trig or Polar Form
This lesson instructs you on how to convert complex numbers into trig form (also known as polar form). Complex numbers, written in the form (a + bi), are an extension of the real numbers obtained by adjoining an imaginary unit, denoted by i, which is the square root of negative 1. To convert complex numbers into trigonometric or polar form, Professor Burger first walks you through sketching a graph of the number and drawing a right triangle. From that, he shows you how to use the trig properties to find the unknown values and the modulus. Then, you plug these falues into the trig form and determine the angle. To illustrate this method, Professor Burger walks you through an example in which he converts (-(3^1/2), +i) to polar or trigonometric form Multiply Complex Numbers-Trig,Polar DeMoivre, Complex Numbers, Powers Power and Roots of Complex Numbers
Professor Burger explains how to find the powers and roots of complex numbers. The equation of a complex number is z= r(cosx + isinx). To raise the complex number to a power, n, the equation is z^n = r^n[cos(nx) + isin(nx)]. In general, if you are raising a complex number to the power of n or 1/n (taking the nth root), you will come up with n solutions, as you will always have one solution for each of the degrees of power. When taking the root of a complex number, you will find one solution for each degree of power. To find the nth root of a complex number the equation is n root of z = (n root r) *[cos ((x + 2 Pi K)/n) + 1 sin ((x + 2 Pi K)/n)] where k = 0, 1, 2,...n-1 More Roots of Complex Numbers Roots of Unity Polar Coordinates Polar & Rectangular Coordinates
You will learn how to convert from polar coordinates to rectangular coordinates (or Cartesian coordinates or coordinates in a Cartesian plane), and vice versa in this lesson. First, Professor Burger gives you an overview of polar and rectangular coordinates. Then, you will learn how to convert a polar cordinate (r, Theta) into a rectangular coordinate (x, y), using the equations x = rcosTheta and y = rsinTheta. To convert from rectangular to polar, you will use the equations r = root(x^2 + y^2) and Theta = arctan (y/x). To illustrate the use of all of these formulae, Professor Burger will walk you through the conversion of (3, pi/6) from polar to rectangular coordinates and the conversion of (-1.1) in rectangular coordinates to equivalent polar coordinates Weighty Graphing Simple Polar Intro to Conic Sections Info from a Parabola's Equation number Writing an Equation for a Parabola
A parabola is a conic section in which the locus of points constructing it are equidistant from the focus and the directrix. To find the formula of this equation when given the vertex (h,k) and the distance from the focus (p), this lesson will show you how to find the equations for the parabola described by these criteria. There will be two formulas depending on whether p is positive or p is negative (which should indicate whether the parabola opens up or down). Ellips Ellipses
An ellipse is a collection of points whose combined distance from two fixed points (both called a focus) are the same. This lesson will show you how to find the equation of an ellipse if you know some information but not all of it. The equation for this type of conic section is typically written as: x^2/a^2 + y^2/b^2 = 1. We start with a situation where you know the x-intercepts and the foci (both focuses) but are looking for the equation of the ellipse. To approach this problem, we'll use triangle formulas and the Pythagorean Theorem to find the y-intercept Apply Ellipse - Satellites Hyperb Hyperbolas
A hyperbola is a collection of points for which the difference in distance between two fixed points (both called a focus) is constant. This type of conic section is typically described by a formula that is written as: x^2/a^2 - y^2/b^2 = 1. In this lesson, we walk through how to find the equation for a hyperbola when we know the x-intercept and the foci (both focus points) of the hyperbola. We also work through a problem in which the y-intercept and both foci are provided and you are asked to find the equation for the hyperbola Hyperbolas in Navigation a of Name ThatSupplementary Files:
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Linear and Integer Programming
This course will cover the very basic ideas in optimization. Topics include the basic theory and algorithms behind linear and integer linear programming along with some of the important applications. We will also explore the theory of convex polyhedra using linear programming.
Sessions
Course at a Glance
5-7 hours/week
English
Instructors
Categories
About the Course
Linear Programming (LP) is arguably one of the most important optimization problems in applied mathematics and engineering. The Simplex algorithm to solve linear programs is widely regarded as one among the "top ten" algorithms of the 20th century. Linear Programs arise in almost all fields of engineering including operations research, statistics, machine learning, control system design, scheduling, formal verification and computer vision. It forms the basis for numerous approaches to solving hard combinatorial optimization problems through randomization and approximation.
At the end of the course, the successful student will be able to cast various problems that may arise in her research as optimization problems, understand the cases where the optimization problem will be linear, choose appropriate solution methods and interpret results appropriately. This is generally considered a useful ability in many research areas.
Course Syllabus
Introductory Material
Introduction to Linear Programming.
Week #1:
The Diet Problem.
Linear Programming Formulations.
Tutorials on using GLPK (AMPL), Matlab, CVX and Microsft Excel.
The Simplex Algorithm (basics).
Week #2:
Handling unbounded problems
Degeneracy
Geometry of Simplex
Initializing Simplex.
Cycling and the Use of Bland's rule.
Week #3:
Duality: dual variables and dual linear program.
Strong duality theorem.
Complementary Slackness.
KKT conditions for Linear Programs.
Understanding the dual problem: shadow costs.
Extra: The revised simplex method.
Week #4:
Advanced LP formulations: norm optimization.
Least squares, and quadratic programming.
Applications #1: Signal reconstruction and De-noising.
Applications #2: Regression.
Week #5:
Integer Linear Programming.
Integer vs. Real-valued variables.
NP-completeness: basic introduction.
Reductions from Combinatorial Problems (SAT, TSP and Vertex Cover).
Approximation Algorithms: Introduction.
Week #6:
Branch and Bound Method
Cutting Plane Method
Week #7:
Applications: solving puzzles (Sudoku), reasoning about systems and other applications.
Other texts: We may borrow material that is covered in some of thetexts below. As you go down this list, the texts become less relevant for this class but remain very important for the broader field of optimization that we seek to introduce through this class.
Convex Optimization by Boyd and Vandenbreghe is a good reference for the more general area of convex optimization.
Numerical Optimization by Nocedal and Wright is a great reference for solving non-linear optimization problems.
Course Format
Each class will consist of many short lecture videos between 10-15 minutes in length with nearly 1.5-2 hours of lectures each week. We will have weekly assignments involving 5-10 problems each week. These assignments will either be multiple choice or ask you to compute something and enter the answer. We will have four programming assignments (one every two weeks) that will incrementally help you build an ILP solver. Programming assignments can be done in virtually any general purpose language.
The course will be structured with two interactive tracks: algorithms/theory for linear optimization problems and applications. The applications will include ideas on modeling real-life optimization problems as linear programs and the appropriate use of concepts like duality in finding and interpreting solutions.
FAQ
Will I get a statement of accomplishment after completing this class?
Yes.
What textbooks will I need for this class?
No textbook is officially required. However, there are numerous great textbooks on this topic. We are hoping you will be able to find one. You can always cross check whether your textbook has adequate coverage for the topics to be taught in this class.
We recommend a great textbook by Prof. Vanderbei (see It is available as a free download. The classic book by Chvatal is an excellent textbook but sadly out-of-print.
What are the pre-requisites?
The prerequisites for this class include:
Basic College Level mathematics: calculus and some knowledge of linear algebra.
Some programming skills: However, students without a programming background can pass the course by solving the weekly assignments.
What will the level of the class be?
Normally, this will be a junior/senior level undergraduate class or even a beginning graduate class, depending on the major and the university. A highly motivated high school student with advanced mathematical skills can follow along.
What skills will I learn?
The class is about Linear and Integer Programming. You will definitely learn:
what optimization problems are, what are linear optimization problems, what are the applications, what are the algorithms used for solving LPs, and how do they work?
In addition, we found through our previous offering in 2013 that the course helps many students think mathematically and improve the programming skills. We were proud of the many students from last year who took on some of the challenging programming assignments and reported a big confidence boost in their skills.
