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Creative Mathematics by H.S. Wall
Book Description
Professor H. S. Wall (1902-1971) developed Creative Mathematics over a period of many years of working with students at the University of Texas, Austin. His aim was to lead students to develop their mathematical abilities, to help them learn the art of mathematics, and to teach them to create mathematical ideas. This book, according to Wall, 'is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over paintings. It is, instead, a sketchbook in which readers try their hands at mathematical discovery.' In less than two hundred pages, he takes the reader on a stimulating tour starting with numbers, and then moving on to simple graphs, the integral, simple surfaces, successive approximations, linear spaces of simple graphs, and concluding with mechanical systems. The book is self contained, and assumes little formal mathematical background on the part of the reader | 677.169 | 1 |
Information design is all around us - from our food labeling, weather charts and all those PowerPoint presentations which we endure. We are perfectly able to decode this information. But how do we...more
statistics, numbers, algebra, geometry, trigonometry and functional maths. The syllabus is modular with 4 exam modules, which may be taken in March or June depending on student gain a level 2 qualification in maths. This is one of the qualifications you will need for entry to level 3 courses and for a course at university. As200
This course provides you with an opportunity to revise your mathematical skills. Topics have been selected as a preparation for the Certificate of Higher Education programmes in science and mathematicsMathematics is one of humanity's great achievements. Explore fundamental ideas about logic, geometry and numbers in early Greek thought and discover modern debates about how maths underpins science. ThisRecent reviews of this course provider
This course will help develop a positive attitude to maths and the ability to solve problems: to read, write, use, apply and talk about the subject with confidence and enjoyment. We run both Foundation...more
Each year we welcome thousands of adult learners on to an extremely wide variety of full time, part time and short courses, from tasters to degrees.
Recent reviews of this course provider
A Level Mathematics is a much sought after qualification for entry to a wide variety of full-time courses in Higher Education. There are also many areas of employment that see Mathematics A Level as maths and achieved a grade D? Have you recently achieved a Level 2...more
At The Manchester College opportunities are endless, think of a subject and we probably offer it.
Recent reviews of this course provider
If you want to go on to study at university then A level Mathematics will open more doors than any other subject because of its many applications. It trains in logical thought and problem solvingIntroduction to Royal Statistical Society
This one day training event provides the foundation to all presentations of statistical information. The basic principles of presenting information in tables, charts, maps and text are explained. These | 677.169 | 1 |
Source Book in Mathematics by David Eugene Smith
Book Description
The writings of Newton, Leibniz, Pascal, Riemann, Bernoulli, and others in a comprehensive selection of 125 treatises dating from the Renaissance to the late 19th century - most unavailable elsewhere. Grouped in five sections: Number; Algebra; Geometry; Probability; and Calculus, Functions, and Quaternions. Includes a biographical-historical introduction for each article.
Buy Source Book in Mathematics book by David Eugene Smith from Australia's Online Bookstore, Boomerang Books.
Other Editions...
Books By Author David Eugene Smith
Classic survey chronicles the development of the Japanese mathematics: use of the abacus; application of counting rods to algebra; Seki Kowa; the circle principle; Ajima Chokuyen; Wada Nei; more. 1914 edition. Includes 74 | 677.169 | 1 |
References
Index
Foreword
The lecture notes have been developed during over thirty years of teaching experience by the authors in the area of Numerical Analysis. They have taught the course on Numerical Methods designed for Science and Engineering students, at the University of Warsaw, University of Jos, Nigeria and the University of Botswana. The content of the notes covers the following topics: Computer Numbers and Roundoff Errors Estimates, Interpolation and Approximation of Functions, Polynomial Splines and Applications, Numerical Integration and Solu- tion of Non-linear Equations. The authors have presented the subjects in exact and comprehensive way with emphasis put on formulation of fundamental theorems with proofs supported by well selected examples. They used Mathematica, System for doing Mathematics, in solving problems specific to the subjects. In the notes, the reader will find interesting algorithms and their implementation in the Mathematica System. The lecture notes are well written and recommended as reading material for a basic course in Numerical Methods for science and engineering students.
O. A. Daman
University of Botswana,
Botswana
Preface
This text is intended for science and engineering students.
It covers most of the topics taught in a first course on numerical
analysis and requires some basic knowledge in calculus,
linear algebra and computing. The text has been
used as recommended handbook for courses taught on numerical
analysis at undergraduate level. Each chapter ends
with a number of questions. It is taken for granted if the
reader has access to computer facilities for solving some of
these questions using Mathematica. There is extensive literature
published on numerical analysis including books on
undergraduate and postgraduate levels. As the text for a
first course in numerical analysis, this handbook contains
classical content of the subject with emphases put on error
analysis, optimal algorithms and their implementation in
computer arithmetic. There is also a desire that the reader
will find interesting theorems with proofs and verity of examples
with programs in Mathematica which help reading
the text. The first chapter is designed for floating point
computer arithmetic and round-off error analysis of simple
algorithms. It also includes the notion of well conditioned
problems and concept of complexity and stability of algorithms.
Within chapter 2, interpolation of functions is discussed.
The problem of interpolation first is stated in its simplest
form for polynomials, and then is extended to generalized
polynomials. Different Chebyshev's systems for generalized
interpolating polynomials are considered.
In chapter 3, polynomial splines are considered for uniform
approximation of an one variable function.
Chapter 5 is an introduction to the least squares method and
contains approximation of functions in the norm of L2(a, b)
space. Also, it contains approximation of discrete data and
an analysis of experimental data.
In the chapter 6, two techniques of numerical integration are
developed, the Newton-Cotes methods and Gauss methods.
In both methods an error analysis is carried out.
For solution of non-linear algebraic equations, the most popular
methods, such as Fixed Point Iterations, NewtonMethod,
SecantMethod and BisectionMethod, are described in chapter
7.
Krystyna STYŠ
University of Warsaw, Poland Tadeusz STYŠ
University of Warsaw, Poland | 677.169 | 1 |
Principles of Mathematics Book 2 Set
Katherine Loop has done the remarkable! She has written a solid math course with a truly Biblical worldview. This course goes way beyond the same old Christian math course that teaches math with a few Scriptures sprinkled in and maybe some church-based word problems. This course truly transforms the way we see math.
Katherine makes the argument that math is not a neutral subject as most have come to believe. She carefully lays the foundation of how math points to our Creator, the God of the Bible. The nature of God, His Creation, and even the Gospel itself is seen through the study of math. Katherine does a marvelous job of revealing His Glory in this one-of-a-kind math course.
Katherine Loop's Principles of Mathematics Biblical Worldview Curriculum is a first of its kind. It takes math to a whole new level students and parents are going to love. It is a guaranteed faith grower!
Now that you know the core principles of arithmetic and geometry, you're ready to move on to learning advanced skills that will allow you to explore more aspects of God's creation. In Book 2, the focus is on the essential principles of algebra, coordinate graphing, probability, statistics, functions, and other important areas of mathematics. Here at last is a math curriculum with a biblical worldview focus that will help you:
Apply what you're learning outside a textbook, and, above all, see God's handiwork in math and His creation
Firm up the foundational concepts and prepare students for upper-level math in a logical, step-by-step way.
Some mathematical terms seem so complex, but don't worry; they're just fancy names to describe useful tools. In Book 2 you'll continue discovering that all of math boils down to a way of describing God's world and is a useful means we can use to serve and worship Him.
A description of this math course would not be complete without mentioning the author's conversational style when teaching math concepts. She presents math in a way that lends understanding to the "why's" in addition to the "how's" in a way that opens up the concept and completes the process of true understanding. This course leaves many of us wondering, "Where was this course when I was learning math?"
The Student Workbook is set up in the same style our customers have come to love and depend on. We have included Daily Schedules to complete the course in either one full year or in one semester. The student worksheets, quizzes & tests, and answer keys are all included in the 3-hole punched, perforate format that keeps everything organized and flexible. | 677.169 | 1 |
in the stock market, in sports, and all over the news. Algebra is all about figuring out the numbers you don't see. You might know how fast you can throw a...
.... Develop a rich understanding of the rudiments of algebra in a relaxed and supportive learning environment. This course will help you understand some of the most important algebraic concepts, including orders of operation, units of measurement, scientific notation, algebraic equations, inequalities... Learn about: Basic IT, Skills and Training, Basic IT training... | 677.169 | 1 |
Student Solutions Manual
Written from the perspective of the applied mathematician, the latest edition of this bestselling book focuses on the theory and practical applications of Differential Equations to engineering and the sciences. Emphasis is placed on the methods of solution, analysis, and approximation. Use of technology, illustrations, and problem sets help readers develop an intuitive understanding of the material. Historical footnotes trace the development of the discipline and identify outstanding individual contributions. This book builds the foundation for anyone who needs to learn differential equations and then progress to more advanced studies.
* A sound and accurate exposition of the elementary theory of differential equations * Self-contained chapters allow instructors to customize the selection, order, and depth of chapters. * More than 450 problems are marked with a technology icon indicating those that are considered to be technology intensive.
Unlike other texts in the market, this second edition presents differential equations consistent with the way scientists and engineers use modern methods in their work. Technology is used freely, with more emphasis on modeling, graphical representation, qualitative concepts, and geometric intuition than on theoretical | 677.169 | 1 |
EnhancingCoreWeb
Enhancing the Mathematical Core Grand Valley State University: Enhancing the Mathematical Core Grand Valley State University Working Groups David Austin Ed Aboufadel Alverna Champion Charlene Beckmann Pam Wells Phyllis Curtiss Will Dickinson John Gabrosek Matt Boelkins Esther Billings David Coffey John Golden Karen Novotny Karen Novotny Clark Wells Steven Schlicker Akalu Tefera
Core Math & Stats Courses: Core Math & Stats Courses
Faculty Teams: Faculty Teams Each course is being enhanced by a team of faculty made up of 1 or 2 mathematicians or statisticians and a mathematics educator
Goals: Goals Maintain a high level of mathematical content for all clients. Motivate, introduce, extend, or apply mathematics content. Make evident to mathematics majors: That the material in undergraduate mathematics courses arises from the mathematics they learned in K-12 and applies to the mathematics prospective teachers will teach. That the level of mathematics required of children (and their teachers) through exemplary K-12 curricula is challenging.
Our Process for Each Course: Our Process for Each Course Become more familiar with how students learn the mathematics encompassed by the course. Review the course content and determine the areas most likely to have links to K-12. Review K-12 materials for motivating activities that lead to higher level mathematics. Adapt or write materials to make the course even more interactive while challenging students at a high level.
Enhanced Courses: Enhanced Courses 01-02: Course studies and first pilot Communicating in Mathematics (first proof course) Euclidean Geometry Probability and Statistics 02-03: Revisions and further piloting of course materials Beginning Fall 2003: Research of efficacy of courses.
Courses to Be Enhanced: Courses to Be Enhanced 02-03: Course studies and first pilot Linear Algebra Modern Algebra Discrete Mathematics 03-04: Revisions and further piloting of course materials Beginning Fall 2004: Research of efficacy of courses.
Active Involvement: Active Involvement Building on the Departments of Mathematics and Statistics philosophy of active learning: Activities engage students in a hands-on way. Students investigate and discover mathematics. Students share their discoveries in small groups then with the full class.
Assessment: Assessment Authentic and constant Through observations as students work in class, Through students' written and oral responses to labs and homework, Through oral presentations, Through exams.
Enhancing Communicating in Mathematics: Enhancing Communicating in Mathematics Team: Ed Aboufadel Pam Wells
Communicating in Mathematics: Communicating in Mathematics A study of proof techniques used in mathematics. Intensive practice in reading mathematics, expository writing in mathematics, and constructing and writing mathematical proofs.
Communicating in Mathematics-Content: Communicating in Mathematics-Content Elementary logic Set theory Congruence arithmetic Functions Equivalence relations and classes
Communicating in Mathematics: Communicating in Mathematics Activities launch discussion of different proof techniques so that students: See how proof techniques are used to reason informally, both by adults and children. Have concrete references for various proof techniques to help them understand and apply the techniques correctly.
Understanding Proof Techniques: Understanding Proof Techniques Several activities were used to help students apply various proof techniques or to assess students' understanding of the mathematical content of the course. Practice with direct proofs using divisibility Practice on functions and associated terminology Portfolio Proof dealing with twin primes Final Exam question related to twin primes portfolio proof.
Proof by Cases: Proof by Cases Sheryl was excited when she arrived home with three small boxes of marbles. She labeled each box with its contents. One box had 2 blue marbles, the second had 2 red marbles, and the third had 1 blue and 1 red marble. Sheryl's mischievous little sister removed all of the labels and replaced them so that each box was labeled incorrectly. Sheryl wanted to put the correct labels back on without opening all the boxes. She wondered if she could figure out the correct labeling by just seeing the color of 1 marble in one box. Is it possible for Sheryl to pull only one marble from only one box and know the correct labels for each of the boxes? If so, how? If not, why not? Source: Teaching Children Mathematics, December 2000.
Twin Primes Theorem: Twin Primes Theorem Theorem #3 in the Portfolio: Twin prime numbers are pairs of prime numbers whose difference is 2. Examples are 5 and 7, 17 and 19, or 41 and 43. Source: Interactive Mathematics Program, Year 3 Theorem: If x and y are twin primes other than 3 and 5, then xy + 1 is a perfect square and is divisible by 36.
Twin Primes Theorem (cont'd): Twin Primes Theorem (cont'd) On the Final Exam: a. In 1993, the world record for the largest known twin primes was x = 459 28529 – 1, and y = 459 28529 + 1. Use your knowledge of the proof of Theorem 3 of the Portfolio to explain what the remainder is when 459 28529 is divided by 6. b. Use modulus arithmetic to determine the remainder when 459 28529 is divided by 6. (Show your work.)
High-Low Differences: High-Low Differences Follow these steps for any three digit number: Arrange the digits from largest to smallest; Arrange the digits from smallest to largest; Subtract the smaller number from the larger. Example: 743 – 347 = 396, 396 is the high-low difference of 473. Source: Year 1, Interactive Mathematics Program
High-Low Differences (cont'd): High-Low Differences (cont'd) Before answering the following questions, create several examples of high-low sequences to develop some intuition about them. For notation, define f(x) to be the high-low difference of x. Decide on sets X and Y so that f: X Y makes sense. Which set is the domain? Which set is the co-domain? Is f a 1-to-1 function? Justify your answer. Is f an onto function? Justify your answer.
Writing Assignment: Writing Assignment Imagine you are teaching calculus to seniors in high school. You ask the students the following question: If f is an even function, then must f' be an odd function? A student gives the response at right:
Questions: Questions Suppose this is the response of a Math 210 student. Discuss the strengths and weaknesses of the response using appropriate technology from this course. Now consider the response to be that of a high school student. What questions might you ask to probe the student's thinking? What explanation would you say or write to the student?
Induction: Induction An elementary school student performs the calculations at right to find the sum of the whole numbers 1 to 100. 10(55)+(100+200+300+400+500+600+700+800+900)=5050
Questions: Questions What is the child doing and why does it work? Use terminology from Math 210 in your discussion. How could you use this child's technique to add the whole numbers from 1-250? From 1-217? Explain. Utilizing the student's technique, can you find a formula for the sum of the whole numbers from 1 to n, where n>1 and n is a multiple of 10? Do any interesting issues arise?
Consecutive Addends: Consecutive Addends The number 15 can be written as a sum of consecutive whole numbers in three different ways: 15 = 7+8 = 1+2+3+4+5 = 4+5+6 The number 9 can be written as a sum of consecutive whole numbers in two ways: 9 = 4+5 = 2+3+4 Adapted from Balanced Assessment Project, Middle Grades Assessment. New Jersey: Dale Seymour Publications, pp. 39-48.
Questions: Questions Look at other numbers and find out all you can about writing them as sums of consecutive whole numbers. Begin with numbers from 1 through 36. Decide what kind of numbers can be written as a sum of 2 consecutive whole numbers; 3 consecutive whole numbers; 4 consecutive whole numbers; and so on. What numbers cannot be written as sums of consecutive whole numbers? Write your answers to the above questions in the form of conjectures and prove each conjecture.
Enhancing Euclidean Geometry: Enhancing Euclidean Geometry Team: David Austin Charlene Beckmann Will Dickinson
How Students Learn Geometry: How Students Learn Geometry Van Hiele Model of Geometric Thought, 5 stages arising from experience (not maturation): Visualization Analysis Informal Deduction Formal Deduction Rigor
Outline of Euclidean Geometry: Outline of Euclidean Geometry Triangle Explorations (2 wks) Quadrilateral Explorations (2 wks) Spherical and Hyperbolic Geometries (1 wk) Axiomatic Systems, Neutral Geometry, Euclidean Geometry (6-7 wks) Transformational Geometry (2-3 wks)
Triangle Activities: Goals: Triangle Activities: Goals Recall how and why congruence works Work towards precise language See geometry in the real world Surprise and challenge Introduce fundamental concepts in the course
Triangle Activities: Triangle Activities Congruence Scavenger Hunt Relationships between angles and sides Similarity What is a straight line Spherical geometry
Triangle Congruence: Triangle Congruence Given Angles: Given Side lengths: How many triangles can be made? Explore.
Student Responses: Student Responses "There is only one triangle that can be built." "They are the same." "There are no differences." "One can be laid on top of the other." "They are congruent." "Congruent means they match up."
Scavenger Hunt: Scavenger Hunt Equilateral triangle Isosceles triangle Scalene triangle Right triangle Equiangular triangle
Relationships Between Sides and Angles: Relationships Between Sides and Angles
Similarity: Similarity
Angle Measures Add to 180: Angle Measures Add to 180
Spherical Geometry: Spherical Geometry Using the Lenart sphere, determine which path is the shortest between two points? Compare "lines" on the sphere and in the plane. Measure the interior angles in a triangle and determine the angle sum. What is the largest angle sum you can find? What is the smallest?
Next Time, We Will: Next Time, We Will Trim and refine activities Spread them throughout the term Have more explicit follow up Include more to challenge better students
Exploring Quadrilaterals: Exploring Quadrilaterals Starts with intuitive definitions of quadrilaterals and works toward discovering their properties and several equivalent definitions.
Quadrilateral Search: Quadrilateral Search Pre-Activity: Search for quadrilaterals around you Square, Rhombus, Kite, Parallelogram Trapezoid, Isosceles Trapezoid, Rectangle and General Quadrilateral Intuitive definitions of these Definition #1 for quadrilaterals Day 1: Constructions with Geometer's Sketchpad
Diagonals Activity: Diagonals Activity Day 2: Definition #2 for quadrilaterals Tree diagram of quadrilaterals
Diagonal Activity : Diagonal Activity Solution to Diagonal Part:
Diagonal Activity: Diagonal Activity Solutions to Tree Diagram:
Diagonal Activity: Observations and Student Progress: Diagonal Activity: Observations and Student Progress Overall the activity went well and students were engaged. Some students assumed too many conditions. Students began to see the role of definition in geometry (e.g. If you choose a trapezoid to have exactly one set of parallel sides then you get a different tree diagram). Students moved up in their Van Hiele level to informal deduction and began to progress into formal deduction Later students proved the new definitions equivalent and most began to perform at the formal deduction level.
Symmetries with Geometer's Sketchpad: Day 3: Discover a 3rd definition for the quadrilaterals using the properties grid sheet: Angle, side, diagonal and symmetry properties for each type of quadrilateral Definition #3 for each quadrilateral in terms of symmetries Symmetries with Geometer's Sketchpad
Quadrilaterals on the Sphere and the Hyperbolic Plane: Quadrilaterals on the Sphere and the Hyperbolic Plane Days 4 and 5: Introduction to Spherical and Hyperbolic Geometry Which of our equivalent definitions make sense in the context of each of these geometries?
Final Project on Quadrilaterals:: Final Project on Quadrilaterals: Pick any three of the definitions for the same quadrilateral and prove they are equivalent. Point-wise definitions of symmetry make this challenging
Next Time, We Will…: Next Time, We Will… Explore the connection between the definitions and the tree diagram more. Some students didn't see property grid sheet connection Prove definition equivalences earlier Definition #1 Definition #2 Use Geometer's Sketchpad more effectively Make sure everyone has a working quadrilateral construction after Day 1 Define Convex versus Non-Convex figures earlier
Axiomatic Systems and Proof: Axiomatic Systems and Proof Develop naturally as students analyze and make conjectures about triangles and quadrilaterals. Undefined terms, axioms and definitions are introduced and discussed as they arise in discussions and through explorations. Proof begins with informal deduction then to formal deduction and finally to rigor.
Transformational Geometry: Transformational Geometry Start with informal discussion via patty paper, GeoTools and Miras Matrix / Functional representation and composition
EnhancingProbability and Statistics: Enhancing Probability and Statistics Team: Alverna Champion Phyllis Curtiss John Gabrosek
Course Syllabus: Course Syllabus Sampling (1 wk) Descriptive Statistics (2 wks) Probability (6 wks) Sampling Distributions (1 wk) Statistical Inference (3 wks) Regression (1 wk)
Elementary Grade Activities: Elementary Grade Activities Investigations in Number, Data, and Space (Grade 5 - Probability) Everyday Mathematics (Grade 5 - Law of Large Numbers)
Middle School Activities: Middle School Activities MathThematics (Book 1 - Regression) Connected Mathematics Project (Grade 6 - Descriptive Statistics)
High School Activities: High School Activities Contemporary Mathematics in Context (Course 3 - Sampling and Confidence Intervals) Math Connections (Volume 3a - Expectation) Activity-Based Statistics (Sampling Distributions) Teaching Statistics (Vol. 15 - Hypothesis Testing)
Sampling Distributions: Key Concepts: Sampling Distributions: Key Concepts Statistics vary from sample to sample For a simple random sample (SRS), as the sample size increases the sampling distribution of the sample mean becomes closer to a normal distribution For SRS, as the sample size increases the variability in the sample mean decreases
Making Cents Out of Pennies: Making Cents Out of Pennies A population of 100 pennies is given to each group of students. Students draw simple random samples of various sizes from the population. Students investigate the effect of sample size on the distribution of the sample mean of the mint date. Source: Activity-Based Statistics
Given a frequency table of the mint dates, students construct a histogram and describe its center, spread, and shape.: Given a frequency table of the mint dates, students construct a histogram and describe its center, spread, and shape.
Each student draws a SRS of size 3 and finds the mean mint date.: Each student draws a SRS of size 3 and finds the mean mint date. Example: Dates: 1986, 1994, 1990
Students create a frequency table and histogram using the class sample means from SRS of size 3. : Students create a frequency table and histogram using the class sample means from SRS of size 3.
Each student draws a SRS of size 10 and finds the mean mint date.: Each student draws a SRS of size 10 and finds the mean mint date. Example: Dates: 1998, 1994, 1981, 1999, 1997, 1996, 1998, 1970, 1998, 2000
Students create a frequency table and histogram using the class sample means from SRS of size 10.: Students create a frequency table and histogram using the class sample means from SRS of size 10.
Students compare histograms and answer questions including:: Students compare histograms and answer questions including: Look at the three histograms. What can you say about the shape of the distribution of the sample means as the sample size increases? What can you say about the mean of the sample means as the sample size increases? What can you say about the standard deviation of the sample means as the sample size increases?
An "A" Student Response: An "A" Student Response As the sample size increases, the values bunch closer to the mean of the population. As the sample size increases, the mean gets closer to the mean of the population, and the standard deviation gets smaller. Gets closer to the normal curve.
A "B" Student Response: A "B" Student Response As the sample size increases the graph becomes more symmetrical. As the sample size increases the sample means seem to stay relatively close. As the sample size increases the standard deviation decreases.
A "C" Student Response: A "C" Student Response SD = smaller as size increases. Mean is about the same. Becomes more symmetric.
Enhancing Linear Algebra: Enhancing Linear Algebra Team: Matt Boelkins John Golden Clark Wells
Our Process : Our Process Identify mathematical goals of the class Identify relevant parts of the standards Find relevant research about student learning of linear algebra Find and adapt activities to meet our goals
Course Outlinethe Big Topics: Course Outline the Big Topics Systems of Linear Equations Matrix Algebra (Matrices as Quantities) Matrices as and in Functions Linear Dependence and Independence Invertible Matrix Theorem (and consequences) Vector Spaces Eigenvalues and Eigenvectors
Relations to NCTM Content Standards: Relations to NCTM Content Standards One of the perceived problems with preservice teachers and linear algebra is the idea that the material is irrelevant, or mostly so, to them. But the NCTM standards make it clear that the content is directly relevant.
Number and operation: develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases Number and operation The high school number and operation standards have explicit linear algebra content such as: develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices
Algebra: but it was also clear how strongly all four strands of the algebra standards for high school and middle school applied. Some specific examples: High School: understand vectors and matrices as systems that have some of the properties of the real-number system; develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices; develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. Algebra
Algebra, cont'd: Algebra, cont'd model and solve contextualized problems using various representations, such as graphs, tables, and equations. Even though middle school students do not use matrix algebra (typically) to address these, it's clear that a linear algebra course could deepen the preservice teacher's understanding. Middle School: represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
How Students Learn Linear Algebra: How Students Learn Linear Algebra The best information we could find has already been collected in a single text: Resources for Teaching Linear Algebra, published by the MAA and edited by Carlson, Johnson, lay, Porter, Watkins and Watkins. The most useful materials (on curriculum planning) are probably those from the Linear Algebra Curriculum Study Group. Ed Dubinsky's response to the LACSG has the most information about student learning of linear algebra.
LA Curriculum Study Group Recommendations: LA Curriculum Study Group Recommendations The LACSG made five recommendations (paraphrased here), as well as making a pretty detailed suggested syllabus. The course must respond to needs of client disciplines. Math dept.s should make the first linear algebra course matrix oriented. Students' needs and interests as learners should be considered. Technology should be utilized. A second course in linear algebra should be a high priority for every math curriculum.
Searching the curricula: Searching the curricula We found relevant problems and examples in all the middle school and high school curricula in which we looked, and eventually drew examples from: Math in Context (8/9): the Toy Factory (Linear combinations) Contemporary Math in Context (Core+) (9): Why I is like 1 (matrices as number-like objects) Core + (10)/David Lay's Text:Owls and Rats (eigenvalues, matrix valued functions) SIMMS (10): Family Snack (Matrix Multiplication) Interactive Mathematics Project (12): Merry Go Round (vectors)
What is missing?: What is missing? We were impressed at the amount of linear algebra content in the curricula. But much of the more abstract content in the course, while it informs the applications, had no direct corollary in the grade 6-12 curricula. There is also so much linear algebra content in the curricula that not all of it is covered in our classes, the most notable and most common being a lot of linear programming. We estimate that we would need at least two weeks in the college course to address it in any depth, and there is not room in the class.
One activity – a contextfor matrix multiplication: One activity – a context for matrix multiplication In general we felt the MS and HS curricula provided excellent concrete applications and contexts. One of the areas we thought needed more concrete examples was the idea of matrices as a quantity. The students spend a lot of time performing algebraic operations on matrices, but the course was missing an element where the matrices represent real life quantities. The following is from the activity on matrix multiplication.
The context: Family Snack: The context: Family Snack A company called Family Snack is in the business of selling nuts, beef jerky and jam. They sell these items individually and in various combinations. Their Snack Pack consists of two boxes of nuts and 6 beef jerkies. The Gift Pack consists of two snack packs and three jars of jam. The Family Pack consists of 3 gift packs and three jars of jam. In the snack industry business the nuts, jerky and jam are referred to as simple components, and the packs as composite products.
A Question: A Question One way of displaying this information is in a matrix, and there are several different ways that could be done. 1Explain how each matrix displays the information. 2 4 12 b) 1 0 0 2 4 12 6 12 36 0 1 0 6 12 36 0 3 12 0 0 1 0 3 12 0 0 0 1 2 6 0 0 0 0 1 3 0 0 0 0 0 1
A Question of Interpretation: A Question of Interpretation Once students are representing such quantities in matrices, questions can be asked both about interpretation and use of the more matrices in compatible ways. Such as: Family Snack has received an order for 20 boxes of nuts, 60 pieces of jerky, 48 jars of jam, 24 Snack Packs, 12 Gift Packs and 2 Family Packs. How many boxes of nuts, pieces of jerky and jars of jam are required to fill this order? Use one of the matrices above, and describe how you would use it to solve this problem.
Further Questions in Context: Further Questions in Context Family Snack employs three sales people, Keyes, Zhang and Troy. In the first two weeks of September, Keyes sold 16 Snack Packs, 28 Gift Packs, and 8 Family Packs. In the same week, Zhang sold 12 Snack Packs, 36 Gift Packs, and 4 Family Packs and Troy sold 18 Snack Packs, 12 Gift Packs, and 11 Family Packs. Calculate (and show your calculations) the total value of the packs they sold. Sales people at Family Snack are paid in part by commission: $2.25, $4.20 and $7.20 for the different size packs. Write this information in a matrix, and use matrix multiplication to figure out the total commission earned by each salesperson in the first two weeks of September.
In Conclusion: In Conclusion
Student Benefits: Student Benefits Students acquire deeper and more connected understanding of mathematics and statistics. Prospective teachers (and parents) experience the level of mathematics embedded in K-12 curricula. Students are motivated to develop profound understanding of the mathematics they will teach (as teachers or parents).
Faculty Benefits: Faculty Benefits Growth in understanding of national standards for school mathematics and content of K-12 mathematics curricula Deeper appreciation for the level of mathematics present in K-12 materials Stronger collegial and collaborative environment Large collection of activities to teach mathematics and statistics with understanding
As We Continue...: As We Continue... We will benefit from your assistance. Questions? Comments? Suggestions?
Thank You!: Thank You! Your participation today is greatly appreciated!
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Maintaining and enhancing core web sites used in the Sales Centre and online.. Read more
JAVA Developer (JAVA SCRIPT, JSP, STRUTS, JAVA, HIBERNATE | 677.169 | 1 |
generate and compare the relationship between two patterns, such as those formed by x + 3 and x + 6. The standard goes on to ask students to graph ordered pairs on a coordinate plane to support their investigation. Students in sixth grade are asked to evaluate expressions as well as write and solve equations derived from real-world contexts. These early expectations lay the foundation for meeting multiple standards outlined in the Common Core standard for high school algebra. However, for many students, progressing from modeling situations with equations such as 3x + 10 = 25 to equations such as 3x + 10 = y creates a seemingly insurmountable problem. The transition from one-variable equations with a single solution to linear equations with two variables and infinitely many solutions presents many challenges. One specific obstacle to making this transition lies in students' misunderstanding of the equals sign. For many students, the equals sign indicates an operation rather than a relationship (Ronda 2009). Once the concept of relational equality is sufficiently developed, students can begin the task of making sense of two-variable equations. Knowledge construction for understanding linear equations occurs in various stages. Ronda (2009) suggests four clearly defined stages of conceptual development, which range from the most elementary level--being able to evaluate variables for specific values--to the most complex level--being able to view the function holistically. This article describes a series of activities that comprises a single, multiphase lesson which incorporates Rhonda's stages and guides students students from single-variable equations to linear relationships. | 677.169 | 1 |
Finite Mathematics Documents
Showing 1 to 28 of 28
Saturday, March 6, 2010
Page 1
Mathematics 1228B
Test 2
CODE 111
PART A (18 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. Let E be any event dened on any sample
Probability Trees
Constructing a Probability Tree
A probability tree models a stochastic process, i.e. a sequence of probabilistic experiments. Each level in the tree corresponds to a dierent stage in the
stochastic process (i.e. a dierent experiment in t
Math 028b
Coins, Dice and Cards
In this course, we will be learning about probability. Certain games are very useful in demonstrating
the concepts involved in probability and are commonly used in any introductory course. It is usually
assumed that student
1
Using Venn Diagrams to demonstrate the Distributive properties
of the union, intersection and set complement operators
Combining the union and intersection operators
Property: Union is distributive over Intersection
That is, given any sets A, B and C , | 677.169 | 1 |
Research in Mathematics Education in Papua New Guinea: 1984. Proceedings of the Mathematics Education Conference, Mathematics Education Centre (4th, Lae, Papua New Guinea. May 1984).
Clarkson, Philip C., Ed.
Seventeen research reports are arranged under several broad topics of mathematics education. Section 1 concerns language and mathematics learning, with one paper on word problems and one on concept formation. Papers on university level mathematics in Section 2 explore problem solving, characteristics of school leavers, some mathematical aspects of surveying education, an experimental approach to teaching numerical analysis, and remediation. In the third section, reports consider mathematical deficits of university entry students and topics to consider in the curriculum. The fourth section concerns post year 10 mathematics, with reports on the quality of school leavers and on distance versus face-to-face learning. Mathematics teaching in various cultures is the focus of Section 5, with secondary mathematics teachers, mathematical abilities, basic skills, and secondary teacher education in Uganda considered. Finally, problem solving with a function graphing program and a report on microcomputer use in Fiji are presented in Section 6. (MNS) | 677.169 | 1 |
The wide-ranging debate brought about by the calculus reform movement has had a significant impact on calculus textbooks. In response to many of the questions and concerns surrounding this debate, the authors have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields. The text is written for the average student -- one who does not already know the subject, whose background is somewhat weak in spots, and who requires a significant motivation to study calculus. The authors follow a relatively standard order of presentation, while integrating technology and thought-provoking exercises throughout the text. Some minor changes have been made in the order of topics to reflect shifts in the importance of certain applications in engineering and science. This text also gives an early introduction to logarithms, exponentials and the trigonometric functions. Wherever practical, concepts are developed from graphical, numerical, and algebraic perspectives (the Rule of Three) to give students a full understanding of calculus. This text places a significant emphasis on problem solving and presents realistic applications, as well as open-ended | 677.169 | 1 |
GRADE 12 FUNCTIONS
Grade 12 Functions can be a very tough for some students. There are a lot of difficult concepts to learn.
That's were we come in! We can elucidate all the tough concepts and help you understand the material.
…and you'll see that Grade 12 Functions can be really interesting and fun!
And that will improve your grades!
What Our Students Will Learn
This course extends students' experience with functions. Students will investigate the
properties of polynomial, rational, logarithmic, and trigonometric functions; develop
techniques for combining functions; broaden their understanding of rates of change; and
develop facility in applying these concepts and skills. Students will also refine their use
of the mathematical processes necessary for success in senior mathematics. This course
is intended both for students taking the Calculus and Vectors course as a prerequisite
for a university program and for those wishing to consolidate their understanding of
mathematics before proceeding to any one of a variety of university programs.
Prerequisite: Functions, Grade 11, University Preparation, or Mathematics for College
Technology, Grade 12, College Preparation
BIG IDEAS
Exponential and Logarithmic Functions
demonstrate an understanding of the relationship between exponential expressions and logarithmic
expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
identify and describe some key features of the graphs of logarithmic functions, make connections
among the numeric, graphical, and algebraic representations of logarithmic functions, and solve
related problems graphically;
solve exponential and simple logarithmic equations
Trigonometric Functions
demonstrate an understanding of the meaning and application of radian measure;
make connections between trigonometric ratios and the graphical and algebraic representations of
the corresponding trigonometric functions and between trigonometric functions and their reciprocals,
and use these connections to solve problems;
demonstrate an understanding of solving polynomial and simple rational inequalities.
Characteristics of Functions
demonstrate an understanding of average and instantaneous rate of change, and determine,
numerically and graphically, and interpret the average rate of change of a function over a given
interval and the instantaneous rate of change of a function at a given point;
determine functions that result from the addition, subtraction, multiplication, and division of two
functions and from the composition of two functions, describe some properties of the resulting
functions, and solve related problems;
compare the characteristics of functions, and solve problems by modelling and reasoning with
functions, including problems with solutions that are not accessible by standard algebraic techniques | 677.169 | 1 |
Prerequisites: Math 8 or Math 8 AdvancedDescription:Explore the characteristics of basic functions using tables, graphs, and simple algebraic techniques; operate with radical, polynomial, and rational expressions, solve a variety of equations, including quadratic equations with a leading coefficient of one, radial equations, and rational equations; use the language of mathematical argument and justification; utilize counting techniques and determine probability; use summary statistics to compare samples to populations; and explore the variability of data; represent and operate with complex numbers; use numerical, graphical, and algebraic techniques to explore quadratic, exponential, and piecewise functions and to solve quadratic, exponential and absolute value equations and inequalities; use algebraic models to represent and explore real phenomena; explore inverses of functions.
Course Name:CCGPS Algebra Support
Course Number:27.0440000Term:Yearlong Grade:9
Prerequisites:Passed Math 8, level 1 on CRCT, take with CCGPS AlgebraDescription: GPS Algebra Support is an elective course that some students take concurrently GPS Algebra. Support gives students an extra hour of class time each day to get additional practice and instruction with the material being covered in CCGPS Algebra. Some days the Support class focuses on prerequisite skills (e.g., reviewing integer arithmetic to help with factoring); other days it mirrors the Math 1 class more exactly (e.g., giving a set of additional factoring problems like the ones students are doing in Math 1). There is no homework in CCGPS Algebra Support, but class work is collected virtually every day.
Other:Students who score a "1" on the 8th grade CRCT (Math) must be enrolled in CCGPS Algebra Support. For other students, placement is based on teacher recommendation. After CCGPS Algebra Support, students may continue with CCGPS Geometry Support as a 10th grade elective, or they may take CCGPS Geometry without CCGPS Geometry Support.Students who want to learn the material but need extra time to grasp it are the ideal fit for the Support class. Students whose struggles are due to lack of effort will not reap the benefits of the Support program.
Description:Represent and operate with complex numbers; explore the characteristics of basic functions utilizing tables, graphs, and simple algebraic techniques; operate with radical, polynomial, and rational expressions; solve equations, including quadratic, radical, and rational equations; investigate properties of geometric figures in the coordinate plane; use the language of mathematical argument and justification; discover, prove, and apply properties of polygons, circles and spheres; utilize counting techniques and determine probability; use summary statistics to compare samples to populations; explore variability of data; and fit curves to data and examine the issues related to curve fitting.
Other:Students may enroll in CCGPS Accelerated Algebra upon successful completion of ADVANCED Math 8. Students transferring from other states should be placed in Accelerated CCGPS Algebra Honors if they have demonstrated a strong understanding of the topics covered in Accelerated Math 8 (all of Algebra I, parts of Geometry, some Algebra II). After Accelerated CCGPS Algebra Honors, students will take CCGPS Accelerated Geometry Honors, then CCGPS Accelerated Precalculus, and then either AP Calculus (AB or BC) or AP Statistics.
Course Name:GPS Geometry
Course Number:27.0820000 Term:Yearlong Grade:10
Prerequisites:GPS Advanced Algebra Description:Use; investigate properties of geometric figures in the coordinate plane; prove, and apply properties of polygons.
Course Name:GPS Geometry Support
Course Number:27.0450000Term:Yearlong Grade:10
Prerequisites:Passed GPS Algebra with a 70 – 74, took GPS Algebra Support, take with GPS Geometry
Description:GPS Geometry Support is an elective course that some students take concurrently with GPS Geometry. Support gives students an extra hour of class time each day to get additional practice and instruction with the material being covered in Math 2. Some days the Support class focuses on prerequisite skills (e.g., reviewing topics from GPS Algebra); other days it mirrors the GPS Geometry class more exactly (e.g., giving a set of additional graphing problems like the ones students are doing in Math 2). There is no homework in GPS Geometry Support, but class work is collected virtually every day. Students enroll in GPS Geometry Support concurrently with GPS Geometry. Students generally personally elect to take GPS Geometry GPS Geometry Honors
Course Number:27.0820040
Term:Yearlong
Grade:9
Prerequisites:GPS Algebra Honors
Description:Because this course offers 7 honors points, it is only for students who are a year or more ahead in mathematics. Use. Investigate properties of geometric figures in the coordinate plane; prove, and apply properties of polygons.
Course Name:GPS Accelerated Geometry Honors
Course Number:27.0920040
Term:Yearlong
Grade:9 – 10
Prerequisites:Passed Accelerated GPS Advanced Algebra H
Description:Accelerated Math 2 is a Honors-level course. Self-motivated students are best suited for the rigor of this class. The course builds on itself and on material from previous courses, so students who are willing to seek out additional practice or help on difficult topics will be better able to stay on top of the material and not fall behind. Because this is an accelerated course, there is little time for review or re-teaching built into the schedule. Students are expected to retain information for significant periods of time to ensure that they can apply their knowledge in new situations encountered later in the semester. Students should be self-motivated to find extra practice or to seek extra help with topics they find difficult. The work-load is moderate with homework 3 to 4 nights per week. Good attendance is essential, as in all math courses. Students in the class are freshmen and sophomores.
Other: Students may enroll in GPS Accelerated Geometry Honors upon successful completion of Accelerated GPS Algebra Honors OR on-level GPS Algebra with the summer bridge course. Even for students who were successful in GPS Algebra and the bridge course sometimes have an "adjustment period" at the beginning of GPS Accelerated Geometry Honors to acclimate themselves to the rigor of the new class. Students transferring from other states should be placed in GPS Accelerated Geometry Honors if they have demonstrated a strong understanding of the topics covered in Accelerated GPS Algebra (most of Geometry, large portions of what used to be called Algebra II, some probability / statistics). After Accelerated GPS Geometry Honors, students will take GPS Accelerated Pre-Calculus), followed the next year by either AP Calculus (AB or BC) or AP Statistics. Topics include exploration of the characteristics of exponential, logarithmic, and higher degree polynomial functions using tables, graphs, and algebraic techniques; explore inverses of functions; use algebraic models to represent and explore real phenomena; solve a variety of equations and inequalities using numerical, graphical, and algebraic techniques with appropriate technology; use matrices to formulate and solve problems; use linear programming to solve problems; use matrices to represent and solve problems involving vertex-edge; use right triangle trigonometry to formulate and solve problems; investigate the relationships between lines and circles; recognize, analyze, and graph the equations of conic sections; investigate planes and spheres; use sample data to make informal inferences about population means and standard deviations; solve problems by interpreting a normal distribution as a probability distribution; and design and conduct experimental and observational studies.
Student Comments:"You have to be willing to put in a lot of time for this course. If you miss one topic, you're going to be confused for the rest of the unit;" "Whenever I fall behind I go see [my teacher] for extra help. I know I can't just ignore it if I don't understand it."
Description:Same as Accelerated GPS Geometry Honors plus a research element
Course Name:GPS Advanced Algebra
Course Number:27.0830000
Term:Yearlong
Grade:9 – 11
Prerequisites:GPS Geometry.
Description:GPS Advanced Algebra Support is an elective course that some students take concurrently with GPS Advanced Algebra. Support gives students an extra hour of class time each day to get additional practice and instruction with the material being covered in GPS Advanced Algebra. Some days the Support class focuses on prerequisite skills (e.g., reviewing topics from GPS Geometry); other days it mirrors the Math 3 class more exactly. There is no homework in GPS Advanced AlgebraSupport, but class work is collected virtually every day. Students enroll in GPS Advanced Algebra Support concurrently with GPS Algebra II. Students generally personally elect to take GPS Advanced AlgebraGPS Advanced Algebra/GHSGT Preparation
Course Number:27.0470000
Term:Yearlong Grade:12
Prerequisites:GPS Geometry or GPS Geometry Support
Description:This is the Course in a sequence of courses designed to provide students with a rigorous program of study in mathematics. It is also intended to help students master GPS Algebra, GPS Geometry, & GPS Advanced Algebrastandards that will appear on the Georgia High School Graduation Test (GHSGT).
Course Name:GPS Advanced Algebra Honors
Course Number:27.0830040
Term:Yearlong
Grade: 9 – 11
Prerequisites:GPS Geometry Honors; Additional topics from Accelerated GPS Pre-Calculus Honors; (individual projects).
Course Name:Accelerated GPS Pre-Calculus Honors
Course Number:27.0930040
Term:Yearlong
Grade:9 – 11
Prerequisites:Passed GPS Accelerated Geometry Honors
Description:Students will investigate and use rational functions; analyze and use trigonometric functions, their graphs, and their inverses; find areas of triangles using trigonometric relationships; use trigonometric identities to solve problems and verify equivalence statements; solve trigonometric equations analytically and with technology; use complex numbers in trigonometric form; understand and use vectors; use sequences and series; explore parametric representations of plane curves; explore polar equations; investigate the Central Limit theorem; and use margins of error and confidence intervals to make inferences from data.
Description:Same as Accelerated GPS Pre-Calculus Honors plus a research element
Course Name:GPS Pre-Calculus
Course Number:27.0840000
Term:Yearlong Grade:12
Prerequisites: GPS Advanced Algebra
Description:This is a fourth year mathematics course designed to prepare students for calculus and similar college mathematics courses. It requires students to investigate and use rational functions; analyze and use trigonometric functions, their graphs, and their inverses; use trigonometric identities to solve problems and verify equivalence statements; solve trigonometric equations analytically and with technology; find areas of triangles using trigonometric relationships; use sequences and series; understand and use vectors;investigate the Central Limit theorem; and use margins of error and confidence intervals to make inferences from data.
Course Name:GPS Mathematics in Finance
Course Number:27.0870000
Term:Yearlong Grade:12
Prerequisites:GPS Advanced Algebra or GPS Accelerated Geometry Honors
Description:This is a fourth year mathematics course designed to follow the completion of GPS Advanced Algebra. The course concentrates on the mathematics necessary to understand and make informed decisions related to personal finance.The mathematics in the course will be based on many topics in prior courses; however, the specific applications will extend the student's understanding of when and how to use these topics.This course does not meet the requirements for admission to four-year colleges in Georgia.
Course Name:GPS Advanced Mathematical Decision Making
Course Number:27.0850000
Term:Yearlong Grade:12
Prerequisites:GPS Advanced Algebra or GPS Accelerated Geometry Honors
Description:This is a fourth year mathematics coursedesigned to follow the completion of GPS Advanced Algebra or Accelerated GPS Geometry Honors.The course will give students further experiences with statistical information and summaries, methods of designing and conducting statistical studies, an opportunity to analyze various voting processes, modeling of data, basic financial decisions, and use network models for making informed decisions.This course meets the requirements for admission to four-year colleges in Georgia.
Course Name:Mathematical Decision Making in Industry and Government
(Available as a virtual course)
Course Number:27.0860000
Term:Yearlong
Grade:12
Prerequisites:GPS Advanced Algebra or GPS Accelerated Geometry Honors
Description:This is a fourth year mathematics coursedesigned to follow the completion of GPS Advanced Algebra or GPS Accelerated Geometry Honors. Modeled after operations research courses,MD in Industry and Government allows students to explore decision making in a variety of industries such as:Airline - scheduling planes and crews, pricing tickets, taking reservations, and planning the size of the fleet; Pharmaceutical - R& D management; Logistics companies - routing and planning; Lumber and wood products - managing forests and harvesting timber; Local government - deployment of emergency services, and Policy studies and regulation - environmental pollution, air traffic safety, AIDS, and criminal justice policy. Students learn to focus on the development of mathematical models that can be used to model, improve, predict, and optimize real-world systems. These mathematical models include both deterministic models such as mathematical programming, routing or network flows and probabilistic models such as queuing, and simulation. This course meets the requirements for admission to four-year colleges in Georgia.