Can I get university course credits for this class?
Unfortunately, not at this time. University of Colorado, Boulder may be considering ways of recognizing Coursera classes, but there is no consensus at this point in time.
However, Prof. Sankaranarayanan teaches an on-campus class (CSCI 5654) at the University of Colorado, Boulder, and it will be available simultaneously on-line for credit through CAETE (see ). You can enroll for our class CSCI 5654 through CAETE from anywhere in the world.
Are we going to do rigorous mathematical proofs? How much programming do you expect to do?
We will cover proofs for interested students. But we will never have proofs in the assignments.
Assignments will test the conceptual ideas in the class including algorithms, and many assignments may ask you to use an existing LP solver (open source or commercial) to solve a LP/ILP and interpret its answer. We do not consider this part as programming: any computer user should be able to manage this.
We will have programming assignments: in fact, students will get to build their own LP and later ILP solver in stages. But the programming is not going to be for everyone. The last assignment (ILP) will be quite intense and definitely challenging to students.
Is there a particular programming language that you expect for the assignments?
No. We will allow students to program in most general purpose languages. However, by the very nature of the assignments, some languages such as python, C/C++, Java, Haskell, Scheme, Lisp, MATLAB/Octave, ... will be more suitable than others. You can judge for yourselves as soon as the assignments are posted on week 1.
What will the coursework involve?
Coursework will involve watching videos, solving weekly assignments and tackling four programming assignments. See below on how the grading will work.
How do I pass? How does distinction work?
To pass the class: you will need to get at least 35% of the total grade. The pass cutoff is designed so that students can pass either by solving some/all of the programming assignments OR by solving some/all of the weekly assignments.
To score a distinction: you will need to get at least 85% of the total grade. To score a distinction, students must do well in ALL aspects of the course: programming as well as weekly assignments. | 677.169 | 1 |
Homework Help Math
Basic algebra is the first in a series of higher-level math classes students need to succeed in college and life. Make sure your tween or teen is on track for high school math with this guide to algebra.
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Details about Worksheets for Classroom or Lab Practice for Beginning Algebra:
Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom-friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems.
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About
"Summarize The Major Mathematical Concepts In This Course" Essays and Research Papers
Math 213
MathematicalConcepts Reflective Paper
Mathematics for Elementary Educators teaches many concepts that... are needed for basic understanding of what you will be teaching in your classroom. There were several ideas covered in thiscourse but there are several of the majormathematicalconcepts that stand out to me. Those concepts are the, National Council of Teachers of Mathematics principals and standards, Whole Numbers and their Operations, Algebraic Thinking, Rational Numbers as Fractions...
issues in the accounting environment.
Apply cost concepts and techniques for planning and control decisions.
2. Introduction to Economics...Thiscourse aims to facilitate students' learning of basic concepts of microeconomics, and their application to the real world. Students would be encouraged to develop their attitude and ability to discover and innovate.
3. College Mathematics
The course is designed to familiarize students with mathematicalconcepts that serve as a foundation of subsequent coursework...
This system in which a child is constantly moving objects with his hands and actively exercising his senses, also takes into account a child's... special aptitude for mathematics. When they leave the material, the children very easily reach the point where they wish to write out the operation. They thus carry out an abstract mental operation and acquire a kind of natural and spontaneous inclination for mental calculations.
Dr. Maria Montessori, The Discovery of the Child, Maria Montessori
Discuss...
Fundamental Concepts of Industrial and Organizational Psychology
Jessica Lindsay
University of Phoenix
PSY 435
Facilitator: Gary Mayhew,... Ed. D.
July 26, 2011
Fundamental Concepts of Industrial and Organizational PsychologyMathematical Connection
Mathematics has had an incredible impact on technology as we know it today. Understanding this impact... aids in understanding the history of how technology has developed so thoroughly and what significant events happened to facilitate such an advanced society. A better understanding can be derived by analyzing the historical background on the mathematicians, the time periods, and the contributions that affected their society and modern society as well as specific examples...
Course Project: Applied Information Technology Project
Objectives | Guidelines | Milestones | Course Project Outline |...Course Project Proposal and Milestones | Course Project Overall Table of Contents | Course Project Technical Areas
| |
Objective | |
Provide an efficient means for students to develop various aspects of a technical/business proposal throughout the course. There are three papers that build on each other through the session to give the students experience in taking a concept...
Heavens
In summer of 1609, Galileo Galilei (1564-1642) pointed his revolutionary astronomical telescope to the heavens under the starry Venetian sky; his... greatly important observations unveiled the mysteries of universe and would end up changing the course of scientific thought forever. Galileo lived in an age where there was much status quo, when scientists and philosophers would accept scientific and religious doctrine that had stood for hundreds, if not thousands, of years instead of challenging...
Political sociology was traditionally concerned with how social trends, dynamics, and structures of domination affect formal political processes, as well as... exploring how various social forces work together to change political policies (Burnham, 2012) MajorConcepts of Political Sociology Political Culture Political culture refers to what people believe and feel about government, and how they think people should act towards it. To understand the relationship of a government to its people, and how those people...
In this Mathematics for Elementary Educators I course, it teaches me many concepts that a professional mathematics... teacher should possess while teaching elementary students. The concepts has influences my own ideas and philosophy of teaching. In this reflection paper, I will also summarize the majormathematicalconcepts, explaining how the learning concepts are relevant to the characteristics of a professional mathematics teacher.
During this five week course, there were five majorconcepts that...
Research Synthesis – Mathematical Tasks and Classroom Implementation
Introduction
For decades, researchers have found that designing... instruction in ways that foster critical thinking and problem solving can improve student learning. Studies have shown that careful implementation of well-designed mathematical tasks can improve student understanding of mathematicalconcepts and connections between topics in mathematics. In short, these tasks can help students to think mathematically. Although...
Individual- MajorConcepts Paper
Indiana Wesleyan University
I currently work for Ivy Tech... Community College as an Adjunct Instructor for the School of Public and Social Services teaching criminal justice courses. In 2008 Ivy Tech Community College became the state of Indiana's largest public college system and is one of the most culturally diverse in the state including students from every race, ethnicity, religion and age group.
Ivy Tech Community College is faced...
through creating and exchanging products and values with others.(Kotler.P 2002 : 5) The goals of marketing is to attract new customers by promising superior... value and keep and grow current customers by delivering satisfaction. There are five core concepts of marketing, which includes needs, wants and demand; products, services and experience; value, satisfaction and quality; exchanges, transactions and relationships; and finally, market and marketing.
After World War II, the variety of products increased...
MARKETING CONCEPTS
Following are the six concepts of marketing
• Production concept
• Product...conceptThis concept...
Your Course Project
Financial Statement Analysis Project -- A Comparative Analysis of Oracle Corporation and Microsoft Corporation
Here is... the link for the financial statements for Oracle Corporation for the fiscal year ending 2011. First, select 2011 using the drop-down arrow labeled for Year on the right-hand side of the page, and then select Annual Reports using the drop-down arrow labeled Filing Type on the left-hand side of the page.
You should select the 10k dated 6/28/2011 and choose...
This paperwork includes MTH 213 Week 5 Reflective Paper
General Questions - General General Questions
Prepare a 700- to 1,050-word... paper synthesizing the majorconcepts addressed in thiscourse. Include the following in your paper:
· Summarize the majormathematicalconcepts of thiscourse.
· Explain how the concepts learned in thiscourse are relevant to the characteristics of a professional mathematics teacher.
· Determine how the courseconcepts have influenced your...
With a sincere sense of gratification, this ensuring statement of purpose is put forth by me, HIMANI SHARMA just to pursue my career & study... in ELECTRONIC ENGINEERING TECHNICIAN- COMMUNICATIONS course at esteemed college like Seneca College of Applied Arts & Technology in Canada. I feel it is a life time opportunity to explore and study in a rich foreign nation like Canada where I can make best possible use of educational resources provided there. In this ever-changing world of science and technology...
Linear Programming Concept Paper
There are two types of linear programming:
1. Linear Programming- involves no more than 2 variables, linear...Abraham Lincoln High School
Advanced Placement Course Descriptions
2012-13
ENGLISH
Eleventh Grade Honors American Literature...Thiscourse uses canonical American texts to chronologically explore the development of America as a nation, an identity and an idea. We investigate themes such as: the importance of self-reliance, action versus inaction, and having the courage of one's convictions. Semester one follows early American history and we read authors such as Miller, Emerson, JeffersonUnit #2: MajorConcepts Essay
No Child Left Behind: Representative Democracy, Bureaucracy, and Accountability
Demetrius... Zeigler
Kaplan University
Representative Democracy has its roots as a concept or principle in the very fabric of the founding of the United States of America. Early settlers were looking for a place to live while being free to choose their leaders. They were eager to say bon voyage to the old way of rule by monarchy or dynastic family rule. The new wave or system...
Choosing a college major for some is easy; some people know exactly what they want to be when they grow up. For others, choosing a college...major is probably one of the hardest decisions they will make in their life. It doesn't help that there are now a lot of college courses and college programs among which you have to choose.
There was once a time when choices were simple: good or evil, ketchup or mayonnaise, Bachelor of Science or Bachelor of Arts. Nowadays, you have to choose between shades...
understanding
• promotion of moral thinking
• feeling and action
• enlargement of the imagination
• fostering of growth, development, and... self-realization
Based on the AIMS concept we are building an online learning system for our employees:
Ideally, the learning outcomes in order of priority are
Translated into course content, resources and an approach to the teaching and learning process that will enable a student to achieve those outcomes.
Once these basic parameters have been thought...
Running head: MATHEMATICAL CONNECTION PROJECT
Mathematical Connection Project
University of Phoenix
MTH 110...
The Impact of Mathematics on Daily Social Activities
In society today people deal with some kind of problem solving method that involves
math. Thanks to the mathematicians from the past and present we are able to evolve as a society
with advancements on medicine, technology and able to travel into space. The impact that
Euclid, Al-Khwarizmi, Rudolf Laban...
Logical-Mathematical Intelligence
The theory of multiple intelligences was thought up by Howard Gardner through his opinion on people having... not only one way of thinking. Howard Gardner is a Professor of Education at Harvard Graduate School of Education and author of Multiple Intelligence: New Horizons and many other books. Gardner defines intelligence as an ability or set of abilities that allow a person to solve a problem or fashion a product that is valued in one or more cultures (Lane, 2005)...
A Summary of the Major Rhetoric Concepts
Kairos (Isocrates)
The fundamental concept of 'Kairos'—fitness for the... occasion, or the right moment/timing for something—is a recurring topic in Isocrates's writings. The consistent advocation and practice of thisconcept may constitute his most significant contribution to rhetoric. A first- handing knowledge of Kairos' rich and elusive meanings can be obtained by reading through Isocrates.
For what has been said by one speaker is not equally useful...
Mathematical modeling is commonly used to predict the
behavior of phenomena in the environment. Basically,
it involves analyzing a set of... points from given data
by plotting them, finding a line of "best fit" through
these points, and then using the resulting graph to
evaluate any given point. Models are useful in
hypothesizing the future behavior of populations,
investments, businesses, and many other things that
are characterized by fluctuations. A mathematical
model usually describes a...
Key Concepts
MHR 3200—Exam 1
Note: This is not necessarily an all-inclusive list, but rather is intended merely to be a... list of the majorconcepts of which you should have a good level of knowledge and understanding. I reserve the right to ask questions on Exam 1 that do not tie directly to material on this list but do fall into the required reading and/or other presentation materials for this section of the course.
Introduction/Course Overview/Introductory Lecture
- Disciplines contributing...
Self-concept is the cognitive thinking aspect of self also related to one's self-image, it's the way we see ourselves in the mirror. We are... grown into our self-concept by what we learn when we are young from our parents or our peers. Self-concept is changed throughout life from how people look at you, how you compare to others, how your traditions and customs differ from other people, and how you feel about yourself. We all have concepts or perceptions of ourselves which continually develop and evolve...
COURSE DESCRIPTIONS
Statistics
Thiscourse has 3 credits.
Thiscourse provided... an introduction to the various methodologies involved with business statistics. Various topics were covered in thiscourse, including probability distributions, testing of hypotheses,correlation analysis, regression, goodness of fit, error analysis, and data summation.
Statistics in International Business
Thiscourse has 3 credits.
When the students have completed thiscourse, they will be able to: Organize...
Mathematical logic is something that has been around for a very long time. Centuries Ago Greek and other logicians tried to make sense out of...mathematical proofs. As time went on other people tried to do the same thing but using only symbols and variables. But I will get into detail about that a little later. There is also something called set theory, which is related with this. In mathematical logic a lot of terms are used such as axiom and proofs. A lot of things in math can be proven, but there...
"Dr Maria Montessori took this idea that the human has a mathematical mind from a French philosopher Pascal and developed a... revolutionary math learning material for children as young as 3 years old. Her mathematical materials allow the children to begin their mathematical journey from a concrete concept to abstract idea".
With reference to the above statement please discuss how these children utilize their mathematical mind as part of their natural progression, to reason, to calculate and estimate...
completion of this programme
you will be awarded the University of
Sunderland BA (Hons) Business Management.
This module... will enable you to understand the
processes of project planning, financing and
implementation using a variety of techniques
covering:
Course Description
The BA (Hons) Business Management will
provide you with the necessary expertise to
critically evaluate business models and concepts
and apply them to real-world situations. Upon
completion of thiscourse you will be...
categorized into different paradigms. Hergenhahn and Olson (2005), define a paradigm as "a viewpoint shared by several scientists that provides a general... framework for empirical research, and is usually more than just one theory" (p. 24). Two of the major paradigms are the functionalistic and associationistic paradigms.
DiscussionWithin the functionalistic paradigm, theorists influenced by Darwin, attempt to explain learning by discovering and researching how mental and behavioral processes are related...
Course Handout
Accounting For Managers (AFM)
Semester-I (June -October, 2013)
Course Overview
The course is... designed to provide the students with a sound understanding of the basic accounting principles, concepts and standards and skill in the preparation and presentation of financial statements. Moreover thiscourse a blend of financial accounting and management accounting. At the end of thiscourse, students will be capable of preparing and understanding financial statements and use financial...
THE EFFECTS OF ACCOUNTING CONCEPTS ON FINANCIAL STATEMENT
5.1 ENTITY CONCEPT
The first accounting concept is... entity concept. These concept shows accounts are kept for entities and not the people who own or run the company. Even in proprietorships and partnerships, the accounts for the business must be kept separate from those of the owners. This is because what whatever amount the company owes to others is not the liabilities of the owners. The maximum amount that the owner is going to lose...
Date:
Course Name:
Unit Number:
Case Name: Apple Inc. Case... Study Analysis
Introduction
This analysis is based on Apple Inc. case study in which the strategic management is analyzed. In the process of analyzing thisconcept, the article also indentifies the issues and problems as they are presented together with the identification of the major issues surrounding the organization and individuals...
Sample course evaluation questions for courses using HPC technology
Courtesy of the LEAD Center,
University of... Wisconsin-Madison
Below is a list of sample survey questions that one might use to evaluate course impact and learning gains in a course that incorporates HPC technology. The list includes questions about student background, the impact the course had on students' interest and confidence in certain skills, and the effectiveness of various elements of the course in promoting studentResponse Paper
Interpersonal Communication
Analyzing the idea of self-concept; which encompasses the ideas of values, attributes, and...people have come to realize that to do this; there is a need to learn another language besides their mother tongue.