Description:Students should expect to do homework daily, have good work/study habits, and must attend class daily. This course is fairly rigorous and should be taken by students who desire to learn about other branches of mathematics. Topics include introduction to statistics; descriptive statistics; probability; probability distributions; normal probability distributions; estimates and sample size; hypotheses testing; inferences from two samples; correlation and regression; multinomial experiments; analysis of variance; statistical process control; nonparametric statistics; design and sampling. Students are required to do a fair amount of reading and are expected to use the text book as a primary source of information. This There is a major emphasis on writing, so students do not have to be extremely strong math students. However, they do need to have a good work ethic and the motivation to seek help when they do not understand a topic.
Description:This course consists of a review of real numbers and the Cartesian plane; review of functions; limits and their properties; derivatives and differentiation applications; anti-derivatives and indefinite integration; area and definite integrals; integration by substitution; the trapezoidal rule; logarithmic, exponential and other transcendental functions; applications and methods of integration; miscellaneous topics in Calculus BC. All students enrolled in AP Calculus AB are required to take the AP Examination in May if they are passing this course prior to the AP Examinations.
Description:This Students need to have a good work ethic and the motivation to seek help when they do not understand a topic. Topics include applications of integration involving work and arc length; parametric equations; analysis of acceleration and velocity vectors; applications of slope fields to differential equations; analysis of geometric, harmonic, p-series and alternating series; and approximations of polynomials with Taylor and Maclaurin series.
Student Comment:"You have to enjoy math, be organized, and diligent so you will be willing to put forth the effort."
Course Name/Course Number:
College Calculus 227.0750405*
College Calculus 327.0750406**co-requisites; students must register for both courses
Description: This course is taught by a Georgia Tech professor through the Distance Learning Lab.Students from several Fulton County High Schools become part of this Georgia Tech classroom through the use of technology. The Calculus 2 course concludes the treatment of single variable calculus, and begins linear algebra—the linear basis of the multivariable theory. The Calculus 3 course involves multivariable calculus: Linear approximation and Taylor's theorems, Lagrange multiples and constrained optimization, multiple integration and vector analysis including the theorems of Green, Gauss and Stokes.
Course Name:SAT Prep
Course Number: 35.0660001
Term:SemesterGrade:11, 12
Prerequisites:Juniors and seniors, Completed or Enrolled in GPS Algebra II or higher
Description: Students receive intense practice and instruction in the areas of problem solving and advanced grammar. The course is team taught by a mathematics and a language arts teacher. Highly recommended for seniors for fall semester and juniors for spring semester | 677.169 | 1 |
Bach/Leitner's progressive text lays a solid foundation for elementary algebra that carefully addresses student needs. The authors' clear, non-intimidating, and humorous style reassures math-anxious readers. Unlike workbook-format Prealgebra texts that stress competence at procedures, this text emphasizes understanding and mastery through careful step-by-step explanations that strengthen students' long-term abilities to conceptualize and solve problems. The text's innovative sequencing builds students' confidence with arithmetic operations early on before extending the basic concepts to algebraic expressions and equations. The authors' unusually thorough introduction to variables eases students through the crucial transition from working with numbers. Throughout the text, interesting applied examples and exercises and math-appreciation features highlight key concepts at work in a wide variety of real-world contexts.
"synopsis" may belong to another edition of this title.
About the Author:
Daniel Bach earned his B.A and M.A degrees in Mathematics from the University of California at Berkeley, specializing in algebraic number theory. He is currently a popular math instructor at Diablo Valley College, having previously taught the subject at Mills College and UC-Berkeley. Dan was head tutor for the Minority Engineering Students Asociation and a Math Specialist at Black Pine Circle Day School, where he coauthored several California Junior High Math Competitions during the 1980s. Dan is the author and programmer of "Dan's Basic Math Clinic," an interactive multimedia courseware in arithmetic, prealgebra, and beginning algebra. In the 1990s, Dan was awarded an NSF developer's grant to write a series of notebooks Precalculus and Mathematica, for the Interactive Math Textbook Project. Since 1997, he has maintained an active website with free math lessons, a weekly contest, feature pages, and an international puzzle-problem community. He is an avid coffee drinker, runner, and bicyclist and enjoys creating 3D animations in his hard-to-find spare time.
Patricia Leitner received her B.A. in Applied Mathematics and her M.A. in Mathematics from the University of California at Berkeley. She has taught Mathematics at the University of California at Berkeley and at City College in San Francisco and currently teaches at Diablo Valley College in Pleasant Hill, California. Patricia has focused her career on making Mathematics understandable and accessible to every student. Many have attributed their success in Math to her teaching style. She was awarded the Nikki Kose Memorial Teaching Prize by the UC-Berkeley, and she has also served on the Mathematical Association of America's National Committee on Two-Year Colleges. Patricia helped design and implement a state-of-the-art computer lab at Diablo Valley College and has written a series of Computer Calculus Tutorials. She has also been a principal developer of the college's self-paced Algebra program and recently created a video series aimed at helping learning-disabled students grasp the concepts of algebra. Patricia has traveled extensively, particularly in France, where she has fun subjecting the locals to her suboptimal French. Most recently, she has begun a second career writing fiction. Patricia currently lives in the San Francisco Bay Area where she enjoys long walks, fitness swimming, and relaxing in cafes with a good book. | 677.169 | 1 |
The 4th Editions of Algebra 1 and Algebra 2 are intended primarily for students who plan on taking Saxon Geometry and are wanting Algebra 1 and Algebra 2 courses with reduced geometry content. The main difference between the 3rd and 4th editions is that much of the previously-integrated geometry content has been removed.
Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. 129 lessons cover topics such as geometric functions, angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring.
Exploring World History Quiz and Exam Book, and the. Quiz and Exam Book contains history quizzes for each unit, and six comprehensive exams in history, English, and Bible that each cover five chapters' worth of materialThe Notgrass. From Adam to Us curriculum is a one-year world history and literature curriculum for students in grades 5-8. TheAnswer Key includes unit-by-unit answers to the Thinking Biblically and Vocabulary assignments given in the textbook, as well as answers for the Lesson Review book, Student Workbook, and Timeline Book.
The Mystery of History Volume III Student Reader continues the spectacular and provocative study of world history from the viewpoint of a Christian author. Spanning the Renaissance, Reformation, Exploration, and some early American history, this volume explores the backdrop to and the significance surrounding the time-honored contributions found in art, music, literature, science, and philosophy of this rich era.
The pages are yellowed with age. There are some marks on a couple of pages. See the photos for examples. This was an obvious mistake that has been corrected. On the top of page 55 (see photo) and 157, the name 'Mudgee High School' has been stamped.
This is a beautiful yet practical coffee-table style book that teaches anyone who doesn't much like maths or struggles with it, to appreciate mathematics and get to grips with the fundamentals of numbers and numeracy. | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
124.33 MB | 30 pages
PRODUCT DESCRIPTION
Functions - Systems - Predicting Outcomes
This is an introduction to systems of equations. Students use their knowledge of functions to construct two equations, complete a function table, and create a graph. The students are required to reflect and analyze every step of the way. This lesson starts with what the kids know and helps them grow painlessly! The students think they are discovering new stuff on their own, and they internalize it. In future, more complex systems, all I have to do is mention the Tortoise and Hare and the kids just "get it"!
I have used this in centers, small groups, and whole class discussions. It is particularly good for students who may struggle with math and need real world (fantasy in this case) applications to get it. My more advanced students have lots of success with this in a center.
This file contains a three page recording sheet or guided notes for the students and three pages of answers and explanations. Along with the recording sheets, I'm including a PowerPoint file with 23 slides that will help lead the students through the lesson process. The slides include discussion questions, brain breaks (videos), tables and graphs for students to "show what they know", standardized test practice, and just plain fun that will engage any middle school student | 677.169 | 1 |
Word Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.17 MB | 4 pages
PRODUCT DESCRIPTION
This is a worksheet that includes topics covering:
Variables and Expressions
Order of Operations
Real numbers and the number line
Properties of Real Numbers
Adding, subtracting, multiplying and dividing Real Numbers
The Distributive Property
Intro to Equations
Patterns, Equations and graphs | 677.169 | 1 |
The mathematics offerings provide a traditional focus on algebra and geometry proficiency as students work toward elective offerings in the advanced levels. Advanced Placement offerings in this area include: Statistics, Calculus AB and Calculus BC and BC (compacted). | 677.169 | 1 |
Monday January 12 − Lecture 4 : Matrices (Refers to section 3.1) Expectations: 1. Define the operation of addition, multiplication, scalar multiplication and transposition on matrices. 2. Use the rules of matrix algebra. (Hand-out) 3. Define identity matrix, zero matrix 4. Express the dot-product of two vectors as a product of two matrices. We have previously spoken of "matrix" but only as related to a system of linear equations (coefficient matrix of a system and augmented coefficient matrix of a system) . In this lecture we define the notion of a matrix (simply a rectangular array of numbers) independently of a system of linear equations. We define operations on matrices and then, once again, find a new way of representing a system of linear equations, namely a matrix equation . 4.1 Definition − Matrices . If m and n are positive integers, a matrix of size m × n (or of dimension m × n ), is a rectangular array of real numbers, arranged in m rows and n columns and is represented as A = [ a ij ] m × n The symbol i is called the row index and j is called the column index . The a ij 's are called the entries or components of the matrix. Example : This is a 3 by 3 matrix A : ⎡ a 11 a 12 a 13 ⎤ ⏐ a 21 a 22 a 23 ⏐ ⎣ a 31 a 32 a 33 ⎦ 4.1.1 Two matrices are said to be equal matrices if their corresponding entries are equal. Clearly matrices of different dimensions cannot be equal. 4.1.2 Remark − Distinction between vectors and matrices :
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This book is about the theory and practice of integer factorization presented in a historic perspective. It describes about twenty algorithms for factoring and a dozen other number theory algorithms that support the factoring algorithms. This book is written for readers who want to learn more about the best methods of factoring integers, many reasons for factoring, and some history of this fascinating subject. | 677.169 | 1 |
Personal website of Roumen Damianoff
Introduction to Algorithms
"Introduction to Algorithms" provides an introduction to mathematical modeling of computational problems. It covers the common algorithms, algorithmic paradigms, and data structures used to solve these problems. | 677.169 | 1 |
This course explores the fundamental connections between mathematics and music. Students will have the opportunity to learn about acoustics and music theory through a mathematical lens. At the core of music are patterns, structures, and relationships that can be understood mathematically. The course will provide opportunities for students to play with and explore some of the concepts, including scales, the Fourier series, beats, rhythm, and harmonicsLearn about acoustics and sound fields by using the concept of impedance. We will start with the fundamental concept of one-dimensional cases, understand the essentials, and also cover extended topics. | 677.169 | 1 |
25% of
users liked this software
Platform:
Windows 7/Vista/XP
Publisher:
VaxaSoftware
Price:
$24
File size:
2.11 Mb
Date added:
June 21, 2011
Description from the Publisher
FUGP - Fungraph - Graphs of mathematical functions
- 5 types of graphs:
- Single
- Piecewise
- Parametric
- Polar
- Multiple
- Print and copy graph into clipboard
- 15 preset examples
- Easy to use - User's manual in PDF format
- At home or in the classroom, FunGraph is suitable both for learning and for teaching. | 677.169 | 1 |
This programed mathematics textbook is for student use in vocational education courses. It was developed as part of a programed series covering 21 mathematical competencies which were identified by university researchers through task analysis of several occupational clusters. The development of a sequential content structure was also based on these mathematics competencies. After completion of this program the student should be able to: (1) know that "difference" indicates the operation of subtraction, (2) order any set of fractions, (3) subtract a small fraction of the form a/b where 0 is less than (a,b) when these are less than 100 from a larger fraction with the same denominator, (4) subtract a small fraction of the form a/b, where 0 is less than (a,b) and these are less than 100, from a larger fraction of the same form with unlike denominators, (5) subtract two literal fractions with common denominators, (6) subtract two literal fractions with unlike denominators, and (7) subtract a small mixed number from a larger one of the form Xa/b where 0 is less than (X,a,b) when these are less than 100. The material is to be used by individual students under teacher supervision. Twenty-six other programed texts and an introductory volume are available as VT 006 882-VT 006 909, and VT 006 975. (EM) | 677.169 | 1 |
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Unformatted text preview: MATH 2043 SYLLABUS: Spring 2011 Instructors Version Prerequisites for Math 2043 are a grade of C or higher in Math 1042 (0086), Calculus II, or transfer credit for a course equivalent to Math 1042. TEXT: Jon Rogawski, Calculus: Early Transcendentals , First Edition, Freeman and Company Syllabus REVIEW OF DIFFERENTION AND INTEGRATION SKILLS (1 hour) Please remind the students to do a review on Calculus 1 and 2 materials, especially in differention rules such as product, quotient, and chain rules. And integration skills such as substitution rule, integration by parts and double angel formula for sine and cosine fuctions. You should give them a quiz about these review materials. CHAPTER 12: VECTOR GEOMETRY (7 hours) Cover Sections 12.1-12.5. Section 12.1 introduces vectors in the plane. Please use the list of homework problems as a guide to the concepts and notations you need to cover in this section. Section 12.2. In addition to vectors in 3 dimensions this section contains parametric equations of lines. Please also show them how to parametrize a line segment since they will need to do that for line integrals later. Section 12.3 covers the dot product. Please show them how to sketch the projection in addition to the fomula. Section 12.4 covers the cross product and its applications. You need not discuss basic properties of the cross product (Theorem 2) here. All you need to do is cover Theorem 1 and discuss the relation between the cross product, areas and volumes. Section 12.5 deals with equations of planes. You can assign Section 12.6 for independent reading (all they need to do is look at the pictures). Section 12.7 covers cylindrical and spherical coordinates. You can use it as a resource when you discuss integration in spherical and cylindrical coordinates in Chapter 15, but please do not cover it now....
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A graph G=(V,E) is composed of V: set of vertices E: set of edges An edge e=(V,V) is a pair of vertices V={A,B,C,D,E} E={(A,B),(A,D),(A,C),(B,E),(C,D),(C,E)} Graphs are useful to model a lot of real life problems-communication network-transport network :streets, intersections etc-electronic circuits-event dependencies
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GRAPH TERMINOLOGY a. Adjacent vertices : vertices connected by an edge B,C,D are adjacent to A b. Degree of a matrix : number of adjacent vertices Degree of A is 3 Degree (V)=2(number of edges) Degree (A)=3 Degree (B)=2 Degree (C)=2 Degree (D)=1------------8 = 2(4) Path is a sequence of vertices V 1 ,V 2 ….V k such that consecutive vertices Vi and Vi+1 are adjacent. A, B, C, D is a path A, B, D is not a path Simple path is a path with no repeated vertices. A cycle is similar to a simple path except that the first vertex is the same as the last vertex. Ex: A,B,C,A.
Connected graph is a graph where any two vertices are connected by the same path. Subgraph is a subset of vertices and edges of a graph Connected components : the maximal connected graphs of an original graph.
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Sample records for calculus
Practice makes perfect-and helps deepen your understanding of calculus 1001 Calculus Practice Problems For Dummies takes you beyond the instruction and guidance offered in Calculus For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in your calculus course. Plus, an online component provides you with a collection of calculus problems presented in multiple-choice format to further help you test your skills as you go. Gives you a chance to practice and reinforce the skills you learn in your calculus courseHelps you refine your understanding of calculusP
Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis.This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics
The Larson CALCULUS program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and professors for its solid and effective pedagogy that addresses the needs of a broad range of teaching and learning styles and environments. Each title is just one component in a comprehensive calculus course program that carefully integrates and coordinates print, media, and technology products for successful teaching and learning.
Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.
Calculus, Third Edition emphasizes the techniques and theorems of calculus, including many applied examples and exercises in both drill and applied-type problems.This book discusses shifting the graphs of functions, derivative as a rate of change, derivative of a power function, and theory of maxima and minima. The area between two curves, differential equations of exponential growth and decay, inverse hyperbolic functions, and integration of rational functions are also elaborated. This text likewise covers the fluid pressure, ellipse and translation of axes, graphing in polar coordinates, pro
We take great notes-and make learning a snap When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core Calculus concepts-from functions, limits, and derivatives to differentials, integration, and definite integrals- and get the best possible grade. At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase- and zero in on the essential information you need to succeed.
We reduce some key calculations of compositions of morphisms between Soergel bimodules ("Soergel calculus") to calculations in the nil Hecke ring ("Schubert calculus"). This formula has several applications in modular representation theory.
Another Calculus book? As long as students find calculus scary, the failure rate in mathematics is higher than in all other subjects, and as long as most people mistakenly believe that only geniuses can learn and understand mathematics, there will always be room for a new book of Calculus. We call it Calculus Light. This book is designed for a one semester course in ""light"" calculus -- mostly single variable, meant to be used by undergraduate students without a wide mathematical background and who do not major in mathematics but study subjects such as engineering, biology or management infor
We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplicatio...This book is unique in English as a refresher for engineers, technicians, and students who either wish to brush up their calculus or find parts of calculus unclear. It is not an ordinary textbook. It is, instead, an examination of the most important aspects of integral and differential calculus in terms of the 756 questions most likely to occur to the technical reader. It provides a very easily followed presentation and may also be used as either an introductory or supplementary textbook. The first part of this book covers simple differential calculus, with constants, variables, functions, inc
Full Text Available Programs with control are usually modeled using lambda calculus extended with control operators. Instead of modifying lambda calculus, we consider a different model of computation. We introduce continuation calculus, or CC, a deterministic model of computation that is evaluated using only head reduction, and argue that it is suitable for modeling programs with control. It is demonstrated how to define programs, specify them, and prove them correct. This is shown in detail by presenting in CC a list multiplication program that prematurely returns when it encounters a zero. The correctness proof includes termination of the program. In continuation calculus we can model both call-by-name and call-by-value. In addition, call-by-name functions can be applied to call-by-value results, and conversely.
The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels, ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.
""This book is a radical departure from all previous concepts of advanced calculus,"" declared the Bulletin of the American Mathematics Society, ""and the nature of this departure merits serious study of the book by everyone interestedEarn College Credit with REA's Test Prep for CLEP* Calculus Everything you need to pass the exam and get the college credit you deserve.Our test prep for CLEP* Calculus and the free online tools that come with it, will allow you to create a personalized CLEP* study plan that can be customized to fit you: your schedule, your learning style, and your current level of knowledge.Here's how it works:Diagnostic exam at the REA Study Center focuses your studyOur online diagnostic exam pinpoints your strengths and shows you exactly where you need to focus your study. Armed with this information, you
In this paper we discuss flipping pedagogy and how it can transform the teaching and learning of calculus by applying pedagogical practices that are steeped in our understanding of how students learn most effectively. In particular, we describe the results of an exploratory study we conducted to examine the benefits and challenges of flipping a…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is presumed.
The Rho-calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, lambda-calculus and non-deterministic computations. This paper describes the calculus from its syntax to its basic properties in the untyped case. We show how it embeds first-order conditional rewriting and lambda-calculus. Finally we use the Rho-calculus to give an operational semantics to the rewrite based language Elan.
Jennifer Ouellette never took maths in the sixth form, mostly because she – like most of us – assumed she wouldn't need it much in real life. But then the English graduate, now an award-winning science-writer, had a change of heart and decided to revisit the equations and formulas that had haunted her youth. The Calculus Diaries is the fun and fascinating account of a year spent confronting her numbers-phobia head on. With wit and verve, Ouellette explains how she discovered that maths could apply to everything from petrol mileages to dieting, rollercoaster rides to winning in Las Vegas.
Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements. The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant. The text ponders on approximate solutions, as wellHow THIS BOOK DIFFERS This book is about the calculus. What distinguishes it, however, from other books is that it uses the pocket calculator to illustrate the theory. A computation that requires hours of labor when done by hand with tables is quite inappropriate as an example or exercise in a beginning calculus course. But that same computation can become a delicate illustration of the theory when the student does it in seconds on his calculator. t Furthermore, the student's own personal involvement and easy accomplishment give hi~ reassurance and en couragement. The machine is like a microscope, and its magnification is a hundred millionfold. We shall be interested in limits, and no stage of numerical approximation proves anything about the limit. However, the derivative of fex) = 67.SgX, for instance, acquires real meaning when a student first appreciates its values as numbers, as limits of 10 100 1000 t A quick example is 1.1 , 1.01 , 1.001 , •••• Another example is t = 0.1, 0.01, in the functio...I formalize important theorems about classical propositional logic in the proof assistant Coq. The main theorems I prove are (1) the soundness and completeness of natural deduction calculus, (2) the equivalence between natural deduction calculus, Hilbert systems and sequent calculus and (3) cut elimination for sequent calculus.
Calculus of One Variable, Second Edition presents the essential topics in the study of the techniques and theorems of calculus.The book provides a comprehensive introduction to calculus. It contains examples, exercises, the history and development of calculus, and various applications. Some of the topics discussed in the text include the concept of limits, one-variable theory, the derivatives of all six trigonometric functions, exponential and logarithmic functions, and infinite series.This textbook is intended for use by college students.
On the Refinement Calculus gives one view of the development of the refinement calculus and its attempt to bring together - among other things - Z specifications and Dijkstra's programming language. It is an excellent source of reference material for all those seeking the background and mathematical underpinnings of the refinement calculus.
Fundamentals of Calculus encourages students to use power, quotient, and product rules for solutions as well as stresses the importance of modeling skills. In addition to core integral and differential calculus coverage, the book features finite calculus, which lends itself to modeling and spreadsheets. Specifically, finite calculus is applied to marginal economic analysis, finance, growth, and decay. Includes: Linear Equations and FunctionsThe DerivativeUsing the Derivative Exponential and Logarithmic Functions Techniques of DifferentiationIntegral CalculusIntegration TechniquesFunctions
Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simply-typed lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of beta-reduction that preserves safety. In the same vein as Schwichtenberg's 1976 characterization of the simply-typed lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the ...
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential calculus part of geometric calculus. The multivector differential is introduced, followed by the multivector derivative and the adjoint of multivector functions. The basic rules of multivector differentiation are derived explicitly, as well as a variety of basic m...
One of the twentieth century's most original mathematicians and thinkers, Karl Menger taught students of many backgrounds. In this, his radical revision of the traditional calculus text, he presents pure and applied calculus in a unified conceptual frame, offering a thorough understanding of theory as well as of the methodology underlying the use of calculus as a tool.The most outstanding feature of this text is the care with which it explains basic ideas, a feature that makes it equally suitable for beginners and experienced readers. The text begins with a ""mini-calculus"" which brings out t
We introduce the Stochastic Quality Calculus in order to model and reason about distributed processes that rely on each other in order to achieve their overall behaviour. The calculus supports broadcast communication in a truly concurrent setting. Generally distributed delays are associated with...
We introduce the Stochastic Quality Calculus in order to model and reason about distributed processes that rely on each other in order to achieve their overall behaviour. The calculus supports broadcast communication in a truly concurrent setting. Generally distributed delays are associated...
Rigorous but accessible text introduces undergraduate-level students to necessary background math, then clear coverage of differential calculus, differentiation as a tool, integral calculus, integration as a tool, and functions of several variables. Numerous problems and a supplementary section of ""Hints and Answers."" 1977 edition.
A main challenge of programming component-based software is to ensure that the components continue to behave in a reasonable manner even when communication becomes unreliable. We propose a process calculus, the Quality Calculus, for programming software components where it becomes natural to plan...This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of theThe $\\rho$-calculus is a new calculus that integrates in a uniform and simple setting first-order rewriting, $\\lambda$-calculus and non-deterministic computations. This paper describes the calculus from its syntax to its basic properties in the untyped case. We show how it embeds first-order conditional rewriting and $\\lambda$-calculus. Finally we use the $\\rho$-calcul- us to give an operational semantics to the rewrite based language ELAN.
Slay the calculus monster with this user-friendly guide Calculus For Dummies, 2nd Edition makes calculus manageable- ""how"" and ""why"" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll sComputing for Calculus focuses on BASIC as the computer language used for solving calculus problems.This book discusses the input statement for numeric variables, advanced intrinsic functions, numerical estimation of limits, and linear approximations and tangents. The elementary estimation of areas, numerical and string arrays, line drawing algorithms, and bisection and secant method are also elaborated. This text likewise covers the implicit functions and differentiation, upper and lower rectangular estimates, Simpson's rule and parabolic approximation, and interpolating polynomials. Other toWe present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior caAn alternative presentation of the π-calculus is given.This version of the π-calculus is symmetric in the sense that communications are symmetric and there is no difference between input and output prefixes.The point of the symmetric π-calculus is that it has no abstract names.The set of closed names is therefore homogeneous.The π-calculus can be fully embedded into the symmetric π-calculus.The symmetry changes the emphasis of the communication mechanism of the π-calculus and opens up possibility for further variations.We argue that the use of differentials in introductory calculus courses is useful and provides a unifying theme, leading to a coherent view of the calculus. Along the way, we meet several interpretations of differentials, some better than others.
The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: (A) Efforts to extend the Curry–Howard isomorphism, established between the simply-typed lambda calculus and intuitionistic logic, to classical logic. (B) Efforts to establish the tacit conjecture that call-by-value (CBV) reduction in lambda calculus is dual to call-by-name (CBN) reduction. This paper initially investigates relations of the Dual Calculus t...
A first order inference system, named R-calculus, is defined to develop the specifications.This system intends to eliminate the laws which are not consistent with users' requirements. TheR-calculus consists of the structural rules, an axiom, a cut rule, and the rules for logical connectives.Some examples are given to demonstrate the usage of the R-calculus. Furthermore, the propertiesregarding reachability and completeness of the R-calculus are formally defined and proved.
Cirquent calculus is a new proof-theoretic framework, originally motivited by the needs of computability logic (see ). Its main distinguishing feature is sharing: unlike the more traditional frameworks that manipulate tree- or forest-like objects such as formulas, sequents or hypersequents, cirquent calculus deals with circuit-style structures called cirquents. The present article elaborates a deep-inference cirquent calculus system CL8 for classical propositional logic and the corresponding fragment of the resource-conscious computability logic. It also shows the existence of polynomial-size analytic CL8-proofs of the pigeonhole principle -- the family of tautologies known to have no such proofs in traditional systems.Comprehensive but concise, this introduction to differential and integral calculus covers all the topics usually included in a first course. The straightforward development places less emphasis on mathematical rigor, and the informal manner of presentation sets students at ease. Many carefully worked-out examples illuminate the text, in addition to numerous diagrams, problems, and answers.Bearing the needs of beginners constantly in mind, the treatment covers all the basic concepts of calculus: functions, derivatives, differentiation of algebraic and transcendental functions, partial different
With its origins stretching back several centuries, discrete calculus is now an increasingly central methodology for many problems related to discrete systems and algorithms analysis. It is based on the authors lecture notes used and revised nearly every year over the last decade. The book contains numerous illustrations and cross references throughout, as well as exercises with solutions at the end of each section
Students can gain a thorough understanding of differential and integral calculus with this powerful study tool. They'll also find the related analytic geometry much easier. The clear review of algebra and geometry in this edition will make calculus easier for students who wish to strengthen their knowledge in these areas. Updated to meet the emphasis in current courses, this new edition of a popular guide--more than 104,000 copies were bought of the prior edition--includes problems and examples using graphing calculators.
The basic component of several-variable calculus, two-dimensional calculus is vital to mastery of the broader field. This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end o
The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambda-calculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly. We present a simply typed version of the rewriting calculus. With a good choice of the type system, we show that the calculus is type preserving and terminating, i.e. verifies the subject reduction and strong normalization properties.
This article is an introduction to Malliavin Calculus for practitioners. We treat one specific application to the calculation of greeks in Finance. We consider also the kernel density method to compute greeks and an extension of the Vega index called the local vega index.A method of using vector analysis is presented that is an application of calculus that helps to find the best angle for tacking a boat into the wind. While the discussion is theoretical, it is seen as a good illustration of mathematical investigation of a given situation. (MP)
Silvestre François Lacroix (Paris, 1765 - ibid., 1843) was a most influential mathematical book author. His most famous work is the three-volume Traité du calcul différentiel et du calcul intégral (1797-1800; 2nd ed. 1810-1819) – an encyclopedic appraisal of 18th-century calculus which remained the standard reference on the subject through much of the 19th century, in spite of Cauchy's reform of the subject in the 1820's. Lacroix and the Calculus is the first major study of Lacroix's large Traité. It uses the unique and massive bibliography given by Lacroix to explore late 18th-century calculus, and the way it is reflected in Lacroix's account. Several particular aspects are addressed in detail, including: the foundations of differential calculus, analytic and differential geometry, conceptions of the integral, and types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions). Lacroix's large Traité... was a...
The ESeal Calculus is a secure mobile calculus based on Seal Calculus. By using open-channels,ESeal Calculus makes it possible to communicate between any two arbitrary seals with some secure restrictions. It improves the expression ability and efficiency of Seal calculus without losing security.
Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students I covers functions, limits, basic derivatives, and integrals...... case studies and it has been extended in several directions. The aim of this paper is to provide a thorough presentation of the logic....
We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, incl...
The paper significantly extends and generalizes our previous paper. Here we discuss explicit general constructions for paragrassmann calculus with one and many variables. For one variable nondegenerate differentiation algebras are identified and shown to be equivalent to the algebra of (p+1)x(p+1) complex matrices. For many variables we give a general construction of the differentiation algebras. Some particular examples are related to the multiparametric quantum deformations of the harmonic oscillators. 18 refs
The concept of Boolean integration is developed, and different Boolean integral operators are introduced. Given the changes in a desired function in terms of the changes in its arguments, the ways of 'integrating' (i.e. realizing) such a function, if it exists, are presented. The necessary and sufficient conditions for integrating, in different senses, the expression specifying the changes are obtained. Boolean calculus has applications in the design of logic circuits and in fault Pre-Calculus reviews sets, numbers, operations and properties, coordinate geometry, fundamental algebraic topics, solving equations and inequalities, functions, trigonometry, exponents
This introduction to Malliavin's stochastic calculus of variations is suitable for graduate students and professional mathematicians. Author Denis R. Bell particularly emphasizes the problem that motivated the subject's development, with detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and descriptions of a variety of applications.The first chapter covers enough technical background to make the subsequent material accessible to readers without specialized knowledge of stochastic analysis. Succe
Calculus with Vectors grew out of a strong need for a beginning calculus textbook for undergraduates who intend to pursue careers in STEM. fields. The approach introduces vector-valued functions from the start, emphasizing the connections between one-variable and multi-variable calculus. The text includes early vectors and early transcendentals and includes a rigorous but informal approach to vectors. Examples and focused applications are well presented along with an abundance of motivating exercises. All three-dimensional graphs have rotatable versions included as extra source materials and may be freely downloaded and manipulated with Maple Player; a free Maple Player App is available for the iPad on iTunes. The approaches taken to topics such as the derivation of the derivatives of sine and cosine, the approach to limits, and the use of "tables" of integration have been modified from the standards seen in other textbooks in order to maximize the ease with which students may comprehend the material. Additio...The lambda mu-calculus is an extension of the lambda-calculus that has been introduced by M. Parigot to give an algorithmic content to classical proofs. We show that Bohm's theorem fails in this calculus.
The divergence theorem, Stokes' theorem, and Green's theorem appear near the end of calculus texts. These are important results, but many instructors struggle to reach them. We describe a pathway through a standard calculus text that allows instructors to emphasize these theorems. (Contains 2 figures.)
Of the most universal applications in integral calculus are those involved with finding volumes of solids of revolution. These profound problems are typically taught with traditional approaches of the disk and shell methods, after which most calculus curriculums will additionally cover arc length and surfaces of revolution. Even in these visibly…
The aim of the present study was to investigate the fluorescence properties of dental calculus in comparison with the properties of adjacent unaffected tooth structure using both lasers and LEDs in the UV-visible range for fluorescence excitation. The influence of calculus color on the informative signal is demonstrated. The optimal spectral bands of excitation and registration of the fluorescence are determined
A simple partial version of the Fundamental Theorem of Calculus can be presented on the first day of the first-year calculus course, and then relied upon repeatedly in assigned problems throughout the course. With that experience behind them, students can use the partial version to understand the full-fledged Fundamental Theorem, with further…
This article presents an example of how middle school teachers can lay a foundation for calculus. Although many middle school activities connect directly to calculus concepts, the authors have decided to look in depth at only one: the concept of change. They will show how teachers can lead their students to see and appreciate the calculusGet 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 Calculus I Super Review includes a review of functions, limits, basic derivatives, the definite integral, combinations, and permutations. Take the Super Review quizzes to see how much you've learned - and where you need more study. Makes an excellent study aid and textbook companion. Great for self-study!DETAILS- From cover to cover, each in-depth topic review is easy-to-follow and easy-to-grasp - Perfect when preparing forWe show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of $O(n)$. By work of Greenlees and Shipley, we see that these layers are classified by torsion $H^*(B SO(n))[O(n)/SO(n)]$-modulesDartmouth College mathematicians have developed a free online calculus course called "Open Calculus." Open Calculus is an exportable distance-learning/self-study environment for learning calculus including written text, nearly 4000 online homework problems and instructional videos. The paper recounts the evaluation of course elements since 2000 in…
An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;… - not just the number crunching - and understand how to perform all pre-calc tasks, from graphing to tackling proofs. You'll also get a new appreciation forWhen first published posthumously in 1963, this book presented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique a
Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the bookThis report presents a new distributed process calculus, called the -calculus. Key insights for the calculus are similar to those laid out by L. Cardelli for its calculus of ambients. Mobile Ambients and other recent distributed process calculi such as the Join calculus or the D-calculus introduce notions of distributed locations or localities, corresponding to a spatial partitioning of computations and embodying different features of distributed computations (e.g. failures, access control, p...
Presents three fundamental ideas of calculus and explains using the coordinate plane geometrically. Uses Cabri Geometry II to show how computer geometry systems can facilitate student understanding of general conic objects and its dynamic algebraic equations. (KHR)
Full Text Available ABSTRACTBackground:TCase hypothesis:Solid testicular lesions in young adults generally correspond to testicular cancer. Differential diagnosis should be done carefully.Future implications:In
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
Neutrosophic Analysis is a generalization of Set Analysis, which in its turn is a generalization of Interval Analysis. Neutrosophic Precalculus is referred to indeterminate staticity, while Neutrosophic Calculus is the mathematics of indeterminate change. The Neutrosophic Precalculus and Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the methods used to deal with such indeterminacy. In this book, the author presents a few examples o...
Mean Value Calculus (MVC)[1] is a real-time logicwhich can be used to specify and verify real-time systems[2]. As aconservative extension of Duration Calculus (DC)[3], MVC increasesthe expressive power but keeps the properties of DC. In this paper wepresent decidability results of MVC. An interesting result is that propositional MVC with chop star operator is still decidable, which develops the results of[4]and[5].
Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. However, in engineering problems, functions of interest are often real-valued and thus not analytic, such as the standard cost function. The HR calculus is a convenient way to calculate formal derivatives of both analytic and non-analytic functions of quaternion variables, however, both the HR and other functional calculus in quaternion analysis have encountered an essential tech...
In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.
The stack calculus is a functional language in which is in a Curry-Howard correspondence with classical logic. It enjoys confluence but, as well as Parigot's lambda-mu, does not admit the Bohm Theorem, typical of the lambda-calculus. We present a simple extension of stack calculus which is for the stack calculus what Saurin's Lambda-mu is for lambda-mu.
Aspect-oriented programming (AOP) has produced interesting language designs, but also ad hoc semantics that needs clarification. We contribute to this clarification with a calculus that models essential AOP, both simpler and more general than existing formalizations. In AOP, advice may intercept......-oriented code. Two well-known pointcut categories, call and execution, are commonly considered similar.We formally expose their differences, and resolve the associated soundness problem. Our calculus includes type ranges, an intuitive and concise alternative to explicit type variables that allows advice...... to be polymorphic over intercepted methods. We use calculus parameters to cover type safety for a wide design space of other features. Type soundness is verified in Coq....
Duration Calculus is a logic for reasoning about requirements for real-time systems at a high level of abstraction from operational detail, which qualifies it as an interesting starting point for embedded controller design. Such a design activity is generally thought to aim at a control device the...... physical behaviours of which satisfy the requirements formula, i.e. the refinement relation between requirements and implementations is taken to be trajectory inclusion. Due to the abstractness of the vocabulary of Duration Calculus, trajectory inclusion between control requirements and controller designs...... relation for embedded controller design and exploit this fact for developing an automatic procedure for controller synthesis from specifications formalized in Duration Calculus. As far as we know, this is the first positive result concerning feasibility of automatic synthesis from dense-time Duration...
This work is devoted to the optimization of fluorescence dental calculus diagnostics in optical spectrum. The optimal wavelengths for fluorescence excitation and registration are determined. Two spectral ranges 620 – 645 nm and 340 – 370 nm are the most convenient for supra- and subgingival calculus determination. The simple implementation of differential method free from the necessity of spectrometer using was investigated. Calculus detection reliability in the case of simple implementation is higher than in the case of spectra analysis at optimal wavelengths. The use of modulated excitation light and narrowband detection of informative signal allows us to decrease essentially its diagnostic intensity even in comparison with intensity of the low level laser dental therapy
The π-Calculus has been developed to reason about behavioural equivalence. Different notations of equivalence are defined in terms of process interactions, as well as the context of processes. There are various extensions of the π-Calculus, such as the SPI calculus, which has primitives to facili...We define a differential lambda-mu-calculus which is an extension of both Parigot's lambda-mu-calculus and Ehrhard- Regnier's differential lambda-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing.
This article describes investigative calculus projects in which students explore a question or problem of their own construction. Three exemplary pieces of student work are showcased. Investigative calculus projects are an excellent way to foster student understanding and interest in calculus. (Contains 4 figures.)We introduce an extension of the pure lambda-calculus by endowing the set of terms with a structure of vector space, or more generally of module, over a fixed set of scalars. Terms are moreover subject to identities similar to usual point-wise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta-reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda-calculus. W...
This first-year calculus book is centered around the use of infinitesimals, an approach largely neglected until recently for reasons of mathematical rigor. It exposes students to the intuition that originally led to the calculus, simplifying their grasp of the central concepts of derivatives and integrals. The author also teaches the traditional approach, giving students the benefits of both methods.Chapters 1 through 4 employ infinitesimals to quickly develop the basic concepts of derivatives, continuity, and integrals. Chapter 5 introduces the traditional limit concept, using approximation p
This report presents a calculus for higher-order distributed components, the Kell calculus. The calculus can be understood as a direct extension of the higher-order -calculus with programmable locations. The report illustrates the expressive power of the Kell calculus by encoding several process calculi with explicit locations, including Mobile Ambients, the Distributed Join calculus and the . The latter encoding demonstrates that the Kell calculus retains the expressive power of the but in a...
Based on cognitive science, the EnergyCalculus in Chinese language segmentation was presented to eliminate segmentation ambiguity. The notion of "EnergyCost" was advanced to denote the extent of the under-standability of a certain segmentation. EnergyCost function was defined with Z-notation. This approcah is effective to all natural language segmentation.
A flexible unified framework for both classical and quantum Schubert calculus is proposed. It is based on a natural combinatorial approach relying on the Hasse-Schmidt extension of a certain family of pairwise commuting endomorphisms of an infinite free Z-module M to its exterior algebra.Describes a rich, investigative approach to multivariable calculus. Introduces a project in which students construct physical models of surfaces that represent real-life applications of their choice. The models, along with student-selected datasets, serve as vehicles to study most of the concepts of the course from both continuous and discrete…
The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position,…
We develop a version of stochastic Pi-calculus with a semantics based on measure theory. We dene the behaviour of a process in a rate environment using measures over the measurable space of processes induced by structural congruence. We extend the stochastic bisimulation to include the concept of...
We introduce ctm, a process calculus which embodies a notion of trust for global computing systems. In ctm each principal (location) is equipped with a policy, which determines its legal behaviour, and with a protocol, which allows interactions between principals and the flow of information from ...
The purpose of this study is to present some of the classical concepts, definitions, and theorems of calculus from the constructivists' point of view in the spirit of the philosophies of L.E.J. Brouwer and Errett Bishop. This presentation will compare the classical statements to the constructivized statements. The method focuses on giving…It is now increasingly recognized that mathematics is not a neutral value-free subject. Rather, mathematics can challenge students' taken-for-granted realities and promote action. This article describes two issues, namely deforestation and income inequality. These were specifically chosen because they can be related to a range of calculus concepts…
Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students' understanding of volume. This study investigated calculus students' understanding of volume. Clinical interview transcripts and written responses to volume…
Since Parigot designed the λμ-calculus to algorithmically interpret classical natural deduction, several variants of λμ-calculus have been proposed. Some of these variants derived from an alteration of the original syntax due to de Groote, leading in particular to the Λμ-calculus of the second author, a calculus truly different from λμ-calculus since, in the untyped case, it provides a Böhm separation theorem that the original calculus does not satisfy. In addition to a survey of some aspects...
Students are entering college having earned credit for college Calculus 1 based on their scores on the College Board's Advanced Placement (AP) Calculus AB exam. Despite being granted credit for college Calculus 1, it is unclear whether these students are adequately prepared for college Calculus 2. College calculus classes are often taught…
The rewriting calculus is a minimal framework embedding lambda calculus and term rewriting systems that allows abstraction on variables and patterns. The rewriting calculus features higher-order functions (from the lambda calculus) and pattern matching (from term rewriting systems). In this paper, we study extensively the decidability of type inference in the second-order rewriting calculus à la Curry.
In the seminal essay, "On the unreasonable effectiveness of mathematics in the physical sciences," physicist Eugene Wigner poses a fundamental philosophical question concerning the relationship between a physical system and our capacity to model its behavior with the symbolic language of mathematics. In this essay, I examine an ambitious 16th and 17th-century intellectual agenda from the perspective of Wigner's question, namely, what historian Paolo Rossi calls "the quest to create a universal language." While many elite thinkers pursued related ideas, the most inspiring and forceful was Gottfried Leibniz's effort to create a "universal calculus," a pictorial language which would transparently represent the entirety of human knowledge, as well as an associated symbolic calculus with which to model the behavior of physical systems and derive new truths. I suggest that a deeper understanding of why the efforts of Leibniz and others failed could shed light on Wigner's original question. I argue that the notion o... integralThis book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large num...
Visualization is key in helping a student understand the fundamentals of Calculus. The new generation of computer literate students, raised in a video-based environment, will expect more than the traditional chalkboard methods in assisting them in this visualization. By integrating computers into the classroom and developing software to assist in mathematics instruction, we can enhance student comprehension of, and ability to apply, mathematics in solving real world problems of interest to th...
Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2) implicit instantiation of implementations of those interfaces. Scala implicits are a GP language mechanism, inspired by type classes, that break with the tradition of coupling implicit instantiation with a special type of interface. Instead, implicits provide only implicit instantiation, which is generalized to work for any types. This turns out to be quite powerful and useful to address many limitations that show up in other GP mechanisms. This paper synthesizes the key ideas of implicits formally in a minimal and general core calculus called the implicit calculus, and it shows how to build source languages supporting implicit instantiation on top of it. A novelty of the calculus is its support for partial resolution and higher-order rules (a feature that has been proposed bef...
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these authors made was that the classical cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely, we show that the cohomology of the Hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and the K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.
We present the MIM calculus, a modeling formalism with a strong biological basis, which provides biologically-meaningful operators for representing the interaction capabilities of molecular species. The operators of the calculus are inspired by the reaction symbols used in Molecular Interaction Maps (MIMs), a diagrammatic notation used by biologists. Models of the calculus can be easily derived from MIM diagrams, for which an unambiguous and executable interpretation is thus obtained. We give...
Programs that transform other programs often require access to the internal structure of the program to be transformed. This is at odds with the usual extensional view of functional programming, as embodied by the lambda calculus and SK combinator calculus. The recently-developed SF combinator calculus offers an alternative, intensional model of computation that may serve as a foundation for developing principled languages in which to express intensional computation, including program transfo...
We revise a monogenic calculus for several non-commuting operators, which is defined through group representations. Instead of an algebraic homomorphism we use group covariance. The related notion of joint spectrum and spectral mapping theorem are discussed. The construction is illustrated by a simple example of calculus and joint spectrum of two non-commuting selfadjoint (n\\times n) matrices. Keywords: Functional calculus, spectrum, intertwining operator, spectral mapping theorem, jet spaces...
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We present a calculus that captures the operational semantics of call-by-need.We demonstrate that the calculus is confluent and standardizable and entails the same observational equivalences as call-by-name lambda calculus.
In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a ...
Solutions Manual to Accompany Fundamentals of Calculus the text that encourages students to use power, quotient, and product rules for solutions as well as stresses the importance of modeling skills. In addition to core integral and differential calculus coverage, the core book features finite calculus, which lends itself to modeling and spreadsheets. Specifically, finite calculus is applied to marginal economic analysis, finance, growth, and decay. Includes: Linear Equations and Functions The Derivative Using the Derivative Exponential and Logarithmic
Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra. Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readersGet 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 Pre-Calculus Workbook For Dummies is the perfect tool for anyone who waCalculus and Mathematica (C&M) by Davis, Porta and Uhl ia a well thought-out method that, when used properly, gives students an intuitive understanding of, and a feeling for, all the major calculus concepts. It is comprised of the following four books: C&M / Derivatives, C&M / Integrals, C&M / Vector Calculus, and C&M / Approximation, known also as Books 1-4. In these books the authors advocate an explore-and-discover method for teaching the basic concepts of Calculus to u...As π-calculus based on the interleaving semantics cannot depict the true concurrency and has few supporting tools,it is translated into Petri nets.π-calculus is divided into basic elements,sequence,concurrency,choice and recursive modules.These modules are translated into Petri nets to construct a complicated system.Petri nets semantics for π-calculus visualize system structure as well as system behaviors.The structural analysis techniques allow direct qualitative analysis of the system properties on the structure of the nets.Finally,Petri nets semantics for π-calculus are illustrated by applying them to mobile telephone systems.
Introduces Novel Applications for Solving Neutron Transport EquationsWhile deemed nonessential in the past, fractional calculus is now gaining momentum in the science and engineering community. Various disciplines have discovered that realistic models of physical phenomenon can be achieved with fractional calculus and are using them in numerous ways. Since fractional calculus represents a reactor more closely than classical integer order calculus, Fractional Calculus with Applications for Nuclear Reactor Dynamics focuses on the application of fractional calculus to describe the physical behavi
The sequent calculus sL for the Lambek calculus L (lambek 58) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus (Morrill and Valent\\'in), which like sL has no structural rules, is also equivalent to an omega-sorted multimodal calculus mD. More concretely, ...
Many business schools or colleges require calculus as a prerequisite for certain classes or for continuing to upper division courses. While there are many studies investigating the relationship between performance in calculus and performance in a single course, such as economics, statistics, and finance, there are very few studies investigating…
The efforts to attract students to precalculus, trigonometry, and calculus classes became more successful at the author's school when projects-based classes were offered. Data collection from an untethered hot air balloon flight for calculus students was planned to maximize enrollment. The data were analyzed numerically, graphically, and…
We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…
This paper presents an extension of the Dpi-calculus due to Hennessy and Riely with constructs for signing and authenticating code and for sandboxing. A sort system, built on Milner's sort systems for the polyadic pi-calculus, is presented and proven sound with respect to an error predicate which...