English as an... international language has become very important to be learned and understood by many people. It is the only one way to communicate between countries, cultural groups, various companies and organizations, communities and friends all over the world. By mastering English, people can enrich their knowledge and survive in this competitive atmosphere. Because of the...
Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, cognitive... and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. In practice "quantifiable behavior" is often constituted by "task performance".
As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology. Mathematical...
key objectives covered throughout thiscourse. It will challenge you to apply your knowledge of cost information when evaluating... the decision to make or buy a product. Please use this outline and grading rubric as a guide to completing your course project. It provides specific details of the required elements of the project, and it will be used by your instructor as a grading guide.
Read Integrative Case 4-61, "Make versus Buy," on pages 151 and 152 of the course text. Assume that you are the general...
Considered Within Freire's "Banking" Concept of Education
Education, the process of taking in and applying information, is an important part... of life that directly impacts an individual's judgement and reasoning of both themselves and society. A person's educational experience has the ability to influence their life despite their intelligence level or home life situation. In Paulo Freire's book, The Pedagogy of the Oppressed, two major learning styles, the "banking" concepts and the "problem-posing" methods...
Mathematical Happenings
Rayne Charni
MTH 110
April 6, 2015
Prof. Charles Hobbs
Mathematical Happenings
Greek... mathematicians from the 7th Century BC, such as Pythagoras and Euclid are the reasons for our fundamental understanding of mathematic science today. Adopting elements of mathematics from both the Egyptians and the Babylonians while researching and added their own works has lead to important theories and formulas used for all modern mathematics and science.
Pythagoras was born in Samonof Management
MBA Program
COURSE OUTLINE
Term: Fall 2011-2012
I. Course Title : Strategic Management 01082
II. Section... : B
III. Nature : Core Course
IV. Credit : 3 credit hours
V. Day and Time : MW 08:00- 9:30 PM
VI. Course Description
Thiscourse is designed to provide an overview of strategic Management concepts with current business practice in a way that is both interesting and effective for the students. Studying thiscourse, students will develop basic knowledge...
Introduction
At the beginning of thiscourse, we wrote an assignment to introduce how we conceive the relationship between... community, community development and community economic development. I formed the initial cognition about the community economic development through writing that paper. Then, I got a clearer idea of CED as the course progressing. In this paper, I would like to explore and analysis the related notions of asset based community economic development and the practical example in...
Self-Esteem and Self-Concept
Self-Esteem is the way we view ourselves, and the acceptance of our own worth. It is the reason we compare... each other, and try to be better than others. We judge every little action we do and thought counts. Self-Esteem is linked to the feelings of pride and discouragement. Self-consciousness is associated with self-esteem as self-consciousness is a sense of awareness. Self-confidence is a feeling of personal capacity and self-respect, which is a feeling of personal...
Principle of Management Course: My Experiences
I believe that the Principles of Management course provided me with
invaluable... information which will help in furthering both my professional as
well as personal life. I believe that learning is a process by which an
individual undergoes certain changes. Also, during the learning process, many of
the beliefs which a person holds are challenged. I underwent various changes
during thiscourse. This paper will explain those changes. Furthermore,...
Mathematical Explanations and Arguments
The chapter focused on the fine art of mathematical explanations and arguments. The... history of mathematical explanations and arguments is complex because what we currently use as reasoning and proof arises from a methodology from 300 BC that was mostly due to the efforts of Euclid of Alexandria. India and China shadowed the mathematical explanations and arguments asserted by Euclid. Pythagorean Theorem, which means that the sum of the squares of the two...
The concept of learning can be defined as the changing of knowledge, skills and behaviors of one person, which due to different...sen@baruch.cuny.edu
COURSE OBJECTIVES
Marketing begins and ends with the consumer. The purpose of thiscourse... is to introduce you to the study of consumer behavior. We will take the perspective of a marketing manager who needs knowledge of consumer behavior in order to develop, implement and evaluate effective marketing strategies. We will examine many concepts and theories from the behavioral sciences and analyze their usefulness for marketing strategies. The goals of thiscourse are for you to:
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Course Description
This professional course deals with nursing theories, concepts, framework and models for nursing. It also includes a review of the development of the discipline of nursing, the origin, development and progress... | 677.169 | 1 |
Graphs and Applications
Discrete Mathematics is one of the fastest growing areas in mathematics today with an ever-increasing number of courses in schools and universities. ...Show synopsisDiscrete Mathematics is one of the fastest growing areas in mathematics today with an ever-increasing number of courses in schools and universities. Graphs and Applications is based on a highly successful Open University course and the authors have paid particular attention to the presentation, clarity and arrangement of the material, making it ideally suited for independent study and classroom use. An important part of learning graph theory is problem solving; for this reason large numbers of examples, problems (with full solutions) and exercises (without solutions) are included | 677.169 | 1 |
In this three-book Pre-Algebra series, Fred acts as the human face of elementary physics, pre-algebra, and then takes a summer vacation! The series includes Pre-Algebra 0 with Physics, Pre-Algebra 1 with Biology, and Pre-Algebra 2 with Economics.
For more fun-filled books on your favorite character, visit the Life of Fred page!
What's the difference between math and physics? Life of Fred Pre-Algebra with Physics answers this question and launches your student into 40 chapters of physics to solve quirky scenarios we encounter everyday. Learn about friction as Fred tries to move a safe across a hallway, calculate the amount of force he needs to move it, the transfer of energy, and more! Fred will make your students want to learn physics with fun stories and math that's geared to help them progress to more advanced concepts!
Pre-Algebra 1 with Biology
Math with meaning is synonymous with Life of Fred Biology. In this book you will learn algebraic math and apply it to life sciences in various scenarios to solve Fred's obstacles. Although unorthodox, Life of Fred Pre-Algebra 1 Biology starts the book by acknowledging the need to pull from different subjects and states, "So this is a pre-algebra book… and a biology book. And we will quote six lines of Italian poetry. We will discuss the difference between a metaphor and a simile. Talk about first-aid for fainting. Balancing some chemical equations. Play with a French phrase in the movie Camelot. And in the meantime, you will learn some algebra and a lot of interesting biology."
Pre-Algebra 2 with Economics
Students will learn improper fractions, interest rates, and the differences between capitalism and communism in economic terms. In Life of Fred Economics, students will learn: domain and codomain of a function, conversion factors, steps in solving word problems, how not to bore your horse if you are a jockey, one-to-one functions, unit analysis, key to a successful business, five qualities that money should have, and so much more!
Even if you only want to add as a reading assignment, these books will keep your attention! As a math curriculum or supplement they are priceless. Wonderful series!
Mindy D - Parent - Member Since October 2015
Great series!
Apr 11, 2016
I love how it intertwines other subjects with math!
Firstname L - Member Since November 2013
Intuitive
Apr 04, 2016
I like that my child can grade his own work. I like that my child is drawn to the stories and doesn't complain about math anymore, but instead will pick up math before any other subject. He is learning to teach himself.
Every age in our house loves the Fred books, and these are no exception. The books build on each other really well, and I love how they tie in different topics. The math is absorbed almost without being aware of it. My kids read them over and over and treat the practice problems as more of a game than work. They're fun!
Dr. Schmidt is great at teaching math, as well as other topics in a fun, engaging manner. These books are not easy. They are as fun as math can get, but they teach your kids what they need to know and most importantly, WHY they need to know it.
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We Love Life OF Fred
Feb 26, 2016
We love Life Of Fred...
My kids are 10 & 8 we started at Apples. We sometimes need to add a Youtube video on what we are learning in the books but they love the story line and its a fun math curriculum that really makes them think outside the box.
The Life of Fred contains fun stories about a child prodigy math genius your students can follow along as he gives real-world examples of everything from math to language arts. In the Life of Fred Elementary Math Series, your students will learn everything from telling time, division, beginning algebra, and more!