The purpose of this study was to compare the perspectives of faculty members who had experience teaching undergraduate calculus and preservice teachers who had recently completed student teaching in regards to a first semester undergraduate calculus course. An online survey was created and sent to recent student teachers and college mathematics…The results from a cross-national study comparing calculus performance of students at East China Normal University (ECNU) in Shanghai and students at the University of Michigan before and after their first university calculus course are presented. Overall, ECNU significantly outperformed Michigan on both the pre- and post-tests, but the Michigan…
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of vector spaces of morphisms between products of generating objects in this category.
In order to answer the challenge of pervasive computing, we propose a new process calculus, whose aim is to describe dynamic systems composed of agents able to move and react differently depending on their location. This Context-Aware Calculus features a hierarchical structure similar to mobileThis remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as limWe demonstrate by explicit calculation of the DeWitt-like measure in two-dimensional quantum Regge gravity that it is highly non-local and that the average values of link lengths $l, $, do not exist for sufficient high powers of $n$. Thus the concept of length has no natural definition in this formalism and a generic manifold degenerates into spikes. This might explain the failure of quantum Regge calculus to reproduce the continuum results of two-dimensional quantum gravity. It points to sev... Differential and Integral Calculus introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to givi
All Access for the AP® Calculus AB & BC Exams Book + Web + Mobile Everything you need to prepare for the Advanced Placement® exam, in a study system built around you! There® Cal... and identity matrices.
This well-thought-out text, filled with many special features, is designed for a two-semester course in calculus for technology students with a background in college algebra and trigonometry. The author has taken special care to make the book appealing to students by providing motivating examples, facilitating an intuitive understanding of the underlying concepts involved, and by providing much opportunity to gain proficiency in techniques and skills.Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, in
A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory - many of the answers are fo
From the very beginning process algebra introduced the dichotomy between channels and processes. This dichotomy prevails in all present process calculi.The situation is in contrast to that with lambda calculus which has only one class of entities——the lambda terms. We introduce in this paper a process calculus called Lamp in which channels are process names. The language is more uniform than existing process calculi in two aspects: First it has a unified treatment of channels and processes. There is only one class of syntactical entities——processes. Second it has a unified presentation of both first order and higher order process calculi. The language is functional in the sense that lambda calculus is functional.Two bisimulation equivalences, barbed and closed bisimilarities, are proved to coincide.A natural translation from Pi calculus to Lamp is shown to preserve both operational and algebraic semantics. The relationship between lazy lambda calculus and Lamp is discussed.
We define an ``enriched effect calculus'' by conservatively extending a type theory for computational effects with primitives from linear logic. By doing so, we obtain a generalisation of linear type theory, intended as a formalism for expressing linear aspects of effects. As a worked example, we...... formulate linearly-used continuations in the enriched effect calculus. These are captured by a fundamental translation of the enriched effect calculus into itself, which extends existing call-by-value and call-by-name linearly-used CPS translations. We show that our translation is involutive. Full...... completeness results for the various linearly-used CPS translations follow. Our main results, the conservativity of enriching the effect calculus with linear primitives, and the involution property of the fundamental translation, are proved using a category-theoretic semantics for the enriched effect calculus...
perform these investigations indicate, that although it is perfectly possible to use process calculus techniques on object oriented languages, such techniques will not come to widespread use, but only be limited to reasoning about critical parts of a language or program design.......This thesis investigates the applicability of techniques known from the world of process calculi to reason about properties of object-oriented programs. The investigation is performed upon a small object-oriented language - The Sigma-calculus of Abadi and Cardelli. The investigation is twofold: We......-calculus turns out to be insufficient. Based on our experiences, we present a translation of a typed imperative Sigma-calculus, which looks promising. We are able to provide simple proofs of the equivalence of different Sigma-calculus objects using this translation. We use a labelled transition system adapted to... (papers)
The so-called discrete approach in calculus instruction involves introducing topics from the calculus of finite differences and finite sums, both for motivation and as useful tools for applications of the calculus. In particular, it provides an ideal setting in which to incorporate computers into calculus courses. This approach has been…
We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) b-calculus.
Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the…
Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable constructions does offer such a possibility. This makes it necessary to have a definition of fractal measures based on the fractional calculus so that the fractals can be naturally incorporated in the calculus. With this motivation a definition of fractal measure...
Calculus: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using calculus. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and integration. Everything you will need to know is here in one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.
Get the confidence and the math skills you need to get started with calculus! Are you preparing for calculus? This easy-to-follow, hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in your cour sework. You get 100s of Problems! Detailed, fully worked-out solutions to problem
AP Calculus AB & BC Crash Course - Gets You a Higher Advanced Placement Score in Less TimeFractional calculus is undergoing rapidly and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in t
We study the Pi-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor "arrow". We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.
Innovation in the services area - especially in the electronic services (e-services) domain - can be systematically developed by first considering the strategic drivers and foci, then the tactical principles and enablers, and finally the operational decision attributes, all of which constitute a process or calculus of services innovation. More specifically, there are four customer drivers (i.e., collaboration,customization, integration and adaptation), three business foci (i.e., creation-focused, solution-focused and competition-focused), six business principles (i.e., reconstruct market boundaries, focus on the big picture not numbers, reach beyond existing demand, get strategic sequence right, overcome organizational hurdles and build execution into strategy), eight technical enablers (i.e., software algorithms, automation, telecommunication, collaboration, standardization, customization,organization, and globalization), and six attributes of decision informatics (i.e., decision-driven,information-based, real-time, continuously-adaptive, customer-centric and computationally-intensive).It should be noted that the four customer drivers are all directed at empowering the individual - that is,at recognizing that the individual can, respectively, contribute in a collaborative situation, receive customized or personalized attention, access an integrated system or process, and obtain adaptive real-time or just-in-time input. The developed process or calculus serves to identify the potential white spaces or blue oceans for innovation. In addition to expanding on current innovations in services and related experiences, white spaces are identified for possible future innovations; they include those that can mitigate the unforeseen consequences or abuses of earlier innovations, safeguard our rights to privacy, protect us from the always-on, interconnected world, provide us with an authoritative search engine, and generate a GDP metric that can adequately measure the growing administering the CCR as a readiness examination in calculus are provided along with data to guide others in using the CCR as a readiness examination for beginning calculus.
For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it. All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of
We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.
This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô's formula, the optional stopping theorem and Girsanov's theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested i...
Full Text Available Loss is an important parameter of Quality of Service (QoS. Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.
In this talk I review some recent results concerning multi-instanton calculus in supersymmetric field theories. More in detail, I will show how these computations can be efficiently performed using the formalism of topological field theories. (author)
This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.
Duration calculus was introduced by Chaochen Zhou et al. (1991) as a logic to specify and reason about requirements for real-time systems. It is an extension of interval temporal logic where one can reason about integrated constraints over time-dependent and Boolean valued states without explicit...... mention of absolute time. Several major case studies have shown that duration calculus provides a high level of abstraction for both expressing and reasoning about specifications. Using timed automata one can express how real-time systems can be constructed at a level of detail which is close to an actual...... implementation. We consider in the paper the correctness of timed automata with respect to duration calculus formulae. For a subset of duration calculus, we show that one can automatically verify whether a timed automaton ℳ is correct with respect to a formula 𝒟, abbreviated ℳ|=𝒟, i.e. one...
Full Text Available A and divergence theorems replaced by the more powerful exterior expression of Stokes' theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The numerous advantages of this calculus, over more traditional machinery, are stressed throughout the article. .
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric functions.
Introductory Analysis addresses the needs of students taking a course in analysis after completing a semester or two of calculus, and offers an alternative to texts that assume that math majors are their only audience. By using a conversational style that does not compromise mathematical precision, the author explains the material in terms that help the reader gain a firmer grasp of calculus concepts.* Written in an engaging, conversational tone and readable style while softening the rigor and theory* Takes a realistic approach to the necessary and accessible level of abstraction for the secondary education students* A thorough concentration of basic topics of calculus* Features a student-friendly introduction to delta-epsilon arguments * Includes a limited use of abstract generalizations for easy use* Covers natural logarithms and exponential functions* Provides the computational techniques often encountered in basic calculus
Coping with ambiguity has recently received a lot of attention in natural language processing. Most work focuses on the semantic representation of ambiguous expressions. In this paper we complement this work in two ways. First, we provide an entailment relation for a language with ambiguous expressions. Second, we give a sound and complete tableaux calculus for reasoning with statements involving ambiguous quantification. The calculus interleaves partial disambiguation steps with steps in a t...
We present a modification of the superposition calculus that is meant to generate consequences of sets of first-order axioms. This approach is proven to be sound and deductive-complete in the presence of redundancy elimination rules, provided the considered consequences are built on a given finite set of ground terms, represented by constant symbols. In contrast to other approaches, most existing results about the termination of the superposition calculus can be carried over to our procedure....
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on $\\Gamma$-convergence, and on weighted finite ele...
The dynamism of the natural world means that it is constantly changing, sometimes rapidly, sometimes gradually. By mathematically interpreting the continuous change that characterizes so many natural processes, analysis and calculus have become indispensable to bridging the divide between mathematics and the sciences. This comprehensive volume examines the key concepts of calculus, providing students with a robust understanding of integration and differentiation. Biographies of important figures will leave readers with an increased appreciation for the sometimes competing theories that informe
Usually the first course in mathematics is calculus. Its a core course in the curriculum of the Business, Engineering and the Sciences. However many students face difficulties to learn calculus. These difficulties are often caused by the prior fear of mathematics. The students today cant live without using computer technology. The uses of computer for teaching and learning can transform the boring traditional methodology of teach to more active and attractive method. In this paper, we will sh...
Full Text Available The paper explores properties of Łukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from Łukasiewicz (fuzzy logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.
Full Text Available We introduce a refutation graph calculus for classical first-order predicate logic, which is an extension of previous ones for binary relations. One reduces logical consequence to establishing that a constructed graph has empty extension, i. e. it represents bottom. Our calculus establishes that a graph has empty extension by converting it to a normal form, which is expanded to other graphs until we can recognize conflicting situations (equivalent to a formula and its negation.
The notion of attribute-based communication seems promising to model and analyse systems with huge numbers of interacting components that dynamically adjust and combine their behaviour to achieve specific goals. A basic process calculus, named AbC, is introduced that has as primitive construct....... An example of how well-established process calculi could be encoded into AbC is given by considering the translation into AbC of a proto-typical π-calculus process.... iAnswering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity ...This book goes beyond the basics of a first course in calculus to reveal the power and richness of the subject. Standard topics from calculus — such as the real numbers, differentiation and integration, mean value theorems, the exponential function — are reviewed and elucidated before digging into a deeper exploration of theory and applications, such as the AGM inequality, convexity, the art of integration, and explicit formulas for π. Further topics and examples are introduced through a plethora of exercises that both challenge and delight the reader. While the reader is thereby exposed to the many threads of calculus, the coherence of the subject is preserved throughout by an emphasis on patterns of development, of proof and argumentation, and of generalization. More Calculus of a Single Variable is suitable as a text for a course in advanced calculus, as a supplementary text for courses in analysis, and for self-study by students, instructors, and, indeed, all connoisseurs of ingenious calculations.
We study an extension of Plotkin's call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this extended calculus provides elegant characterizations of many semantic properties, as for example solvability. We prove a standardization theorem for this calculus by generalizing Takahashi's approach of parallel reductions. The standardization property allows us to prove that our calculus is conservative with respect to the Plotkin's one...
The giant calculus within the prostatic urethra is a rare clinical entity in the young population. Most of the calculi within the urethra migrate from the urinary bladder and obliterate the urethra. These stones are often composed of calcium phosphate or calcium oxalate. The decision of treatment strategy is affected by the size, shape and position of the calculus and by the status of the urethra. If the stone is large and immovable, it may be extracted via the perineal or the suprapubic approach. In most cases, the giant calculi were extracted via the transvesical approach and external urethrotomy. Our case is the biggest prostatic calculus, known in the literature so far, which was treated endoscopically by the combination of laser and the pneumatic lithotriptor. PMID:21188583
This book examines fuzzy relational calculus theory with applications in various engineering subjects. The scope of the text covers unified and exact methods with algorithms for direct and inverse problem resolution in fuzzy relational calculus. Extensive engineering applications of fuzzy relation compositions and fuzzy linear systems (linear, relational and intuitionistic) are discussed. Some examples of such applications include solutions of equivalence, reduction and minimization problems in fuzzy machines, pattern recognition in fuzzy languages, optimization and inference engines in textile and chemical engineering, etc. A comprehensive overview of the authors' original work in fuzzy relational calculus is also provided in each chapter. The attached CD-Rom contains a toolbox with many functions for fuzzy calculations, together with an original algorithm for inverse problem resolution in MATLAB. This book is also suitable for use as a textbook in related courses at advanced undergraduate and graduate level...
Calculi with control operators have been studied as extensions of simple type theory. Real programming languages contain datatypes, so to really understand control operators, one should also include these in the calculus. As a first step in that direction, we introduce lambda-mu-T, a combination of Parigot's lambda-mu-calculus and G\\"odel's T, to extend a calculus with control operators with a datatype of natural numbers with a primitive recursor. We consider the problem of confluence on raw terms, and that of strong normalization for the well-typed terms. Observing some problems with extending the proofs of Baba at al. and Parigot's original confluence proof, we provide new, and improved, proofs of confluence (by complete developments) and strong normalization (by reducibility and a postponement argument) for our system. We conclude with some remarks about extensions, choices, and prospects for an improved presentation.
We study geometric properties of dynamical Regge calculus which is a hybridization of dynamical triangulation and quantum Regge calculus. Lattice diffeomorphisms are generated by certain elementary moves on a simplicial lattice in the hybrid model. At the semiclassical level, we discuss a possibility that the lattice diffeomorphisms give a simple explanation for the Bekenstein-Hawking entropy of a black hole. At the quantum level, numerical calculations of 3D pure gravity show that a fractal structure of the hybrid model is the same as that of dynamical triangulation in the strong-coupling phase. In the weak-coupling phase, on the other hand, space-time becomes a spiky configuration, which often occurs in quantum Regge calculus
Offering a concise collection of MatLab programs and exercises to accompany a third semester course in multivariable calculus, A MatLab Companion for Multivariable Calculus introduces simple numerical procedures such as numerical differentiation, numerical integration and Newton''s method in several variables, thereby allowing students to tackle realistic problems. The many examples show students how to use MatLab effectively and easily in many contexts. Numerous exercises in mathematics and applications areas are presented, graded from routine to more demanding projects requiring some programming. Matlab M-files are provided on the Harcourt/Academic Press web site at Computer-oriented material that complements the essential topics in multivariable calculus* Main ideas presented with examples of computations and graphics displays using MATLAB * Numerous examples of short code in the text, which can be modified for use with the exercises* MATLAB files are used to implem...
Full Text Available Giant vesical calculus is a rare entity. Vesical calculi can be primary (stones form de novo in bladder or secondary to the migrated renal calculi, chronic UTI, bladder outlet obstruction, bladder diverticulum or carcinoma, foreign body and neurogenic bladder. We report a case of an 85year old male patient who presented with history of recurrent episodes of burning micturition, pain abdomen, straining at micturition and diminished stream. Ultrasonography and X ray KUB showed a large vesical calculus. Patient underwent a n Open Cystolithomy and a large calculus of size 9x13cm weighing 310gms was removed. Bladder wall hypertrophy was seen with signs of inflammation. Bladder mucosal biopsy was taken which was normal on histopathological examination. Post - operative recovery was uneventful
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.
Full Text Available WS-BPEL is way to define business processes that interact with external entities through webservice operations using WSDL. We have proposed BPEL-TC, an extension to existing WS-BPEL whichuses temporally customized Web Services (WSDL-TC as a model for process decomposition and assembly.WSDL-TC handles both backward compatible and incompatible changes and also maintains variousversions of the artifacts that results due to changes over time and customizations desired by the users. Inthis paper, we are using pi-calculus to formalize Business Process Execution Language- TemporalCustomization (BPEL-TC process. π -calculus is a model of computation for concurrent systems alongwith changing connectivity of interactive systems. Pi-calculus is an extension of the process algebra CCS,with added mobility to CCS while preserving its algebraic properties.Full Text Available The purpose of this study was to investigate the effects of using writing activities on students' understanding and achievement in Calculus. The design of this study was quasi-experimental. The subjects of this study consisted of two secondary schools in one of the states in Malaysia. Each school was assigned one intact class of Form Four to be the experimental group and another one intact class as the control. The experimental group learned mathematics by using the writing activities for five weeks, while the control group learned mathematics by using traditional whole-class instruction. A 20-item Calculus Achievement test was designed with reliability .87. The findings showed that the experimental group exhibited significantly greater improvement on calculus achievement. The students showed positive reaction towards the use of writing. Findings of this study provide information to schools to take advantage of writing activities to promote understanding.
Through the comparison of syntactic structure,operational semantics and algebraic semantics between χ-calculus and π-calculus, this paper concludes that χ-calculus has more succinct syntactic structure,more explicit operational semantics,more intuitionistic algebraic semantics and more favorable algebraic property. And a translation from π-calculus to χ-calculus is presented.
We)$. Domain of constructed calculus isdense in the Banach space.
Full Text Available We$. Domain of constructed calculus isdense in the Banach space.Three years ago our mathematics department rearranged the topics in second and third semester calculus, moving multivariable calculus to the second semester and series to the third semester. This paper describes the new arrangement of topics, and how it could be adapted to calculus curricula at different schools. It also explains the benefits we…
Full Text Available We present a case of a 21 year old male who presented with symptomatic right upper ureteric calculus measuring 5 cm × 1.5 cm fulfilling the criteria to be named as giant ureteric calculus. Laparoscopic right ureterolithotomy was performed and the giant ureteric calculus was retrieved. PMID:26793529
How is calculus used in science? That might seem like an odd question to answer in a magazine intended primarily for elementary school teachers. After all, how much calculus gets used in elementary science? Here the author guesses that quite a few readers of this column do not know a whole lot about calculus and have not taken a course in…
Full Text Available It is presented the calculus of the two-phase ejector for carbon dioxideNakano's "later" modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this...... decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence...
Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner, while more involved, in-depth projects may be used for group learning. Each task draws on special mathematical topics and applications from subjects including medicine, engineering, economics, ecology, physics, and biology.Subjects including:* Medicine* Engineering* Economics* Ecology* Physics* Biology
The covariant loop calculus provides and efficient technique for computing explicit expressions for the density on moduli space corresponding to arbitrary (bosonic string) loop diagrams. Since modular invariance is not manifest, however, we carry out a detailed comparison with known explicit 2- and 3- loop results derived using analytic geometry (1 loop is known to be ok). We establish identity to 'high' order in some moduli and exactly in others. Agreement is found as a result of various non-trivial cancellations, in part related to number theory. We feel our results provide very strong support for the correctness of the covariant loop calculus approach. (orig.)
@@ The calculus of pseudo-differential operators on singular spaces and theconcept of ellipti-city in operator algebras on manifolds with singularitieshave become an enormous challenge for analysists. The so-called cone algebras(with discrete and continuous asymptotics) are investigated by manymathematicians, especially by B. W. Schulze, who developed and enrichedcone and edge pseudo-differential calculus, see Schulze[4-7], Rempel and Schulze [2, 3]. In this note,we construct a cone pseudo-differentialcalculus for operators which respect conormal asymptotics of a prescribedasymptotic typeExplicit substitution calculi can be classified into several dis- tinct categories depending on whether they are confluent, meta-confluent, strong normalization preserving, strongly normalizing, simulating, fully compositional, and/or local. In this paper we present a variant of the λσ-calculus......, which satisfies all seven conditions. In particular, we show how to circumvent Mellies counter-example to strong normalization by a slight restriction of the congruence rules. The calculus is implemented as the core data structure of the Celf logical framework. All meta-theoretic aspects of this work...
We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifier-free SIL....... We present a mechanization of SIL in the generic proof assistant Isabelle and consider techniques for automated reasoning. Many of the results and ideas of this report are also applicable to traditional (non-signed) interval logic and, hence, to Duration Calculus....
We propose a hybrid approach to lattice quantum gravity by combining simultaneously the dynamical triangulation with the Regge calculus, called the dynamical Regge calculus (DRC). In this approach lattice diffeomorphism is realized as an exact symmetry by some hybrid (k, l) moves on the simplicial lattice. Numerical study of 3D pure gravity shows that an entropy of the DRC is not exponetially bounded if we adopt the uniform measure Πidli. On the other hand, using the scale-invariant measure Πidli/li, we can calculate observables and observe a large hysteresis between two phases that indicates the first-order nature of the phase transition
We consider a fragment of the Quality Calculus, previously introduced for defensive programming of software components such that it becomes natural to plan for default behaviour in case the ideal behaviour fails due to unreliable communication. This paper develops a probabilistically based trust...... analysis supporting the Quality Calculus. It uses information about the probabilities that expected input will be absent in order to determine the trustworthiness of the data used for controlling the distributed system; the main challenge is to take accord of the stochastic dependency between some of the...
Let V be a nite dimensional real vector space on which a root system is given. Consider a meromorphic function ' on VC = V +iV , the singular locus of which is a locally nite union of hyperplanes of the form f 2 VC j h; i = sg, 2 , s 2 R. Assume ' is of suitable decay in the imaginary directions, so that integrals of the form R +iV '() d make sense for generic 2 V . A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley{Wiener t...
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and geometrical settings. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers the majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in classical mechanics for E = TM
Full Text Available As modern architectures introduce additional heterogeneity and parallelism, we look for ways to deal with this that do not involve specialising software to every platform. In this paper, we take the Join Calculus, an elegant model for concurrent computation, and show how it can be mapped to an architecture by a Cartesian-product-style construction, thereby making use of the calculus' inherent non-determinism to encode placement choices. This unifies the concepts of placement and scheduling into a single task.
Introduction; Overture: Newton's published work on the calculus of fluxions; Part I. The Early Period: 1. The diffusion of the calculus (1700-1730); 2. Developments in the calculus of fluxions (1714-1733); 3. The controversy on the foundations of the calculus (1734-1742); Part II. The Middle Period: 4. The textbooks on fluxions (1736-1758); 5. Some applications of the calculus (1740-1743); 6. The analytic art (1755-1785); Part III. The Reform: 7. Scotland (1785-1809); 8. The Military Schools (1773-1819); 9. Cambridge and Dublin (1790-1820); 10. Tables; Endnotes; Bibliography; IndexFull Text Available D to identify and remove calculus deposits present on the root surface. The purpose of this review was to compile the various methods and their advantages for the detection and removal of calculus. administe...
Homeschooling in the United States has grown considerably over the past several decades. This article presents findings from the Factors Influencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high…We develop an analog of Jones' planar calculus for II_1We develop an analog of Jones' planar calculus for II 1This advanced undergraduate textbook is based on a one-semester course on single variable calculus that the author has been teaching at San Diego State University for many years. The aim of this classroom-tested book is to deliver a rigorous discussion of the concepts and theorems that are dealt with informally in the first two semesters of a beginning calculus course. As such, students are expected to gain a deeper understanding of the fundamental concepts of calculus, such as limits (with an emphasis on ε-δ definitions), continuity (including an appreciation of the difference between mere pointwise and uniform continuity), the derivative (with rigorous proofs of various versions of L'Hôpital's rule) and the Riemann integral (discussing improper integrals in-depth, including the comparison and Dirichlet tests). Success in this course is expected to prepare students for more advanced courses in real and complex analysis and this book will help to accomplish this. The first semester of advanced calculus...
Regge calculus configuration superspace can be embedded into a more general superspace where the length of any edge is defined ambiguously depending on the 4-tetrahedron containing the edge. Moreover, the latter superspace can be extended further so that even edge lengths in each the 4-tetrahedron are not defined, only area tensors of the 2-faces in it are. We make use of our previous result concerning quantization of the area tensor Regge calculus which gives finite expectation values for areas. Also our result is used showing that quantum measure in the Regge calculus can be uniquely fixed once we know quantum measure on (the space of the functionals on) the superspace of the theory with ambiguously defined edge lengths. We find that in this framework quantization of the usual Regge calculus is defined up to a parameter. The theory may possess nonzero (of the order of Planck scale) or zero length expectation values depending on whether this parameter is larger or smaller than a certain value. Vanishing length expectation values means that the theory is becoming continuous, here dynamically in the originally discrete framework....... dense time, thus allowing exploitation of discrete-time (semi-)decision procedures on dense-time properties....
Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball,… functions au into one big algebra, the ''Cartan Calculus.''
The tcc model is a formalism for reactive concurrent constraint programming. In this paper we propose a model of temporal concurrent constraint programming which adds to tcc the capability of modeling asynchronous and non-deterministic timed behavior. We call this tcc extension the ntcc calculus...
Giant vesical calculus is a rare entity. Vesical calculi can be primary (stones form de novo in bladder) or secondary to the migrated renal calculi, chronic UTI, bladder outlet obstruction, bladder diverticulum or carcinoma, foreign body and neurogenic bladder. We report a case of an 85year old male patient who presented with history of...The benefits of high-level mathematics packages such as Matlab include both a computer algebra system and the ability to provide students with concrete visual examples. This paper discusses how both capabilities of Matlab were used in a multivariate calculus class. Graphical user interfaces which display three-dimensional surfaces, contour plots,……
In the study of process calculi, encoding between different calculi is an effective way to compare the expressive power of calculi and can shed light on the essence of where the difference lies. Thomsen and Sangiorgi have worked on the higher-order calculi (higher-order Calculus of Communicating Systems (CCS) and higher-order It-calculus, respectively) and the encoding from and to first-order π-calculus. However a fully abstract encoding of first-order π-calculus with higher-order CCS is not available up-today. This is what we intend to settle in this paper. We follow the encoding strategy, first proposed by Thomsen, of translating first-order π-calculus into Plain CHOCS. We show that the encoding strategy is fully abstract with respect to early bisimilarity (first-order π-calculus) and wired bisimilarity (Plain CHOCS) (which is a bisimulation defined on wired processes only sending and receiving wires), that is the core of the encoding strategy. Moreover from the fact that the wired bisimilarity is contained by the well-established context bisimilarity, we secure the soundness of the encoding, with respect to early bisimilarity and context bisimilarity. We use index technique to get around all the technical details to reach these main results of this paper. Finally, we make some discussion on our work and suggest some future work.alIn the last few years appeared pedagogical propositional natural deduction systems. In these systems, one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Godel system T. Finally, guided by the logical limitations of CCr, we propose a formal and general definition of what a pedagogical calculus of constructions should be.
In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the "Fourth Moment Theorem" in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre, giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper "Stein's method on Wiener chaos" (written in collaboration with Peccati) in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein's methFull Text Available Until 20th century, bladder stones were one of the most prevalent disorders among the poor class and the incidence was especially high in childhood and adolescent. 1 The decrease in incidence of bladder calculi is attributed mainly to dietary and nutritional progress especially in children. 2 A solitary bladder calculus is usual, although multiple stones are found in 25% of cases. 3 Bladder stones are rare, and they constitute about 5% of all urinary stones, 4, 5 it is classified as migrated from upper urinary tract, primary idiopathic, or secondary calculi. 6 Bladder stones are managed by Extracorporeal Shockwave Lithotripsy (ESWL, endourology procedures, or open surgery. We report an unusual case of giant vesical calculus weighing 600grams in a 55 year old female with no evidence of hematuria, urinary retention, and dysuria.
Understanding Maxwell's equations in differential form is a prerequisite to study the electrodynamic phenomena that are Itô–Stratonovich dilemma is revisited from the perspective of the interpretation of Stratonovich calculus using shot noise. Over the long time scales of the displacement of an observable, the principal issue is how to deal with finite/zero autocorrelation of the stochastic noise. The former (non-zero) noise autocorrelation structure preserves the normal chain rule using a mid-point selection scheme, which is the basis Stratonovich calculus, whereas the instantaneous autocorrelation structure of Itô's approach does not. By considering the finite decay of the noise correlations on time scales very short relative to the overall displacement times of the observable, we suggest a generalization of the integral Taylor expansion criterion of Wong and Zakai (1965 Ann. Math. Stat. 36 1560–4) for the validity of the Stratonovich approach. (paper)
Two cosmological solutions of Regge calculus are presented which correspond to the flat Friedmann-Robertson-Walker and the Kasner solutions of general relativity. By taking advantage of the symmetries that are present, I am able to show explicitly that a limit of Regge calculus does yield Einstein's equations for these cases. The method of averaging these equations when taking limits is important, especially for the Kasner model. I display the leading error term that arises from keeping the Regge equations in discrete form rather than using their continuum limit. In particular, this work shows that for the ''Reggeized'' Friedmann model the minimum volume is a velocity-dominated singularity as in the continuum Friedmann model. However, unlike the latter, the Regge version has a nonzero minimum volume intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.
Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transformations in...
Full Text Available An eight-year old male was admitted with complaints of right scrotal swelling, dysuria and intermittent retention of urine for 10 days. On per-rectal examination, a hard mass was palpable in the posterior urethra. An X-ray (KUB of the abdomen revealed a double dumb-bell calculus at the base of bladder, extending into the posterior urethra. A cystolithotomy via the suprapubic approach was successfully curative.
Excellent text provides basis for thorough understanding of the problems, methods and techniques of the calculus of variations and prepares readers for the study of modern optimal control theory. Treatment limited to extensive coverage of single integral problems in one and more unknown functions. Carefully chosen variational problems and over 400 exercises. ""Should find wide acceptance as a text and reference.""-American Mathematical Monthly. 1969 edition. Bibliography. investigate Calculus teaching at university mathematics departments and in particular research math-ematicians' teaching practice in the context of lectures. We are interested in how lecturers draw mathematics students into mathematical culture. In this paper, we focus on the teaching of a lecturer of a large cohort of students that we analyse using grounded techniques and the Teaching Triad construct (Jaworski, 1994). In spite of the lecture format, the analysis suggests that this lecture...In this paper we illustrate the LU representation of fuzzy numbers and present an LU-fuzzy calculator, in order to explain the use of the LU-fuzzy model and to show the advantage of the parametrization. The model can be applied either in the level-cut or in generalized LR frames. The hand-like fuzzy calculator has been developed for the MSWindows platform and produces the basic fuzzy calculus: the arithmetic operations (scalar multiplication, addition, subtraction, multiplication, division) a...
Higher spin theories can be efficiently described in terms of auxiliary St\\"uckelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for op...
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces t... functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geo ca...
We present the general framework of \\'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \\'Ecalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then p l...
The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta method is derived and is applied to academic examples, with known closed form solutions, and a sample problem from mathematical modelling in biology. The comparison of the results of the Bigeometric Runge-Kutta method with the ordinary Runge-Kutta method shows...
This article discusses an experience of teaching Calculus classes for the freshmen students enrolled at Sungkyunkwan University, one of the private universities in South Korea. The teaching and learning approach is a balance combination between the teacher-oriented traditional style of lecturing and other activities that encourage students for active learning and classroom participation. Based on the initial observation during several semesters, some anecdotal evidences show that students' le...
Objective: Emphysematous pyelonephritis (EPN) with calculus is well recognized but with very few reports on its treatment. Our aim is to elucidate our experience in its successful management. Materials and Methods: Over four years, we diagnosed seven cases (eight renal units) of EPN, out of which two patients (three renal units) had EPN with urinary calculi. After the initial conservative management of EPN, the stones were tackled appropriately. Results: EPN was initially managed effectively ...Full Text Available BACKGROUND Vesical calculi are the most common manifestation of lower urinary tract lithiasis. Urinary infections play an important role in aetiopathogenesis of vesical calculi. OBJECTIVE Aim of this study was proposed to establish the bacteriology of stone and urine in an attempt to evaluate the role of infection in the formation of stone. Associated factors like age, sex, site of infection, obstruction, diet were also evaluated. DESIGN Prospective cohort study. METHODS The patients were admitted in surgical ward as provisional diagnosed cases of vesical calculus, were subjected to investigations including CBC, RBS, urine analysis, renal function test, x-ray KUB region and ultrasonography. Patients who were fit for surgery, various surgical procedures were done. Gross examination and core culture of stone was done to establish their aetiology. RESULTS Ninety-four patients with vesical calculus were evaluated. Incidence of vesical calculus was 1.13%. Majority of cases were from rural areas (92.55%. Urinary tract infection was present in 37.2% of cases, majority of cases urine culture was positive (30.95%. Core culture of stone was positive in 18 cases (25.17%. E. coli was the predominant organism both in urine culture (19.04% and core culture of stone (25.71%. CONCLUSIONS There is significant association regarding the presence of vesical calculi and the development of urinary infections. E. coli was the predominant organism found both in urine and core culture of stone.
This paper introduces a formal metalanguage called the lambda-q calculus for the specification of quantum programming languages. This metalanguage is an extension of the lambda calculus, which provides a formal setting for the specification of classical programming languages. As an intermediary step, we introduce a formal metalanguage called the lambda-p calculus for the specification of programming languages that allow true random number generation. We demonstrate how selected randomized algD t...
We derive a numerical method for Darcy flow, hence equa...
A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical Engineering graduate level course on the use of complex gradients and complex Hessian matrices, and has been successfully used in teaching at UC San Diego. Going beyond the commonly encountered treatments of the first-order complex vector calculus, second-...
This paper investigates the reasoning of first year non-mathematics students in non-routine calculus tasks. The students in this study were accustomed to imitative reasoning from their primary and secondary education. In order to move from imitative reasoning toward more creative reasoning, non-routine tasks were implemented as an explicit part of the students' calculus course. We examined the reasoning of six students in the middle of the calculus course and at the end of the course. The ana...
We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations.
This article introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; e.g. the linear usage of imperative features such...... calculus. The second half of the article investigates models for the enriched effect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard effect calculi (such as those based on monads), and to models of intuitionistic linear logic. We...
Prepare for calculus the smart way, with customizable pre-calculus practice 1,001 Pre-Calculus Practice Problems For Dummies offers 1,001 opportunities to gain confidence in your math skills. Much more than a workbook, this study aid provides pre-calculus problems ranked from easy to advanced, with detailed explanations and step-by-step solutions for each one. The companion website gives you free online access to all 1,001 practice problems and solutions, and you can track your progress and ID where you should focus your study time. Accessible on the go by smart phone, tablet, o
This paper is about a categorical approach to model a very simple Semantically Linear lambda calculus, named Sll-calculus. This is a core calculus underlying the programming language SlPCF. In particular, in this work, we introduce the notion of Sll-Category, which is able to describe a very large class of sound models of Sll-calculus. Sll-Category extends in the natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to soundly interpret all the constructs of Sll-calculu...
The BP-calculus is a formalism based on the π-calculus and encoded in WS-BPEL. The BP-calculus is intended to specificaly model and verify Service Oriented Applications. One important feature of SOA is the ability to compose services that may dynamically evolve along runtime. Dynamic...... reconfiguration of services increases their availability, but puts accordingly, heavy demands for validation, verification, and evaluation. In this paper we formally model and analyze dynamic reconfigurations and their requirements in BP-calculus and show how reconfigurable components can be modeled using...
Calculus. For some of us, the word conjures up memories of ten-pound textbooks and visions of tedious abstract equations. And yet, in reality, calculus is fun, accessible, and surrounds us everywhere we go. In Everyday Calculus, Oscar Fernandez shows us how to see the math in our coffee, on the highway, and even in the night sky. Fernandez uses our everyday experiences to skillfully reveal the hidden calculus behind a typical day's events. He guides us through how math naturally emerges from simple observations-how hot coffee cools down, for example-and in discussions of over fifty familia
Formal models of communicating and concurrent systems are one of the most important topics in formal methods,and process calculus is one of the most successful formal models of communicating and concurrent systems.In the previous works,the author systematically studied topology in process calculus,probabilistic process calculus and pi-calculus with noisy channels in order to describe approximate behaviors of communicating and concurrent systems as well as randonmess and noise in them.This article is a brief survey of these works.
A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of um...
Fluent description of the development of both the integral and differential calculus. Early beginnings in antiquity, medieval contributions, and a century of anticipation lead up to a consideration of Newton and Leibniz, the period of indecison that followed them, and the final rigorous formulation that we know today.
In education theory, Bloom's taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is…
This paper describes how one university mathematics department was able to improve student success in Calculus I by requiring a co-requisite lab for certain groups of students. The groups of students required to take the co-requisite lab were identified by analyzing student data, including Math ACT scores, ACT Compass Trigonometry scores, andThis article describes some reflections from the first Calculus I undergraduate teaching assistant in our department as she explored the various ways in which she was able to support both novice and experienced Calculus teachers and the effect of her experience on her academic and career plans.Describes the results of a teacher's exploration of the effects of using graphing calculators in calculus instruction in sections other than those that are experimental. Two experimental and two traditional sections of Calculus I and II participated in the study. (DDR)
Summing powers of integers is presented as an example of finite differences and antidifferences in discrete mathematics. The interrelation between these concepts and their analogues in differential calculus, the derivative and integral, is illustrated and can form the groundwork for students' understanding of differential and integral calculus.…
Pre-calculus concepts such as working with functions and solving equations are essential for students to explore limits, rates of change, and integrals. Yet many students have a weak understanding of these key concepts which impedes performance in their first year university Calculus course. A series of online learning objects was developed to…
We present a proof system for determining satisfaction between processes in a fairly general process algebra and assertions of the modal μ-calculus. The proof system is compositional in the structure of processes. It extends earlier work on compositional reasoning within the modal μ-calculus and ...
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both fractional and classical derivatives are proved. The fundamental problem of the calculus of variations with mixed integer and fractional order derivatives as well as isoperimetric problems are consideredThis paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
Homer is a higher order process calculus with locations. In this paper we study Homer in the setting of the semantic finite control property, which is a finite reachability criterion that implies decidability of barbed bisimilarity. We show that strong and weak barbed bisimilarity are undecidable...... finite control π-calculus in Homer....
Rene…This paper reports the results of a survey study of clicker use and mathematics anxiety among students enrolled in an undergraduate calculus course during the Fall 2013 semester. Students in two large lecture sections of calculus completed surveys at the beginning and end of the course. One class used clickers, whereas the other class was taught…
Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…
The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overviewOver the course of two years, 2012-2014, we have implemented a "flipping" the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of…
This study explored first-semester calculus students' understanding of tangent lines as well as how students used tangent lines within the context of Newton's method. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and…
We present a model checking framework for a spi-calculus dialect which uses a linear time temporal logic for expressing security properties. We have provided our spi-calculus dialect, called SPID, with a semantics based on labeled transition systems (LTS), where the intruder is modeled in the Dolev-We present the case of a 53-year-old man, with a history of alcohol abuse, requiring intensive care unit admission, with an obstructing right upper renal calculus and Klebsiella pneumoniae urosepsis. Ureteroscopic treatment of this obstruction displayed a small calculus within the renal pelvis completely encapsulated within a fungal bezoar. Laboratory analysis of the fungal mass found it to be Candida dubliniensis.In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are basic and co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical studies in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems for fractional differential equations (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising first in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. The operators of the Generalized Fractional Calculus - GFC (integrals and derivatives) are based on commuting m-tuple (m = 1, 2, 3, …) compositions of operators of the classical FC with power weights (the so-called Erdélyi-Kober operators), but represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C-type) operators, where the kernels are special functions of most general hypergeometric kind. The foundations of this theory are given in Kiryakova 18. In this survey we present the genesis of the definitions of the GFC - the generalized fractional integrals and derivatives (of fractional multi-order) of R-L type and Caputo type, analyze their properties and applications. Their special cases are all the known operators of classical FC, their generalizations introduced by other authors, the hyper-Bessel differential operators of higher integer
The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology. This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist - mathematicians, physicists, engineers, students or researchers - in discovering the subjects most important problems, results and techniques. Despite the aim of addressing non-spe
A previously developed formalism for the bosonic string is extended to the Neveu-Schwarz-Ramond string using 2-d superspace techniques throughout. 3-string vertices for NS- and R-strings are constructed, sewing rules developed, and the technique of quasi-superconformal modes is set up for constructing the measure on super moduli space. Symmetries, such as superconformal invariance and BRST-invariance, are guaranteed ab initio. Picture changing and bosonization are avoided. Examples are given. The formalism should allow a superstring loop calculus based on supermoduli. Results concerning the ensuing super-Schottky description are given. (orig.)
This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormaiization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormaiization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variablesThe electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume $V_d\\sim r^d$ with the fractal dimension $d$ and a complementary host volume $V_h=V_3-V_d$. Integrations over these fractal volumes correspond to the convolution int... PMID:19583533
The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer QM, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations. (paper)
Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their courses. Unfortunately, due to the basal conceptions of what it means to…
We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semilocal Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term 'CV graph state' currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the 'closest' CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.
During the past fifty years , Fractional Calculus has become an original and renowned mathematical tool for the modelling of diffusion Partial Differential Equations and the design of robust control algorithms. However, in spite of these celebrated results, some theoretical problems have not yet received a satisfying solution. The mastery of initial conditions, either for Fractional Differential Equations (FDEs) or for the Caputo and Riemann-Liouville fractional derivatives, remains an open research domain. The solution of this fundamental problem, also related to the long range memory property, is certainly the necessary prerequisite for a satisfying approach to modelling and control applications. The fractional integrator and its continuously frequency distributed differential model is a valuable tool for the simulation of fractional systems and the solution of initial condition problems. Indeed, the infinite dimensional state vector of fractional integrators allows the direct generalization to fractional calculus of the theoretical results of integer order systems. After a reminder of definitions and properties related to fractional derivatives and systems, this presentation is intended to show, based on the results of two recent publications [1,2], how the fractional integrator provides the solution of the initial condition problem of FDEs and of Caputo and Riemann-Liouville fractional derivatives. Numerical simulation examples illustrate and validate these new theoretical concepts.
In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus). Algorithms based in our definitions yields decision methods for these calculi.
This study investigated students' conceptual and procedural understanding of calculus within the context of an engineering mechanics course. Four traditional calculus students were compared with three students from one of the calculus reform projects, Calculus & Mathematica. Task-based interviews were conducted with each participant throughout the…
We present a Riesz-like hyperholomorphic functional calculus for a set of non-commuting operators based on the Clifford analysis. Applications to the quantum field theory are described. Keywords: Functional calculus, Weyl calculus, Riesz calculus, Clifford analysis, quantization, quantum field theory. AMSMSC Primary:47A60, Secondary: 81T10
A calculus of sequences started in 1936 opened the way for future extensions of umbral calculus in its finite operator form. Because of historically established notation we call it the psi-calculus.It appears in parts to be almost automatic extension of the standard classical finite operator calculus.
The advantage of COOZ(Complete Object-Oriented Z) is to specify large scale software,but it does not support refinement calculus.Thus its application is comfined for software development.Including refinement calculus into COOZ overcomes its disadvantage during design and implementation.The separation between the design and implementation for structure and notation is removed as well .Then the software can be developed smoothly in the same frame.The combination of COOZ and refinement calculus can build object-oriented frame,in which the specification in COOZ is refined stepwise to code by calculus.In this paper,the development model is established.which is based on COOZ and refinement calculus.Data refinement is harder to deal with in a refinement tool than ordinary algorithmic refinement,since data refinement usually has to be done on a large program component at once.As to the implementation technology of refinement calculus,the data refinement calculator is constructed and an approach for data refinement which is based on data refinement calculus and program window inference is offered.
The advantage of COOZ (Complete Object-Oriented Z) is to specify large scale software, but it does not support refinement calculus. Thus its applica tion is confined for software development. Including refinement calculus into COOZ overcomes its disadvantage during design and implementation. The separation be tween the design and implementation for structure and notation is removed as well. Then the software can be developed smoothly in the same frame. The combina tion of COOZ and refinement calculus can build object-oriented frame, in which the specification in COOZ is refined stepwise to code by calculus. In this paper, the development model is established, which is based on COOZ and refinement calculus. Data refinement is harder to deal with in a refinement tool than ordinary algorithmic refinement, since data refinement usually has to be done on a large program compo nent at once. As to the implementation technology of refinement calculus, the data refinement calculator is constructed and an approach for data refinement which is based on data refinement calculus and program window inference is offered.
The network calculus is a powerful tool to analyze the performance of wireless sensor networks. But the original network calculus can only model the single-mode wireless sensor network. In this paper, we combine the original network calculus with the multimode model to analyze the maximum delay bound of the flow of interest in the multimode wireless sensor network. There are two combined methods A-MM and N-MM. The method A-MM models the whole network as a multimode component, and the method N...
Inference systems for observation equivalences in the pi-calculus with recursion are proposed, and their completeness over the finite-control fragment with guarded recursions are proven. The inference systems consist of inference rules and equational axioms. The judgments are conditional equations which characterise symbolic bisimulations between process terms. This result on the one hand generalises Milner's complete axiomatisation of observation equivalence for regular CCS to the pi-calculus, and on the other hand extends the proof systems of strong bisimulations for guarded regular pi-calculus to observation equivalences.
A Many-Sorted Calculus Based on Resolution and Paramodulation emphasizes the utilization of advantages and concepts of many-sorted logic for resolution and paramodulation based automated theorem proving.This book considers some first-order calculus that defines how theorems from given hypotheses by pure syntactic reasoning are obtained, shifting all the semantic and implicit argumentation to the syntactic and explicit level of formal first-order reasoning. This text discusses the efficiency of many-sorted reasoning, formal preliminaries for the RP- and ?RP-calculus, and many-sorted term rewrit
We propose an extension of the Chronological Calculus, developed by Agrachev and Gamkrelidze for the case of $C^\\infty$-smooth dynamical systems on finite-dimensional $C^\\infty$-smooth manifolds, to the case of $C^m$-smooth dynamical systems and infinite-dimensional $C^m$-manifolds. Due to a relaxation in the underlying structure of the calculus, this extension provides a powerful computational tool without recourse to the theory of calculus in Fr\\'echet spaces required by the classical Chron...
This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.
This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control. Calculus of Variations and Optimal Control Theory
This paper examines the behavior of closed "lattice universes" wherein masses are distributed in a regular lattice on the Cauchy surfaces of closed vacuum universes. Such universes are approximated using a form of Regge calculus originally developed by Collins and Williams to model closed Friedmann-Lemaître-Robertson-Walker universes. We consider two types of lattice universes, one where all masses are identical to each other and another where one mass gets perturbed in magnitude. In the unperturbed universe, we consider the possible arrangements of the masses in the Regge Cauchy surfaces and demonstrate that the model will only be stable if each mass lies within some spherical region of convergence. We also briefly discuss the existence of Regge models that are dual to the ones we have considered. We then model a perturbed lattice universe and demonstrate that the model's evolution is well behaved, with the expansion increasing in magnitude as the perturbation is increased Through3+1) (continuous time) Regge calculus is reduced to Hamiltonian form. The constraints are classified, classical and quantum consequences are discussed. As basic variables connection matrices and antisymmetric area tensors are used supplemented with appropriate bilinear constraints. In these variables the action can be made quasipolinomial with $\\arcsin$ as the only deviation from polinomiality. In comparison with analogous formalism in the continuum theory classification of constraints changes: some of them disappear, the part of I class constraints including Hamiltonian one become II class (and vice versa, some new constraints arise and some II class constraints become I class). As a result, the number of the degrees of freedom coincides with the number of links in 3-dimensional leaf of foliation. Moreover, in empty space classical dynamics is trivial: the scale of timelike links become zero and spacelike links are constant.