You can purchase one book at a time or the entire set at a discounted price! Order more as their skills progress such as the Life of Fred Intermediate Series that incorporate biology, physics, and economics using pre-algebra formulas.
The stories carry on through the exercises, and many of the books have a series of quizzes that bridge them to the next chapter. Some books have all the problem solutions in them, while others have separate answer keys or companions.
Written by Dr. Stanley F. Schmidt, these affordable books were designed to keep your students engaged in math while also following along a storyline. Perfect for independent study, each unit teaches concepts that can be applied using real-world examples. Find out more by reading this blog post!
Shipping: Flat rates apply per set ordered, not per book. For example, if you order a set of 3 books, you will be charged $5.99 if this is the only item in your cart. If you order a set of 3 books, plus an educational toy or any shipped item on the Educents site, you will be charged $5.99 for the set of books, then $1.99 to ship the additional item. | 677.169 | 1 |
Details about Prentice Hall Math Course 1:
Prentice Hall Mathematics Course 1:
A combination of rational numbers, patterns, geometry and integers in preparation for one- and two-step equations and inequalities.
Guided Problem Solving strategies throughout the text provide students with the tools they need to be effective and independent learners. An emphasis on fractions solidifies student understanding of rational number operations preparing them to apply these skills to algebraic equations. Activity Labs throughout the text provide hands-on, minds-on experiences reaching all types of learners.
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Rent Prentice Hall Math Course 1 1st edition today, or search our site for other textbooks by Randall I. Charles. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
Need help ASAP? We have you covered with 24/7 instant online tutoring. Connect with one of our tutors now. | 677.169 | 1 |
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Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Physics Super Review includes vectors and scalars, plane motion, dynamics of a particle, work and energy, conservation of…read more
The Math Made Nice & Easy series simplifies the learning and use of math and lets you see that math is actually interesting and fun. This series is for people who have found math scary, but nevertheless need some understanding of math without having to deal with the complexities found in most…read more
Specifically designed to meet the needs of high school students, REA's High School Biology Tutor presents hundreds of solved problems with step-by-step and detailed solutions. Almost any imaginable problem that might be assigned for homework or given on an exam is covered. Topics include…read more
The Problem Solvers are an exceptional series of books that are thorough, unusually well-organized, and structured in such a way that they can be used with any text. No other series of study and solution guides has come close to the Problem Solvers in usefulness, quality, and effectiveness.…read more
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Need help with Basic Math and Pre-Algebra? Want a quick review or refresher for class? This is the book for you! REA's Basic Math and Pre-Algebra Super Review gives you everything you need to know! This Super Review can be used as a supplement to your high school or college textbook, or as a handyEach Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more…read more | 677.169 | 1 |
"Algebra, Trigonometry, and Statistics" helps in explaining different theorems and formulas within the three branches of mathematics. Use this guide in helping one better understand the properties and rules within Algebra, Trigonometry, and Statistics. | 677.169 | 1 |
Administrative Assistant:
I currently work at Scranton Preparatory School as an algebra and geometry teacher. My college math courses made me aware of the bigger picture in mathematics. They helped me to not only grasp each individual subject on a higher level, but also to understand their connection to one another so I can convey their practical relationships and importance in the greater field of mathematics to my high school students.
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Math Club
The Math Club, SEMI-Group, promotes mathematics in the community. Each year, its members participate in many on- and off-campus events.
Professional Conferences
We usually attend the regional or national meeting of NCTM (National Council of Teachers of Matheamtics), the PCTM (Pennsylvania Council of Teachers of Mathematics), or the MAA (Mathematical Association of America) national conference.
Moravian Student Mathematics Conference
Members attend the annual Moravian Student Mathematics Conference, where undergraduate students present independent research. Marywood math majors often make presentations at this conference. The following is a short list of the most recent ones:
On Feb. 26, 2011, Michael Kuniega gave his presentation titled "The Dihedral Group D_12 and Musical Transformations on Triads".
On Feb. 20, 2010, Kayla Troast presented her work on the stereotype threat for female students on their mathematical performance
On Feb. 20, 2010, Bradley Fenstermaker presented his research on modeling the outbreak of zombies and studying the chance of survival of humans in such a scenario.
On Feb. 17, 2007, Colin Conrad presented his work concerning k-semismooth integers. Colin's research has deep connections to cryptology, the science of code encryption and decryption.
High School Math Contest
The Math Club hosts Marywood's annual High School Math Contest. The contest consists of two 40-problem tests created by Math Club members under faculty supervision. Prizes are awarded for the top three scorers at each level. | 677.169 | 1 |
Details about Applied Trigonometry:
'Applied Trigonometry' teaches the basic trigonometic concepts and skills needed by students in a variety of science and technical programs. The 'programmed approach' used in the text is based on a task analysis of each concept. The calculator is now used as a tool in the study of trigonometry.
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Rent Applied Trigonometry 1st edition today, or search our site for other textbooks by Thomas J. McHale. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson. | 677.169 | 1 |
math nsu mst
1.
I.<br />Mathematics ~ Best PracticesNorthwood High School<br />
2.
I.I<br />NCTM Standards<br /> "The Standards for high school students are ambitious. The demands made on high school teachers in achieving the Standards will require extended and sustained professional development and a large degree of administrative support." (NCTM, 2010)<br />
3.
Grades 9-12 Mathematics <br />Number and Number Relations<br />Understand numbers, ways of representing numbers, relationships among numbers, and number systems<br />Understand meanings of operations and how they relate to one another<br />Compute fluently and make reasonable estimates <br />Algebra<br />Understand patterns, relations, and functions <br />Represent and analyze mathematical situations and structures using algebraic symbols<br />Use mathematical models to represent and understand quantitative relationships <br />Analyze change in various contexts <br />Measurement<br />Understand measurable attributes of objects and the units, systems, and processes of measurement <br />Apply appropriate techniques, tools, and formulas to determine measurements <br />Geometry<br />Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships <br />Specify locations and describe spatial relationships using coordinate geometry and other representational systems <br />Apply transformations and use symmetry to analyze mathematical situations <br />Use visualization, spatial reasoning, and geometric modeling to solve problems <br />Data Analysis, Probability, and Discrete Math<br />Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them <br />Select and use appropriate statistical methods to analyze data <br />Develop and evaluate inferences and predictions that are based on data<br />Understand and apply basic concepts of probability <br />Problem Solving, Reasoning & Proof, Communication, Connections, and Representation<br />build new mathematical knowledge, apply and adapt strategies, and monitor and reflect on the process of problem solving <br />analyze and evaluate the mathematical thinking and strategies of others, communicate mathematical knowledge to others and use the language of mathematics to express mathematical ideas precisely. <br />understand how mathematical ideas interconnect and build on one another to produce a coherent whole <br />recognize and apply mathematics in contexts outside of mathematics<br />select, apply, and translate among mathematical representations to solve problems and create and use representations to organize, record, and communicate mathematical ideas <br />use representations to model and interpret physical, social, and mathematical phenomena. <br />I.II<br />NCTM Standards - Summarized<br />Through Content & Process, Students will:<br />Learn to value all mathematicsBecome confident in the ability to do mathBecome mathematical problem solversLearn to reason mathematically<br />
4.
I.III<br />The NCTM Standards Overview<br />Students should:<br />"Experience the interplay of algebra, geometry, statistics, probability, and discrete mathematics. They need to understand the fundamental mathematical concepts of function and relation, invariance, and transformation. They should be adept at visualizing, describing, and analyzing situations in mathematical terms. And they need to be able to justify and prove mathematically based ideas." (NCTM, 2010)<br />
5.
"Students of teachers who conduct hands-on learning activities outperform their peers by more than 70 percent of a grade level in math and 40 percent of a grade level in science."<br /> (National Council for Accreditation of Teacher Education, 2010)<br />Researched-Based Strategies<br />I.IV<br />
6.