This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers functional calculus approach is applied to the derivation of evolution equations for the moments of the magnetization dynamics of systems subject to stochastic fields. It allows us to derive a general framework for obtaining the master equation for the stochastic magnetization dynamics, that is applied to both, Markovian and non-Markovian dynamics. The formalism is applied for studying different kinds of interactions, that are of practical relevance and hierarchies of evolution equations for the moments of the distribution of the magnetization are obtained. In each case, assumptions are spelled out, in order to close the hierarchies. These closure assumptions are tested by extensive numerical studies, that probe the validity of Gaussian or non--Gaussian closure Ans\\"atze.
This completely self-contained text is intended either for a course in honors calculus or for an introduction to analysis. Beginning with the real number axioms, and involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate math majors. This fourth edition includes an additional chapter on the fundamental theorems in their full Lebesgue generality, based on the Sunrise Lemma. Key features of this text include: • Applications from several parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more; • A heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero; • Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals; • A self-contained t (Burby et al., 2013),The quality of education in science, technology, engineering, and mathematics (STEM) fields is an issue of particular educational and economic importance, and Calculus I is a linchpin course in STEM major tracks. A national study is currently being conducted examining the characteristics of successful programs in college calculus (CSPCC, 2012). In work related to the CSPCC program, this study examines the effects on student outcomes of four different teaching strategies used at a single institution. The four classes were a traditional lecture, a lecture with discussion, a lecture incorporating both discussion and technology, and an inverted model. This dissertation was guided by three questions: (1) What impact do these four instructional approaches have on students' persistence, beliefs about mathematics, and conceptual and procedural achievement in calculus? (2) How do students at the local institution compare to students in the national database? And (3) How do the similarities and differences in opportunities for learning presented in the four classes contribute to the similarities and differences in student outcomes? Quantitative analysis of surveys and exams revealed few statistically significant differences in outcomes, and students in the inverted classroom often had poorer outcomes than those in other classes. Students in the technology-enhanced class scored higher on conceptual items on the final exam than those in other classes. Comparing to the national database, local students had similar switching rates but less expert-like attitudes and beliefs about mathematics than the national average. Qualitative analysis of focus group interviews, classroom observations, and student course evaluations showed that several implementation issues, some the result of pragmatic constraints, others the result of design choice, weakened affordances provided by innovative features and shrunk the differences between classes. There were substantial differences between the
As a variant of process algebra, π-calculus can describe the interactions between evolving processes. By modeling activity as a process interacting with other processes through ports, this paper presents a new approach: representing workflow models using π-calculus. As a result, the model can characterize the dynamic behaviors of the workflow process in terms of the LTS (Labeled Transition Semantics) semantics of π-calculus. The main advantage of the workflow model's formal semantic is that it allows for verification of the model's properties, such as deadlock-free and normal termination. Moreover, the equivalence of workflow models can be checked through weak bisimulation theorem in the π-calculus, thus facilitating the optimization of business processes.
We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.
This paper concerns retracts in simply typed lambda calculus assuming βη-equality. We provide a simple tableau proof system which characterises when a type is a retract of another type and which leads to an exponential decision procedure.
This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes. Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations. This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to pract...
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
Since the seminal work of Zhou Chaochen, M. R. Hansen, and P. Sestoft on decidability of dense-time Duration Calculus [Zhou, Hansen, Sestoft, 1993] it is well-known that decidable fragments of Duration Calculus can only be obtained through withdrawal of much of the interesting vocabulary of this...... logic. While this was formerly taken as an indication that key-press verification of implementations with respect to elaborate Duration Calculus specifications were also impossible, we show that the model property is well decidable for realistic designs which feature natural constraints on their...... suitably sparser model classes we obtain model-checking procedures for rich subsets of Duration Calculus. Together with undecidability results also obtained, this sheds light upon the exact borderline between decidability and undecidability of Duration Calculi and related logics....
A realization of Poincare-Lie algebra in terms of noncommutative differential calculus is constructed. Corresponding relativistic quantum mechanics is considered. The important conclusion is that field equations appear in the integral form
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved.
We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory.
We extend the pi-calculus with polyadic synchronisation, a generalisation of the communication mechanism which allows channel names to be composite. We show that this operator embeds nicely in the theory of pi-calculus, we suggest that it permits divergence-free encodings of distributed calculi......, and we show that a limited form of polyadic synchronisation can be encoded weakly in pi-calculus. After showing that matching cannot be derived in pi-calculus, we compare the expressivity of polyadic synchronisation, mixed choice and matching. In particular we show that the degree of synchronisation...... of a language increases its expressive power by means of a separation result in the style of Palamidessi's result for mixed choice....
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative that involve functions of more than one variable, and thus enjoy a course on Multivariate Calculus. We serve this article as a culinary guide to differentiating and integrating functions of more than one variable -- using differential forms which are the basis for de Rham Cohomology.
This paper applies comparative textbook analysis to studying the mathematical development of differential calculus in northern German states during the eighteenth century. It begins with describing how the four textbooks analyzed presented the foundations of calculus and continues with assessing the influence each of these foundational approaches exerted on the resolution of problems, such as the determination of tangents and extreme values, and even on the choice of coordinates for both algebraic and transcendental curves. PMID:19244874 wer...
This study is part f a larger one whose general objective is to investigate and to develop a new strategy for teaching Differential and Integral Calculus I, specifically for physics majors, through a possible integration with the teaching of General and Experimental Physics I. With the specific objective of identifying physics problem-situations that may help in making sense of the mathematical concepts used in Calculus I, and languages and notations that might be used in the teaching of Calc...Only a subset of adults acquires specific advanced mathematical skills, such as integral calculus. The representation of more sophisticated mathematical concepts probably evolved from basic number systems; however its neuroanatomical basis is still unknown. Using fMRI, we investigated the neural basis of integral calculus while healthy subjects were engaged in an integration verification task. Solving integrals activated a left-lateralized cortical network including the horizontal intrapariet...
The LU-model for fuzzy numbers has been introduced in [4] and applied to fuzzy calculus in [9]; in this paper we build an LU-fuzzy calculator, in order to explain the use of the LU-fuzzy representation and to show the advantage of the parametrization. The calculator produces the basic fuzzy calculus: the arithmetic operations (scalar multiplication, addition, subtraction, multiplica- tion, division) and the fuzzy extension of many univariate functions (power with integer positive or negative ...
Full Text Available ThisThisA graphical notation for the propositionalμ-calculus, called modal graphs, ispresented. It is shown that both the textual and equational presentations of theμ-calculus canbe translated into modal graphs. A model checking algorithm based on such graphs is proposed.The algorithm is truly local in the sense that it only generates the parts of the underlyingsearch space which are necessary for the computation of the final result. The correctness of thealgorithm is proven and its complexity analysed.
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie grup and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.
We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, ``cubic'' approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define We...
Background Renal vein thrombosis (RVT) with flank pain, and hematuria, is often mistaken with renal colic originating from ureteric or renal calculus. Especially in young and otherwise healthy patients, clinicians are easily misled by clinical presentation and calcified RVT. Case presentation A 38-year-old woman presented with flank pain and hematuria suggestive of renal calculus on ultrasound. She underwent extracorporeal shock wave lithotripsy that failed, leading to the recommendation that...
This lucid and balanced introduction for first year engineers and applied mathematicians conveys the clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functions. Short and fundamental diagnostic exercises at the end of each chapter test comprehension before moving to new material.Provides a clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functionsIncludes short, useful diagnostic exercises at the end of each chapter
In this paper we present a minimal object oriented core calculus for modelling the biological notion of type that arises from biological ontologies in formalisms based on term rewriting. This calculus implements encapsulation, method invocation, subtyping and a simple formof overriding inheritance, and it is applicable to models designed in the most popular term-rewriting formalisms. The classes implemented in a formalism can be used in several models, like programming libraries.
We discuss the statistics of spikes trains for different types of integrate-and-fire neurons and different types of synaptic noise models. In cotnrast with the usual approaches in neuroscience, mainly based on statistical physics methods such as the Fokker-Planck equation or the mean-field theory, we chose the point of the view of the stochastic calculus theory to characterize neurons in noisy environments. We present four stochastic calculus techniques that can be used to find the probabilit...
This investigative research focuses on the level of readiness of Science, Technology, Engineering, and Mathematics (STEM) students entering Historically Black Colleges and Universities (HBCU) in the college Calculus sequence. Calculus is a fundamental course for STEM courses. The level of readiness of the students for Calculus can very well play a…
Circumscription and logic programs under the stable model semantics are two well-known nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical form...
The main condition of periodontitis prevention is the full calculus removal from the teeth surface. This procedure should be fulfilled without harming adjacent unaffected tooth tissues. Nevertheless the problem of sensitive and precise estimating of tooth-calculus interface exists and potential risk of hard tissue damage remains. In this work it was shown that fluorescence diagnostics during calculus removal can be successfully used for precise noninvasive detection of calculus-tooth interface. In so doing the simple implementation of this method free from the necessity of spectrometer using can be employed. Such a simple implementation of calculus detection set-up can be aggregated with the devices of calculus removing.
Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.
The guiding idea in this paper is that, from the point of view of physics, functions and fields are more important than the (space time) manifold over which they are defined. The line pursued in these notes belongs to the general framework of ideas that replaces the space M by the ring of functions on it. Our essential observation, underlying this work, is that much of mathematical physics requires only a few differential operators (Lie derivative, d, δ) operating on modules of sections of suitable bundles. A connection (=gauge potential) can be described by a lift of vector fields from the base to the total space of a principal bundle. Much of the information can be encoded in the lift without reference to the bundle structures. In this manner, one arrives at an 'algebraic differential calculus' and its graded generalization that we are going to discuss. We are going to give an exposition of 'algebraic gauge theory' in both ungraded and graded versions. We show how to deal with the essential features of electromagnetism, Dirac, Kaluza-Klein and 't Hooft-Polyakov monopoles. We also show how to break the symmetry from SU(2) to U(1) without Higgs field. We briefly show how to deal with tests particles in external fields and with the Lagrangian formulation of field theories. (orig./HSI)
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ɛ Hörmander class symbols are proven as (i) ɛ ≪ 1 and λ ≪ 1, (ii) ɛ ≪ 1 and λ = 1, as well as (iii) ɛ = 1 and λ ≪This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ε Hoermander class symbols are proven as (i) ε<< 1 and λ<< 1, (ii) ε<< 1 and λ= 1, as well as (iii) ε= 1 and λ<<High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are 'flipping' (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model of mathematics known as the time scales time scales, prove a key theorem about them, and derive the backpropagation weight update equations for a feedforward multilayer neural network architecture. By drawing together the time scales calculus and the area of neural network learning, we present the first connection of two major fields of research. PMID:20615808
In this work the construction of functional calculus for strongly continuous semigroups of operators in Schwartz distribution algebra on some cone is generalized. The partial case of vector valued calculus on the base of modification operator Fourier transformation is researched.
This is the first report to identify and show that bacteria from subgingival calculus under anaerobic conditions are involved in the formation of dental calculus. [Arch Clin Exp Surg 2014; 3(3.000: 153-160
OBJECTIVES: Archaeological dental calculus is a rich source of host-associated biomolecules. Importantly, however, dental calculus is more accurately described as a calcified microbial biofilm than a host tissue. As such, concerns regarding destructive analysis of human remains may not apply as strongly to dental calculus, opening the possibility of obtaining human health and ancestry information from dental calculus in cases where destructive analysis of conventional skeletal remains is not ...
In this work, the Z-graded differential calculus of the extended quantum 3d space is constructed. By using this differential calculus, weobtain the algebra of Cartan-Maurer forms and the corresponding quantum Lie algebra. To give a Z-graded Cartan calculus on the extendedquantum 3d space, the noncommutative differential calculus on thisspace is extended by introducing inner derivations and Lie derivatives.
A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generalization...The reduction of the structure theory of the operator algebras of quantum projective (sl(2, C)-invariant) field theory (QPFT operator algebras) to a commutative exterior differential calculus by means of the operation of renormalization of a pointwise product of operator fields is described. In the first section, the author introduces the concept of the operator algebra of quantum field theory and describes the operation of the renormalization of a pointwise product of operator fields. The second section is devoted to a brief exposition of the fundamentals of the structure theory of QPT operator algebras. The third section is devoted to commutative exterior differential calculus. In the fourth section, the author establishes the connection between the renormalized pointwise product of operator fields in QPFT operator algebras and the commutative exterior differential calculus. 5 refs
Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables.
This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.
As a variant of process algebra, π-calculus can describe the interactions between evolving processes. By modeling activity as a process interacting with other processes through ports, this paper presents a new approach: representing workilow models using ~-calculus. As a result, the model can characterize the dynamic behaviors of the workflow process in terms of the LTS ( Labeled Transition Semantics) semantics of π-calculus. The main advantage of the worktlow model's formal semantic is that it allows for verification of the model's properties, such as deadlock-free and normal termination. Moreover, the equivalence of worktlow models can be checked thlx)ugh weak bisimulation theorem in the π-caleulus, thus facilitating the optimizationof business processes. monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli's work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange's δ-calculus, including: • The isoperimetrical problems • Shortest lines and geodesics • Euler's Methodus Inveniendi and the two Addi...
A binding time analysis imposes a distinction between the computations to be performed early (e.g. at compile-time) and those to be performed late (e.g. at run-time). For the lambda-calculus this distinction is formalized by a two-level lambda-calculus. The authors present an algorithm for static...... analysis of the binding times of a typed lambda-calculus with products, sums, lists and general recursive types. Given partial information about the binding times of some of the subexpressions it will complete that information such that (i) early bindings may be turned into late bindings but not vice versa......, (ii) the resulting two-level lambda-expression reflects our intuition about binding times, e.g. that early bindings are performed before late bindings, and (iii) as few changes as possible have been made compared with the initial binding information. The results can be applied in the implementation...
Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as: •Gauss and Stokes' theorems in the differential geometry, tensor calculus, and theory of ...
We have studied the context and development of the ideas of physical forces and differential calculus in ancient India by studying relevant literature related to both astrology and astronomy since pre-Greek periods. The concept of Naisargika Bala (natural force) discussed in Hora texts from India is defined to be proportional to planetary size and inversely related to planetary distance. This idea developed several centuries prior to Isaac Newton resembles fundamental physical forces in nature especially gravity. We show that the studies on retrograde motion and Chesta Bala of planets like Mars in the context of astrology lead to development of differential calculus and planetary dynamics in ancient India. The idea of instantaneous velocity was first developed during the 1st millennium BC and Indians could solve first order differential equations as early as 6th cent AD. Indian contributions to astrophysics and calculus during European dark ages can be considered as a land mark in the pre-renaissance history ...
Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group.Many sub-elliptic partial differential operators can be inverted by Laguerre calculus.In this article,we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation.The Paneitz operator which plays an important role in CR geometry can be written as follows:Here{Zj}n j=1 is an orthonormal basis for the subbundle T(1,0)of the complex tangent bundle TC(Hn) and T is the"missing direction".The operator Lα is the sub-Laplacian on the Heisenberg group which is sub-elliptic ifαdoes not belong to an exceptional setΛα.We also construct projection operators and relative fundamental solution for the operator Lα whileα∈Λα.
We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying rij = θiθj+Bklijθkθl=0 i, j=1, 2, ..., n. and (θi)2=(θj)2=...=(θn)2=0, where Bklij is the most general matrix defined in terms of complex deformation parameters. Following considerations analogous to those of Wess and Zumino, we are able to exhibit covariance of our calculus under (2n)+1 parameter deformation of GL(n) and explicitly check that the non-anticommutative differential calculus satisfies the general constraints given by them, such as the 'linear' conditions drij≅0 and the 'quadratic' condition rijxn≅0 where xn=dθn are the differentials of the variables. (orig.)
The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = {phi} and Q = -1/{phi}, and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n {yields} {infinity} it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated. (paper)
This article discusses preparation assignments used in a Calculus II course that cover material from prerequisite courses. Prior to learning new material, students work on problems outside of class involving concepts from algebra, trigonometry, and Calculus I. These problems are directly built upon in order to answer Calculus II questions,…
The purpose of this study was to investigate higher education mathematics departments' credit granting policies for students with high school calculus experience. The number of students taking calculus in high school has more than doubled since 1982 (NCES, 2007) and it is estimated that approximately 530,000 students took a calculus course in high…
This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We…
The notion of receptiveness arises in the pi-calculus as a guarantee of determinacy in the behaviour of callable entities and was first investigated by Sangiorgi. The DpiF process calculus, introduced by Francalanza and Hennessy, extends the pi-calculus with located processes and location and link...
Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student…
We derive a numerical method method is then derived by using the framework provided by DEC methodFull Text Available The were participated in the study. This study found that students who taught by using Microsoft Mathematics had higher achievement and has a positive effect on students' confidence of mathematicsTranslation of our paper "Ein Redehandlungskalkül. Ein pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at . Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations ...
The Quality Calculus uses quality binders for input to express strategies for continuing the computation even when the desired input has not been received. The Stochastic Quality Calculus adds generally distributed delays for output actions and real-time constraints on the quality binders for inp...... based on stochastic model checking and we compute closed form solutions for a number of interesting scenarios. The analyses are applied to the design of an intelligent smart electrical meter of the kind to be installed in European households by 2020....
, that supports incremental and contextual reasoning with equality and fixpoints in the setting of linear logic. This system allows deductive and computational steps to mix freely in a continuum which integrates smoothly into the usual versatile rules of multiplicative-additive linear logic in deep......The standard proof theory for logics with equality and fixpoints suffers from limitations of the sequent calculus, where reasoning is separated from computational tasks such as unification or rewriting. We propose in this paper an extension of the calculus of structures, a deep inference formalism...
Full Text Available The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quantum entanglement. In particular the graphical calculus of complementary observables and corresponding phases due to Duncan and one of the authors is used to construct representative members of the two genuinely tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely illustrates the respectively pairwise and global tripartite entanglement found in the W- and GHZ-class states. A new concept of supplementarity allows us to characterise inhabitants of the W class within the abstract diagrammatic calculus; these method extends to more general multipartite qubit states.
Full Text Available We develop a timed calculus for Mobile Ad Hoc Networks embodying the peculiarities of local broadcast, node mobility and communication interference. We present a Reduction Semantics and a Labelled Transition Semantics and prove the equivalence between them. We then apply our calculus to model and study some MAC-layer protocols with special emphasis on node mobility and communication interference. A main purpose of the semantics is to describe the various forms of interference while nodes change their locations in the network. Such interference only occurs when a node is simultaneously reached by more than one ongoing transmission over the same channelWe study the problem of defining normal forms of terms for the algebraic -calculus, an extension of the pure -calculus where linear combinations of terms are first-class entities: the set of terms is enriched with a structure of vector space, or module, over a fixed semiring. Towards a solution to the problem, we propose a variant of the original reduction notion of terms which avoids annoying behaviours affecting the original version, but we find it not even locally confluent. Finally, we co...
This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But thi...This paper studies how to describe the real-time behaviour of programs using duration calculus. Since program variables are interpreted as functions over time in real-time programming, and it is inevitable to introduce quantifications over program variables in order to describe local variable declaration and declare local channel and so on. Therefore to establish a higher-order duration calculus (HDC) is necessary. We first establish HDC, then show some real-time properties of programs in terms of HDC, and then, prove that HDC is complete on abstract domains under the assumption that all program variables vary finitely.
This book, for the first time, provides laymen and mathematicians alike with a detailed picture of the historical development of one of the most momentous achievements of the human intellect ― the calculus. It describes with accuracy and perspective the long development of both the integral and the differential calculus from their early beginnings in antiquity to their final emancipation in the 19th century from both physical and metaphysical ideas alike and their final elaboration as mathematical abstractions, as we know them today, defined in terms of formal logic by means of the idea of a
A main challenge in the development of distributed systems is to ensure that the components continue to behave in a reasonable manner even when communication becomes unreliable. We propose a process calculus, the Quality Calculus, for programming software components where it becomes natural to plan...... for default behaviour in case the ideal behaviour fails due to unreliable communication and thereby to increase the quality of service offered by the system. The development is facilitated by a SAT-based robustness analysis to determine whether or not the code is vulnerable to unreliable communication...
We present a Mobile-Ambients-based process calculus to describe context-aware computing in an infrastructure-based Ubiquitous Computing setting. In our calculus, computing agents can provide and discover contextual information and are owners of security policies. Simple access control to contextual...... information is not sufficient to insure confidentiality in Global Computing, therefore our security policies regulate agents' rights to the provision and discovery of contextual information over distributed flows of actions. A type system enforcing security policies by a combination of static and dynamic...
Offering a fresh take on laser engineering, Laser Modeling: A Numerical Approach with Algebra and Calculus presents algebraic models and traditional calculus-based methods in tandem to make concepts easier to digest and apply in the real world. Each technique is introduced alongside a practical, solved example based on a commercial laser. Assuming some knowledge of the nature of light, emission of radiation, and basic atomic physics, the text:Explains how to formulate an accurate gain threshold equation as well as determine small-signal gainDiscusses gain saturation and introduces a novel pass
The purpose of this study was to investigate the ways in which a multi-layered women's calculus course influenced the participants' learning of mathematics. This study, conducted in a state university in the Midwestern region of the United States, revealed not only that women in this particular section of calculus were likely to select careers that involved mathematics, but that the focus on peer support, psychosocial issues such as self-confidence, and pedagogy helped the young women overcome gender barriers, as well as barriers of class, poverty, and race. In this article we provide some of the relevant quantitative statistics and relate the stories of two particular women through excerpts from interviews, student artefacts, and participant observation data. We selected these young women because they faced multiple barriers to success in Calculus I and might not have completed the course or taken additional mathematics courses without the support structures that were fundamental to the course.
Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus, including:- Limits of a function- Derivatives of a function- Monomials and polynomials- Calculating maxima and minima- Logarithmic differentials- Integrals- Finding the volume of irregularly shaped objectsBy breaking down challenging concepts a
Enables readers to apply the fundamentals of differential calculus to solve real-life problems in engineering and the physical sciences Introduction to Differential Calculus fully engages readers by presenting the fundamental theories and methods of differential calculus and then showcasing how the discussed concepts can be applied to real-world problems in engineering and the physical sciences. With its easy-to-follow style and accessible explanations, the book sets a solid foundation before advancing to specific calculus methods, demonstrating the connections between differential calculus
Full Text Available This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.
The paper reports on first preliminary results and insights gained in a project aiming at implementing the fluent calculus using methods and techniques based on binary decision diagrams. After reporting on an initial experiment showing promising results we discuss our findings concerning various techniques and heuristics used to speed up the reasoning process.Near the conclusion of their final term in the calculus sequence at The United States Military Academy, cadets are given a week long group project. At the end of the week, the project is briefed to their instructors, classmates, and superior officers. From a teaching perspective, the goal is to encapsulate as much of the course as possible in one… | 677.169 | 1 |
NOVA leads viewers on a mathematical mystery tour–a provocative exploration of math's astonishing power across the centuries. We discover math's signature in the swirl of a nautilus shell, the whirlpool of a galaxy, and the spiral in the center of a sunflower. Math was essential to everything from the first wireless radio transmissions to the successful landing of rovers on Mars. But where does math get its power?