Qualities of Best Practice Strategies<br />II.I<br />According to a Research Brief from The Principal's Partnership, "Exceptional high school math programs seem to share some or all of the following characteristics:<br />Connected to standards<br />Reflect high expectations<br />Connected to students' lives & cultures<br />Make learning interactive<br />Connect math to other disciplines<br />Teach math in context<br />Use mathematical modeling"<br /> (Principal's Partnership, 2010)<br />
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III.VII<br />Social<br />Peer Interactions<br />Discussions are flowing among group members<br />Collaboration<br />Ideas are being exchanged with each other<br />Example<br />Respectful behavior among peers<br />
19.
IV.I<br />Exemplary Lesson<br />Questions & Activities are designed to guide lesson direction<br />Structured math stations<br />The students are expected to actively in developing knowledge<br />Mobility to move about the room & take ownership of the learning<br />The goal is to merge conceptual understanding and reasoning<br />Cooperative groups & Individual learning<br />
20.
IV.II<br />Exemplary Lesson <br />Connections to other subject areas<br />Students relate to other lessons or interests<br />Using real-world, familiar materials<br />Board Games<br />Lunch Menus<br />Recipes<br />Are you surprised that this is in a 2nd Grade Classroom?<br /> Exemplary lessons can be adapted to all learners.<br />
21.
V.I<br />"Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals." The National Academies Press<br />
22.
V.II<br />Use Technology Tools to Involve All Stakeholders<br />Develop a Plan with Principals to Implement Best Practices<br />Use Websites to Inform (Pictures, Blogs, Wikis)<br />Create Partnerships with Community Businesses (Guest Speakers)<br />Involve Parents in Lessons (PBL, Digital Diaries)<br />Partner with Experts in the Field (NASA, Siemens)<br />Share Ideas and Information with other Educators<br />
24.
V.IV<br />"Teachers and other educational leaders should consistently help students and parents to understand that an increased emphasis on the importance of effort is related to improved mathematics performance."<br />(Foundations for Success: The Final Report of the National Mathematics Advisory Panel, 2010)<br /> | 677.169 | 1 |
Latest EECS News
EECS 212 - Mathematical Foundations of Computer Science
COURSE OBJECTIVES: In this course, students should develop mathematical thinking and problem-solving skills associated with writing proofs. Students should also be exposed to a wide variety of mathematical concepts that are used in the Computer Science discipline, which may include concepts drawn from the areas of Number Theory, Graph Theory, Combinatorics, and Probability.
PREREQUISITES: EECS 110 or EECS 111, and MATH 230
DETAILED COURSE TOPICS: All sections will deal with topics from Part I (Logic, Proofs, and Mathematical Preliminaries), as well as a selection of topics from Parts II-IV.
Part I: Logic, Proofs, and Mathematical Preliminaries 1. Transform statements between English and formal logic, and manipulate logical expressions according to logical principles such as modus ponens, modus tollens, disjunctive syllogism, transposition, De Morgan's Law, and existential and universal instantiation and generalization. 2. Apply sound logic to develop and prove theorems about mathematical phenomena. 3. Reason about the strengths and weaknesses of various proof techniques, including direct proof, proof by contradiction, proof by cases, and both strong and weak induction. 4. Understand the notation and properties of sets, including set builder notation, union, intersection, complement, difference, De Morgan's Law, power set, and Cartesian product. 5. Understand the definition, notation, and properties of functions, including domain, range, one-to-one, onto, bijections, arithmetic, and composition.
Part II: Number Theory 1. Understand the notation and properties of modular arithmetic, and be able to solve basic congruences. 2. Understand the definition and properties of prime numbers as well as the Sieve of Eratosthenes. 3. Compute the Greatest Common Divisor of two values using the Euclidean algorithm. 4. Solve systems of linear congruences through the use of the Chinese Remainder Theorem. 5. Apply Fermat's Little Theorem to solve problems involving modular arithmetic. 6. Encode and decode messages using the RSA cryptosystem.
Part III: Graph Theory 1. Reason about relations with certain properties, which may include relations that are reflexive, irreflexive, symmetric, anti-symmetric, or transitive. 2. Compute the closure of a given relation. 3. Solve problems that can be modeled as n-ary relations. 4. Reason about graphs with certain properties, which may include min or max degree, diameter, radius, partiteness, planarity, chromatic number, clique or independence number, or the number of vertices, edges, or connected components. 5. Prove whether or not two given graphs are isomorphic. 6. Solve problems that can be modeled by Euclidean or Hamiltonian walks or cycles. 7. Solve problems that can be modeled as a directed graph or DAG, which may include topological sorting, network flow, or strongly or weakly connected components. 8. Solve problems that can be modeled as rooted or unrooted trees. 9. Use Hall's Theorem to solve problems that can be modeled by graph matching. 10. Apply the Mating Ritual to solve problems that can be modeled by stable matchings. 11. Prove whether or not a given graph is planar. 12. Prove that all planar graphs can be colored using no more than five colors.
Part IV: Combinatorics and Probability 1. Apply basic counting principles, including the addition, multiplication, division, and Pigeonhole principles, in order to count or reason about objects of a given description. 2. Produce closed-form expressions for linear and geometric sums and linear recurrences. 3. Use generating functions to produce closed-form expressions for nonlinear recurrences. 4. Apply the Inclusion-Exclusion principle to count objects that can be expressed as a union of sets. 5. Compute the probability of a given event that can be counted. 6. Utilize properties of probabilities, which may include combinations or sequences of events, complementary events, conditional probabilities, or Bayes Rule, to compute the probability of a given event. 7. Establish whether or not two events are conditionally independent. 8. Compute the expected value of a given random variable. | 677.169 | 1 |
Course Description
MAED 550 Interactions of Science and Mathematics
2 cr.
Investigation of unifying concepts and central themes common to mathematics and physical science. Emphasis on developing understanding of fundamental concepts and principles, on problem solving in an interdisciplinary environment and on laboratory activities appropriate for junior and senior high school classes. | 677.169 | 1 |
Master Algebra Lite
This is for students in High School/College learning algebra. If you are a beginner in algebra you might be thinking X+Y=XY, Is not it? But it's not.
The beauty of algebra is, it deals with variables, expressions & equations. You will come to know various formulas.
For example if you know (a+b)^3= a^3+b^3+3a^2b+3ab^2.
You can calculate any number to the powers 2,3,4…in a fraction of seconds.
In the above equation a ,b are variables. So you can calculate (1.034)^3 also using that formula. Just feed a=1& b=.034
IMathPractice Algebra's 3 steps method of teaching has sections like Tutorial, Practice Skills, Practice Test & Algebra Challenge. Under tutorial it teaches you.
Numbers
Types of Number like real number, integer, negative number, complex number
Addition, Subtraction, Multiplication & Division of Real Number
Addition, Subtraction, Multiplication & Division of Negative Number
Addition, Subtraction, Multiplication & Division of Complex Number
Properties of | 677.169 | 1 |
A Biologist's Guide to Mathematical Modeling in Ecology and Evolution
Overview Troy Day provide biology students with the tools necessary to both interpret models and to build their own.
The book starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. Otto and Day then gradually build in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models. The authors provide primers with instructive exercises to introduce readers to the more advanced subjects of linear algebra and probability theory. Through examples, they describe how models have been used to understand such topics as the spread of HIV, chaos, the age structure of a country, speciation, and extinction.
Ecologists and evolutionary biologists today need enough mathematical training to be able to assess the power and limits of biological models and to develop theories and models themselves. This innovative book will be an indispensable guide to the world of mathematical models for the next generation of biologists.
Editorial Reviews
Quarterly Review of Biology
A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature. — Donald L. DeAngelis
Siam Review — Sanjay Basu and Alison P. Galvani
Current Engineering Practice
[.
Basic and Applied Ecology
I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models. — Volker Grimm
Quarterly Review of Biology - Donald L. DeAngelis
A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature.
Siam Review - Sanjay Basu and Alison P. Galvani
Basic and Applied Ecology - Volker Grimm
I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models.
From the Publisher
Honorable Mention for the 2007 Best Professional/Scholarly Book in Biological Sciences, Association of American Publishers
"A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who . . . finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature."—Donald L. DeAngelis, Quarterly Review of Biology
"."—Sanjay Basu and Alison P. Galvani, Siam Review
"[."—Current Engineering Practice
"I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models."—Volker Grimm, Basic and Applied Ecology
"I cannot help but think that future textbook authors will want to have Otto and Day front and center on the work desk, for this is a valuable source of material. . . . This book stands out, and its contribution is quite apparent. In sum, this book is a valuable contribution to the literature, and one to which I expect to refer regularly in connection with my teaching and writing duties."—Steven G. Krantz, UMAP Journal
Biologist's Guide to Mathematical Modeling in Ecology and Evolution
Princeton University Press
Chapter One
Mathematical Modeling in Biology
1.1 Introduction
Mathematics permeates biology. Unfortunately, this is far from obvious to most students of biology. While many biology courses cover results and insights from mathematical models, they rarely describe how these results were obtained. Typically, it is only when biologists start reading research articles that they come to appreciate just how common mathematical modeling is in biology. For many students, this realization comes long after they have chosen the majority of their courses, making it difficult to build the mathematical background needed to appreciate and feel comfortable with the mathematics that they encounter. This book is a guide to help any student develop this appreciation and comfort. To motivate learning more mathematics, we devote this first chapter to emphasizing just how common mathematical models are in biology and to highlighting some of the important ways in which mathematics has shaped our understanding of biology.
Let's begin with some numbers. According to BIOSIS, 886,101 articles published in biological journals contain the keyword "math" (including math, mathematical,mathematics, etc.) as of April 2006. Some of these articles are in specialized journals in mathematical biology, such as the Bulletin of Mathematical Biology, the Journal of Mathematical Biology, Mathematical Biosciences, and Theoretical Population Biology. Many others, however, are published in the most prestigious journals in science, including Nature and Science. Such a coarse survey, however, misses a large fraction of articles describing theoretical models without using "math" as a keyword.
We performed a more in-depth survey of all of the articles published in one year within some popular ecology and evolution journals (Table 1.1). Given that virtually every statistical analysis is based on an underlying mathematical model, nearly all articles relied on mathematics to some extent. With a stricter definition that excludes papers whose only use of mathematics is through statistical analyses, 35% of Evolution and Ecology articles and nearly 60% of American Naturalist articles reported predictions or results obtained using mathematical models. The extent of mathematical analysis varied greatly, but mathematical equations appeared in almost all of these articles. Furthermore, many of the articles used computer simulations to describe changes that occur over time in the populations under study. Such simulations can be incredibly helpful, allowing the reader to "see" what the equations predict and allowing authors to obtain results from even the most complicated models.
An important motivation for learning mathematical biology is that mathematical equations typically "say" more than the surrounding text. Given the space constraints of many journals, authors often leave out intermediate steps or fail to state every assumption that they have made. Being able to read and interpret mathematical equations is therefore extremely important, both to verify the conclusions of an author and to evaluate the limitations of unstated assumptions.
To describe all of the biological insights that have come from mathematical models would be an impossible task. Therefore, we focus the rest of this chapter on the insights obtained from mathematical models in one tiny, but critically important, area of biology: the ecology and epidemiology of the human immunodeficiency virus (HIV). As we shall see, mathematical models have allowed biologists to understand otherwise hidden aspects of HIV, they have produced testable predictions about how HIV replicates and spreads, and they have generated forecasts that improve the efficacy of prevention and health care programs.
1.2 HIV
On June 5, 1981, the Morbidity and Mortality Weekly Report of the Centers for Disease Control reported the deaths of five males in Los Angeles, all of whom had died from pneumocystis, a form of pneumonia that rarely causes death in individuals with healthy immune systems. Since this first report, acquired immunodeficiency syndrome (AIDS), as the disease has come to be known, has reached epidemic proportions, having caused more than 20 million deaths worldwide (Joint United Nations Programme on HIV/AIDS 2004b). AIDS results from the deterioration of the immune system, which then fails to ward off various cancers (e.g., Karposi's sarcoma) and infectious agents (e.g., the protozoa that cause pneumocystis, the viruses that cause retinitis, and the bacteria that cause tuberculosis). The collapse of the immune system is caused by infection with the human immunodeficiency virus (Figure 1.1). HIV is transmitted from infected to susceptible individuals by the exchange of bodily fluids, primarily through sexual intercourse without condoms, sharing of unsterilized needles, or transfusion with infected blood supplies (although routine testing for HIV in donated blood has reduced the risk of infection through blood transfusion from 1 in 2500 to 1 in 250,000 [Revelle 1995]).
Once inside the body, HIV particles infect white blood cells by attaching to the CD4 protein embedded in the cell membranes of helper T cells, macrophages, and dendritic cells. The genome of the virus, which is made up of RNA, then enters these cells and is reverse transcribed into DNA, which is subsequently incorporated into the genome of the host. (The fact that normal transcription from DNA to RNA is reversed is why HIV is called a retrovirus.) The virus may then remain latent within the genome of the host cell or become activated, in which case it is transcribed to produce both the proteins necessary to replicate and daughter RNA particles (Figure 1.2). When actively replicating, HIV can produce hundreds of daughter viruses per day per host cell (Dimitrov et al. 1993), often killing the host cell in the process. These virus particles (or virions) then go on to infect other CD4-bearing cells, repeating the process. Eventually, without treatment, the population of CD4+ helper T cells declines dramatically from about 1000 cells per cubic millimeter of blood to about 200 cells, signaling the onset of AIDS (Figure 1.3).
Normally, CD4+ helper T cells function in the cellular immune response by binding to fragments of viruses and other foreign proteins presented on the surface of other immune cells. This binding activates the helper T cells to release chemicals (cytokines), which stimulate both killer T cells to attack the infected cells and B cells to manufacture antibodies against the foreign particles. What makes HIV particularly harmful to the immune system is that the virus preferentially attacks activated helper T cells; by destroying such cells, HIV can eliminate the very cells that recognize and fight other infections.
Early on in the epidemic, the median period between infection with HIV-1 (the strain most common in North America) and the onset of AIDS was about ten years (Bacchetti and Moss 1989). The median survival time following the onset of an AIDS-associated condition (e.g., Karposi's sarcoma or pneumocystis) was just under one year (Bacchetti et al. 1988). Survival statistics have improved dramatically with the development of effective antiretroviral therapies, such as protease inhibitors, which first became available in 1995, and with the advent of combination drug therapy, which uses multiple drugs to target different steps in the replication cycle of HIV. In San Francisco, the median survival after diagnosis with an AIDS-related opportunistic infection rose from 17 months between 1990 and 1994 to 59 months between 1995 and 1998 (San Francisco Department of Public Health 2000). Unfortunately, modern drug therapies are extremely expensive (typically over US$10,000 per patient per year) and cannot be afforded by the majority of individuals infected with HIV worldwide. Until effective therapy or vaccines become freely available, HIV will continue to take a devastating toll (Figure 1.4; Joint United Nations Programme on HIV/AIDS 2004a).
1.3 Models of HIV/AIDS
Mathematical modeling has been a very important tool in HIV/AIDS research. Every aspect of the natural history, treatment, and prevention of HIV has been the subject of mathematical models, from the thermodynamic characteristics of HIV (e.g., Hansson and Aqvist 1995; Kroeger Smith et al. 1995; Markgren et al. 2001) to its replication rate both within and among individuals (e.g., Funk et al. 2001; Jacquez et al. 1994; Koopman et al. 1997; Levin et al. 1996; Lloyd 2001; Phillips 1996). In the following sections, we describe four of these models in more detail. These models were chosen because of their implications for our understanding of HIV, but they also illustrate the sorts of techniques that are described in the rest of this book.
1.3.1 Dynamics of HIV after Initial Infection After an individual is infected by HIV, the number of virions within the bloodstream skyrockets and then plummets again (Figure 1.3). This period of primary HIV infection is known as the acute phase; it lasts approximately 100 days and often leads to the onset of flu-like symptoms (Perrin and Yerly 1997; Schacker et al. 1996). The rapid rise in virus particles reflects the infection of CD4+ cells and the replication of HIV within actively infected host cells. But what causes the decline in virus particles? The most obvious answer is that the immune system acts to recognize and suppress the viral infection (Koup et al. 1994). Phillips (1996), however, suggested an alternative explanation: the number of virions might decline because most of the susceptible CD4+ cells have already been infected and thus there are fewer host cells to infect. Phillips developed a model to assess whether this alternative explanation could mimic the observed rise and fall of virions in the blood stream over the right time frame. In his model, there are four variables (i.e., four quantities that change over time): R, L, E, and V. R represents the number of activated but uninfected CD4+ cells, L represents the number of latently infected cells, E represents the number of actively infected cells, and V represents the number of virions in the blood stream. The dynamics of each variable (i.e., how the variable changes over time) depend on the values of the remaining variables. For example, the number of viruses changes over time in a manner that depends on the number of cells infected with actively replicating HIV. In the next chapter, we describe the steps involved in building models such as this one (see Chapter 2, Box 2.4).
Phillips' model contains several parameters, which are quantities that are constant over time (see Chapter 2, Box 2.4). In particular, the death rate of actively infected cells ([delta]) and the death rate of viruses ([sigma]) are parameters in the model and are not allowed to change over time. ([delta] and [sigma] are the lower-case Greek letters "delta" and "sigma." Greek letters are often used in models, especially for terms that remain constant ("parameters"). See Table 2.1 for a complete list of Greek letters.) Thus, Phillips built into his model the crucial assumption that the body does not get better at eliminating infected cells or virus particles over time, under the null hypothesis that the immune system does not mount a defense against HIV during the acute phase. To model the progression of HIV within the body, Phillips then needed values for each of the parameters in the model. Unfortunately, few data existed at the time for many of them. To proceed, Phillips chose plausible values for each parameter and numerically ran the model (a technique that we will describe in Chapter 4). The numerical solution for the number of virus particles, V, predicted from Phillips' model is plotted in Figure 1.5 (compare to Figure 1.3). Phillips then showed that similar patterns are observed under a variety of different parameter values. In particular, he observed that the number of virus particles typically rose and then fell by several orders of magnitude over a period of a few days to weeks. (An order of magnitude refers to a factor of ten. The number 100 is two orders of magnitude larger than one.)
Phillips thus came to the counterintuitive conclusion that "the reduction in virus concentration during acute infection may not reflect the ability of the HIV-specific immune response to control the virus replication" (p. 497, Phillips 1996). The wording of this conclusion is critical and insightful. Phillips did not use his model to prove that the immune system plays no role in viral dynamics during primary infection. In fact, his model cannot say one way or the other whether there is a relevant HIV-specific immune response during this time period. What Phillips can say is that an immune response is not necessary to explain the observed data. This result illustrates an important principle in modeling: the principle of parsimony. The principle of parsimony states that one should prefer models containing as few variables and parameters as possible to describe the essential attributes of a system. Paraphrasing Albert Einstein, a model should be as simple as possible, but no simpler. In Phillips' case, he could have added more variables describing an immune response during acute infection, but his results showed that adding such complexity was unnecessary. A simpler hypothesis can explain the rise and fall of HIV in the bloodstream: as infection proceeds, a decline in susceptible host cells reduces the rate at which virus is produced. Without having a good reason to invoke a more complex model, the principle of parsimony encourages us to stick with simple hypotheses.
Phillips' model accomplished a number of important things. First, it changed our view of what was possible. Without such a model, it would seem unlikely that a dramatic viral peak and decline could be caused by the dynamics of a CD4+ cell population without an immune response. Second, it produced testable predictions. One prediction noted by Phillips is that the viral peak and decline should be observed even in individuals that do not mount an immune response (i.e., do not produce anti-HIV antibodies) over this time period. Indeed, this prediction has been confirmed in several patients (Koup et al. 1994; Phillips 1996). Employing a more quantitative test, Stafford et al. (2000) fitted a version of Phillips' model to data on the viral load in ten patients from several time points during primary HIV infection; they found a good fit to the data within the first 100 days following infection. Third, Phillips' model generated a useful null hypothesis: viral dynamics do not reflect an immune response. This null hypothesis might be wrong, but at least it can be tested.
Phillips acknowledged that this null hypothesis can be rejected as a description of the longer-term dynamics of HIV. His model predicts that the viral load should reach an equilibrium (as described in Chapter 8), but observations indicate that the viral load slowly increases over the long term as the immune system weakens (the chronic phase in Figure 1.3). Furthermore, Schmitz et al. (1999) directly tested Phillips' hypothesis by examining the role of the immune system in rhesus monkeys infected with the simian immunodeficiency virus (SIV), the equivalent of HIV in monkeys. By injecting a particular antibody, Schmitz et al. were able to eliminate most CD8+ lymphocytes, which are the killer T cells thought to prevent the replication of HIV and SIV. Compared to control monkeys, the experimentally treated monkeys showed a much more shallow decline in virus load following the peak. This proves that, at least in monkeys, an immune response does play some role in the viral dynamics observed during primary infection. Nevertheless, the peak viral load was observed at similar levels in antibody-treated and untreated monkeys. Thus, an immune response was not responsible for stalling viral growth during the acute phase, which is best explained, instead, by a decline in the number of uninfected CD4+ cells (the targets of HIV and SIV).
What People are saying about this
Adler
This book has the ambitious and worthy goal of teaching biologists enough about modeling and about mathematical methods to be both intelligent consumers of models and competent creators of their own models. Its concentration on the process of building rather than analyzing models is its strongest point. — Frederick R. Adler, author of "Modeling the Dynamics of Life: Calculus and Probability for Life Scientists"
Phillips
This book is an amazing teaching resource for developing a comprehensive understanding of the methods and importance of biological modeling. But more than that, this book should be read by every student of evolutionary biology and ecology so that they can come to a deeper appreciation of the fundamental ideas and models that underlie these fields. — Patrick C. Phillips, University of Oregon
Holt
There is an increasing use of mathematics throughout the biological sciences, yet the training of most biologists still woefully lacks crucial mathematical tools. Sally Otto and Troy Day are themselves two masters at the deft use of theoretical models to crystallize conceptual insights about ecological and evolutionary problems, and in this wonderful book they make accessible to a broad audience the essential mathematical tool kit biologists need, both to read the literature and to craft and analyze models themselves. — Robert D. Holt, University of Florida
Levin
A wonderfully pedagogical introduction to mathematical modeling in population biology: an ideal first course for biologists. — Simon A. Levin, Princeton University
Frank
I am often asked by biologists to recommend a book on mathematical modeling, but I must tell them that there is no single good book that will guide them through the difficult first stages of learning to make models. Otto and Day's book fills the gap. The quality is high throughout, the scholarship is sound, the book is comprehensive. The authors are both first-rate scientists. I think this will be a classic. — Steven A. Frank, author of "Immunology and Evolution of Infectious Disease"
Carl Bergstrom
This book provides a general introduction to mathematical modeling—in particular, to population modeling—in the biological sciences. This past year I taught a 400-level course in mathematical modeling of biological systems, and I had to do so without a textbook because no adequate text existed. Otto and Day's book would have met my needs beautifully. This book is an important addition to the field. — Carl Bergstrom, University of Washington | 677.169 | 1 |
Find a Tamina, TX MathThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science | 677.169 | 1 |
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