Matrix Algebra usually gives students problems in the beginning because although it has applications in algebra, it looks completely different from any algebra the student has used up to this point. The material on these DVDs is covered in most advanced high school algebra courses and is definitely covered in a university linear algebra course.
This DVD teaches students how to easily tackle basic math word problems, and builds upon the foundation laid by the 1st - 7th Grade Math Tutor DVD. After students learn a math skill such as multiplication or division, many are frequently confused on how to apply these skills to solve word problems. Word problems present the problem to be solved in sentence form, and in these types of problems the student must pull the information out of the problem and decide the best way to solve it. The only way to get good at solving these types of problems is to practice, and that is what this 8 hour DVD course provides.
Algebra 1 is one of the most intimidating subjects for math students. The reason is that prior to this point students have been dealing with numbers such as "3" and "43" and now suddenly in Pre-Algebra And Algebra 1 they begin to deal with variables such as "x" and "y". This concept at first can seem a bit daunting with all of those letters running around!
In this course, you will learn all of the old and modern security systems that have been used and are currently being used. You also learn how to crack each one and understand why certain security systems are weak and why others are strong. We will even go into RSA, AES and ECC which are the three main modern cryptosystems used today | 677.169 | 1 |
Definition of a Derivative - Using the Rule of 4
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This activity incorporates the graphing calculator to help students understand the definition of a derivative. Students study the definition of a derivative graphically, numerically, verbally and analytically or use the rule of 4 | 677.169 | 1 |
This is old but good. It has a similar concept to "Numerical Recipes" by Press, et. al. but covers less ground, and at a more elementary level. Contrary to another reviewer, it does give plenty of examples at an elementary level, which I find quite helpful.
University mathematics departments have for many years offered courses with titles such as Advanced Calculus or Introductory Real Analysis. These courses are taken by a variety of students, serve a number of purposes, and are written at various levels of sophistication. The students range from ones who have just completed a course in elementary calculus to beginning graduate students in mathematics.
This book is a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating vector algebra. | 677.169 | 1 |
Algebra by Design
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Algebra by Design contains 40 activity pages covering topics such as solving equations with a single variable, multiplying and dividing monomials, and solving word problems. The book employs a search-and-shade technique that rewards students for their efforts, while allowing them to self-check their work. Each page contains exercises with shading codes that students use to shade a grid labeled with the answers. If the answers are correct, a symmetrical design emerges. Teachers have permission to copy the pages for classroom use and an answer key is provided. | 677.169 | 1 |
Courses in the Math Emporium
Developmental Math Courses (Residential)
Math 100: Fundamentals of Mathematics
A review of basic arithmetic and elementary algebra. Open to all students but required of students with low scores on Liberty University placement tests and inadequate preparation in mathematics. A grade of C or better is required in order to go on to a higher-numbered mathematics course. This course may not be used in meeting General Education requirements in mathematics. (3 credit hours)
Math 110: Intermediate Algebra
Review of exponents, polynomials, factoring, roots and radicals, graphing, rational expressions, equations and inequalities, systems of linear equations and problem solving. This course may not be used to meet the General Education requirement. (3 credit hours)
General Studies Math Courses (Residential)
MATH 115: Mathematics for Liberal Arts
Prerequisite: MATH 110, minimum grade of "C". A survey course for liberal arts majors including a review of algebra and an introduction to logic, probability and statistics, mathematical structure, problem solving, number theory, geometry and consumer applications.
Online Course Offerings
The Distance Learning mathematics courses all use the MyMathLab computer learning system. All work (homework, quizzes, and tests) are done in the MyMathLab system. The textbook and the MyMathLab tutorials and other resources are available to help the students to learn the material and each class is proctored by a mathematics teacher who can give assistance as needed. Only students in residential MATH 100, 110, 115, 116, 121, 201 and BUSI 230 courses may use the Math Emporium.
MATH 100, 110, 115, 116, 121, 201 and BUSI 230 are currently offered through Liberty University Online. Residential students may not take MATH 100 or MATH 110 online. | 677.169 | 1 |
The only book of its kind, the author presents complete, thorough units that use such motivating methods as workshops, cooperative learning groups, and laboratory group experiments to promote active learning in which students do math by manipulating real world objects. Each chapter presents a complete, class tested lesson showing how a particular manipulative can be incorporated to teach mathematics in the classroom.
Book Description Allyn & Bacon, 1991. Book Condition: GOOD. book was well loved but cared for. Possible ex-library copy with all the usual markings and stickers. Some light textual notes, highlighting and underling. Bookseller Inventory # 2640423971 | 677.169 | 1 |
Prepare to Complete the Square Chart
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0.02 MB | 2 pages
PRODUCT DESCRIPTION
Students who have historically struggled in math may find it difficult recognizing the relationships between a perfect square trinomial and a binomial squared. This chart helps them understand the connections. When it's time to completing the square, re-writing a prefect square trinomial should be easier60. | 677.169 | 1 |
Product Description
Jacobs' Geometry: Seeing, Doing, Understanding, 3rd Edition features a unique, proof-based approach to geometry that integrates discussions, cartoons, anecdotes, examples, and plenty of exercises. With a greater emphasis on problems, rather than long introductory instructions, this is a great text for learning-through-doing. Sixteen units plus a final review are included and together cover: lines, angles, direct and indirect proofs, congruence, inequalities, parallel lines, quadrilaterals, transformations, area, similarity, triangles, circles, concurrence theorems, non-Euclidean geometries, and more. A section for the postulates and theorems is included in the back of the book, along with answers to selected exercises. 780 pages with full-color illustrations; indexed; hardcover. Non-consumable text. 3rd Edition. Grades 7-12. | 677.169 | 1 |
TI-30Xa is an easy way to get into scientific calculators. It has a full suite of basic scientific and trigonometric functions suitable for use in general math, pre- algebra, algebra 1&2, trigonometry and biology classes. It can do basic fractional functions in standard numerator/denominator format so you don't have to worry about converting to decimal notation. Its 10 digit display also has a 2 digit exponent display.
Powerful Functionality The Texas Instruments TI-30Xa calculator combines fraction capabilities with basic scientific and trigonometric functions and more to help students explore math and science concepts | 677.169 | 1 |
About this product
Description
Description
A teachers' book for mathematics covering 'Platonic Solids' and 'Rhythm and Cycles' which includes full colour illustrations and diagrams throughout. A resource for Steiner-Waldorf teachers for maths for Class 8 (age 13-14).
Author Biography
John Blackwood, who died in 2015, worked in mechanical engineering design for nearly 30 years and was inspired by Lawrence Edwards' work with plant geometry. He became a teacher at the Glenaeon Rudolf Steiner School in Sydney, Australia. There he designed a maths course for Classes 11 and 12 which was accepted by the school board of New South Wales. He lived in Sydney and was also the author of Geometry in Nature (Floris Books). | 677.169 | 1 |
Introduction to Integration (Integral Calculus)
A great way to start learning Calculus through video lectures and quizzes an introductory course on Integral Calculus. It comprises of a total of 5 hours of videos and quizzes. This is perfect for secondary school students seeking a good primer on Integral Calculus. It is also great as a refresher for everyone else. However prior knowledge in Differential Calculus is a MUST before learning this topic.
The course is arranged from the very basic introduction and progresses swiftly with increasing depth and complexity on the subject. It is recommended that the students do not skip any part of the lectures, or jump back and forth, because good understanding of the fundamental is important as you progress.
Quizzes are included on 7 subtopics to strengthen your understanding and fluency on this topic. So it is advisable that you attempt all the questions.
The course is delivered by an experienced teacher with five years of experience teaching students on a one to one basis. The instructor understands the difficulties that students normally face to become competent in mathematics. So words and examples were carefully chosen to ensure that everybody gets the most out of this series of lectures. This is a MUST course for all secondary school students.
Have fun learning!
What are the requirements?
Students should already be familiar with Basic Algebra
Students should have good understanding of Differential Calculus
What am I going to get from this course?
Understand the relationship between Integral and Differential Calculus
Perform Integration on single variable polynomial expressions
Perform Integration on Composite Functions with Linear Factors
Perform Indefinite and Definite Integration
Determine Area Under a Curve using Definite Integral
Determine Volume of Revolution using Definite Integral
Who is the target audience?
Secondary and High Schools students taking Calculus 1 (Integral Calculus)
Students preparing for Calculus 1 (Integral Calculus) tests / exams at 'O' Level or its equivalence
This video describes how to estimate Volumes of Revolution by calculating the volumes of thin cylinders constructed inside the revolved area under a curve. This video is a sequence of the previous video.
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Instructor Biography
A Penang based Engineer cum Tutor, a father of four, with a passion to teach and help students excel in Math. He is also known as MrMaths by his students.
He was graduated with Bachelors Degree in Electronics Engineering from the University of Sheffield in England. He has fourteen years of working experience in electronics industry with Intel Corp and Dell Corp and years of Mathematics teaching and home tutoring experience. In the past he has also taught Physics. Today he is a full time Math tutor Since January 2012, he started tutoring outside of Penang and outside of Malaysia using Video Conferencing on the internet. Teaching in both Bahasa Melayu (Malay) and in English.
He is friendly and always treat his students as if they are his own children. | 677.169 | 1 |
In this undergraduate/graduate textbook, the authors introduce ODEs and PDEs through 50 class-tested lectures. Mathematical concepts are explained with clarity and rigor, using fully worked-out examples and helpful illustrations. Exercises are provided at the end of each chapter for practice.
Details
The importance of partial differential equations (PDEs) in modeling phenomena in engineering as well as in the physical, natural, and social sciences is well known by students and practitioners in these fields.
Details | 677.169 | 1 |
Calculus Review Find the Error Task Cards
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3.08 MB | 10 task cards pages
PRODUCT DESCRIPTION
Calculus Review
In this packet you will find 10 task cards that students can use to Find the Error. Each card contains a short calculus problem and a proposed solution. Students are expected to find the error in the solution and then give the correct solution.
The 10 cards contain problems about the following topics:
+ equation of a normal line
+ find the derivative of a trig function (chain rule)
+ find the value of the derivative that includes radicals
+ set up and find the volume of a space rotated around a line other than the x or y axis
+ an integral that needs to be solved by using u-substitution
+ find the maximum value of a function over a specific interval
+ tricky limit that is actually a derivative
+ set up and find the volume of a space defined by cross sections
+ find the area between two curves
+ use implicit differentiation to find the derivative
Correct answers provided.
I have also included an answer sheet that students can use | 677.169 | 1 |
Algebra refers to your ability to manipulate variables and unknowns based on rules and properties. Matrix algebra is extremely similar to the algebra you already know for numbers with a few important differences. What are these differences?
Algebra with Matrices
Addition and Subtraction
Two matrices of the same order can be added by summing the entries in the corresponding positions.
Two matrices of the same order can be subtracted by subtracting the entries in the corresponding positions.
Multiplication
You can find the product of matrix and matrix if the number of columns in matrix matches the number of rows in matrix . Another way to remember this is when you write the orders of matrix and matrix next to each other they must be connected by the same number. The resulting matrix has the number of rows from the first matrix and the number of columns from the second matrix.
To compute the first entry of the resulting matrix you should match the first row from the first matrix and the first column of the second matrix. The arithmetic operation to combine these numbers is identical to taking the dot product between two vectors.
The entry in the first row first column of the new matrix is computed as .
The entry in the second row first column of the new matrix is computed as .
The entry in the first row second column of the new matrix is computed as
The entry in the second row second column of the new matrix is computed as
Continue this pattern and you will find that the solution to this multiplication is:
Other Properties of Matrix Algebra
Commutativity holds for matrix addition. This means that when matrices and can be added (when they have matching orders), then:
Examples
Example 1
Earlier, you were asked what the differences between matrix and regular algebra are. The main difference between matrix algebra and regular algebra with numbers is that matrices do not have the commutative property for multiplication. There are other complexities that matrices have, but many of them stem from the fact that for most matrices .
Example 2
Show the commutative property does not hold by demonstrating
Example 3
Compute the following matrix arithmetic: .
When a matrix is multiplied by a scalar (such as with ), multiply each entry in the matrix by the scalar.
Since the associative property holds, you can either distribute the ten or multiply by matrix next.
Example 4
Use your calculator to input and compute the following matrix operations.
Most graphing calculators like the TI-84 can do operations on matrices. Find where you can enter matrices and enter the two matrices.
Then type in the appropriate operation and see the result. The TI-84 has a built in Transpose button.
The actual numbers on this guided practice are less important than the knowledge that your calculator can perform all of the matrix algebra demonstrated in this concept. It is useful to fully know the capabilities of the tools at your disposal, but it should not replace knowing why the calculator does what it does.
Example 5
Matrix multiplication can be used as a transformation in the coordinate system. Consider the triangle with coordinates (0, 0) (1, 2) and (1, 0) the following matrix:
What does the new picture look like?
The matrix simplifies to become:
When applied to each point as a transformation, a new point is produced. Note that is a matrix representing each original point and is the new point. The is read as " prime" and is a common way to refer to a result after a transformation.
Notice how the matrix transformation rotates graphs in a counterclockwise direction .
The matrix transformation applied in the following order will rotate a graph clockwise .
Review
Do #1-#11 without your calculator.
1. Find . If not possible, explain.
2. Find . If not possible, explain.
3. Find . If not possible, explain.
4. Find . If not possible, explain.
5. Find . If not possible, explain.
6. Find . If not possible, explain.
7. Find . If not possible, explain.
8. Find . If not possible, explain.
9. Find . If not possible, explain.
10. Show that .
11. Show that .
Practice using your calculator for #12-#15.
12. Find .
13. Find .
14. Find .
15. Find .
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.4. | 677.169 | 1 |
Departments
Special Areas
Elementary Math Functions (EMF)
Course Description:
This course is an extension of the concepts learned in Algebra II and an introduction to pre-calculus concepts. The course is comprised of the following three units of study: algebra, trigonometry, and statistics. The algebra unit will include a study of linear relations and functions, systems of equations and inequalities, polynomial and rational functions, exponential and logarithmic functions, and the nature of graphs. The trigonometric unit will include a study of trigonometric functions, graphs and inverses of trigonometric functions, and trigonometric identities and equations. The statistics unit will include a study of descriptive statistics and probability. Real-world applications will be incorporated when appropriate. Students will use graphing calculators, computers, and other appropriate technology. There is no SOL test given for this course. | 677.169 | 1 |
Description The first of a two-course series covering the basics of algebra (MATH 065, 095). Topics include variables and their applications working with algebraic expressions, solving equations, and an introduction to graphing linear and quadratic functions. Prerequisite: Appropriate placement score or grade of C- or higher in MATH 050, or permission of Mathematics Department.
Intended
Skills and Attitude Outcomes
A. Perform basic arithmetic calculations without a calculator.
B. Determine if a number is divisible by 2, 3, 4, 5, 9, 10 using tests for divisibility.
C. Use the method of prime factorization to find the least common multiple.
D. Solve problems involving whole numbers, fractions, and decimals.
E. Compute perimeter and area of simply polygons including finding the perimeter and area of the classroom.
F. Interpret data from circle graphs, bar graphs, and histograms.
G. Find the median, mode and mean.
H. Convert within and between the Metric and English measuring systems.
I. Solve mathematical problems involving fractions, decimals, percentages, ratios and proportions.
J. Use estimate to determine reasonableness of problems.
K. Express mathematical ideas orally and in writing.
L. Apply interpersonal skills through collaborative work.
M. Devise appropriate responses to basic application problems.
N. Students will be able to add, subtract, multiply, and divide integers.
O. Translate algebraic equations into written form. | 677.169 | 1 |
Course Summary
This course can help prepare your students for the SBA Grade 6 Mathematics assessment with lessons on basic arithmetic, algebra, geometry, and statistics. The quizzes and tests can help you gauge your students' readiness for the SBA.
About This Course
In this series of lessons we cover all content areas expected to be tested in the Smarter Balanced Assessments (SBA) 6th grade mathematics assessment. Teachers and parents can have their students complete these courses to enhance and/or instruct them in the basic mathematical principles students are expected to know in the Common Core State Standards (CCSS). All lessons come with a fun, educational video containing instruction from an experienced mathematics teacher. Quizzes follow each lesson and every chapter, giving you many opportunities to assess your students' progress and readiness for the SBA while giving them practice in test-taking skills and the kinds of questions they will see during the SBA. The math topics covered in this course include:
Number theory, mathematical reasoning, and basic arithmetic
Fractions, percents, ratios, proportions and rates
Graphs and charts
Basic algebra and algebraic expressions
Perimeter and area of 2-dimensional objects
Surface area and volume of 3-dimensional objects
Basic statistics
About the Exam
The SBA is a comprehensive evaluation of students' academic competence relative to the CCSS. It consists of computer adaptive testing (CAT), performance tasks (PT), and classroom activities. The CAT is a system of multiple-choice questions which respond to the student's performance in real time, giving them more difficult questions as they answer correctly. The PT is also administered on a computer, but it is not adaptive and requires students to use critical thinking, research, analysis, and reasoning skills to answer complex questions dealing with real-world experiences. An additional in-class activity may also be performed to test hands-on performance in the subject. These assessments combined are called summative assessments. The assessments are not timed, allowing educators and parents to get a complete look at the student's knowledge, not their test-taking skills.
Preparing & Registering for the SBA Grade 6 Mathematics assessment
Parents and teachers can use this preparation and practice course to augment instruction received in regular 6th grade math classes. Students can review areas where they or their teachers have recognized a need for additional practice. Teachers can assign lessons and/or assessments as homework or present the lessons during their classroom instruction. There are tools within the lessons to assign work to students, report scores to teachers, and seek help from our instructors on difficult topics (available to paid subscribers only). The practice tests will give the students' parents and teachers a good idea of the students' understanding and synthesis of core concepts, pointing them toward areas where an individual student is struggling. There will be approximately 30-35 CAT questions, 2-7 PT questions, and 1 classroom activity taking about 2 hours, 1 hour, and 30 minutes to complete, respectively. The assessment is divided into four basic categories, described in detail below.
There is no individual registration process. Schools administering the SBA will do so according to their established schedule for all students. The assessment is completed at no cost to students or families.
Scoring the SBA Grade 6 Mathematics assessment
The SBA uses a scaled scoring system that is tied to level of academic achievement and achievement levels. Usually the scores are computed by a computer, and are intended to give an indication of a student's overall competence in the subject area relative to the CCSS standards. There are four achievement levels--which might loosely be described as ranging from novice to advanced--and they should be used not as a pass/fail assessment, but rather as a reference to be taken in the context of other, non-tested considerations of the students' development. The point-values of individual questions are determined by the difficulty of the question.
What you'll study: The initial chapters in this course have students work with basic arithmetic, mathematical reasoning, number theory, fractions, and ratios. Statistics and algebraic equations are covered in several chapters.
SBA Grade 6 Math Claim 2: Problem Solving
Estimated questions: 4-5
Exam topics covered: Applying math to problems of everyday life, strategic use of tools, contextual interpretation, identifying quantities in a situation and chart/graph the relationships between them.
What you'll study: Several lessons in this course explore the use of charts and graphs. The chapter on math reasoning and logic describes the tools used to analyze and interpret situations mathematically and contains a lesson on working with word problems. All lessons use real-world examples during the instruction.
SBA Grade 6 Math Claim 3: Communicating Reasoning
Estimated questions: 8-10
Exam topics covered: Testing propositions and conjectures, developing cases within arguments, refuting or justifying conjectures and propositions using reasoning, recognizing correct logic and describing logical flaws, describing logic being used, using solid referents as the foundation of arguments, determining when conditions do or do not apply to certain arguments.
What you'll study: The logic and math reasoning chapter discusses propositions, conjunctions, conditional statements, logical equivalence, and reasoning in mathematics, among other topics.
SBA Grade 6 Math Claim 4: Modeling and Data Analysis
Estimated questions: 4-6
Exam topics covered: Using mathematics to address everyday problems; using context to interpret results; creating chains of mathematical reasoning to justify models; identifying interpretations and solutions used to address complicated problems; developing, analyzing, and making improvements to models representing phenomena in the real world; describing logic and assumptions which are being used; charting, mapping, or graphing quantities to represent their relationships | 677.169 | 1 |
For a long time I have subscribed to the educational theory that can be summarized in the phrase, "Any way to get it done." The pedagogical method, whether it be formal textbook, comic book or play is irrelevant as long as the proper instruction with suitable retention takes place. That background belief must be an essential part of the reader's approach if they are to consider this book to be educationally effective.
Fred is the newborn baby of Mr. and Mrs. Gauss and he is a calculus genius from the moment he is born. Within a short time, Fred is exploring the concept of the function and discovering the principles of calculus from his baby crib. After he has thought through the concepts of a function Fred moves on to the limit, derivatives, integrals and all of the other topics such as conics, infinite series, and differential equations considered essential coverage in a two-semester calculus sequence.
It is all presented in a whimsical manner, for example at the age of nine months Fred is teaching calculus at KITTENS (Kansas Institute for Teaching Technology, Engineering and Natural Sciences). Well-liked by his students, they never tell him that he should act his age, for they learn a great deal from his classes. There are references to the Kansas depicted by Baum, with Auntie Em, Uncle Henry and Dorofred included in the stories. There is a great deal of humor using wordplay and references to popular culture.
The obvious question that a reader of the review would ask is, "Can a student learn calculus from this book?" The answer depends on your definition of "learn calculus." If your definition is to learn the basics of calculus in a non-rigorous manner then the answer is clearly yes. However, if you mean a rigorous form where students learn specifics such as how to do an epsilon-delta proof then the answer is definitely no. Students will be amused when reading this book and the humor does not mask the serious learning of calculus.
The most astounding aspect of the book was found on page 138, where an eight-line program written in BASIC (with line numbers!) is given. It has been years since I have seen such a program in a recently published book. There are a few other uses of BASIC in the book so it is not a one-off joke, although to many people including BASIC is itself a joke.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business. | 677.169 | 1 |
Meadows or Malls?
Students explore the use of matrices and graphing calculators to solve a complex linear programming problem. They begin by reviewing homework problems in which they relate the solvability of a system of linear equations to the invertibility of the coefficient matrix. Students use graphing calulators to solve linear systems. | 677.169 | 1 |
Math for Liberal Arts Students is a textbook for students that are majoring in liberal arts. Topics include problem solving, logic, sets, numeration systems, measurement, geometry, statistics,... More > probability, normal distribution, and finance. A print copy of this book is available at a low cost on Lulu.com.< Less
Math for Liberal Arts Students is a textbook for students that are majoring in liberal arts. Topics include problem solving, logic, sets, numeration systems, measurement, geometry, statistics,... More > probability, normal distribution, and finance. This book is also available as an ebook at no cost on Lulu.com | 677.169 | 1 |
The third edition of this textbook continues to use an applications approach as the vehicle for explaining theoretical concepts, and promoting real-world problem solving in the exercises. It aims to use the visual nature of applications to bridge the gap between concrete and abstract understanding. A new early introduction of eigenvalues and eigenvectors is included, allowing students time to master these concepts, and there are optional graphing calculator sections at the end of the first two chapters, designed to introduce students who are already experienced with other uses of the graphing calculator to uses in linear algebra | 677.169 | 1 |
Synopsis
Interactive Book Foundation 1 by
ERA awards finalist 2011. Bring Yr 10 Foundation GCSE Maths to life with engaging and easy-to-use whiteboard resources. This interactive maths resource is packed full of extra practice, real-life examples, functional skills, exciting whiteboard lessons and an interactive Scheme of Work fully integrated into the Collins New GCSE Maths scheme. New GCSE Maths Interactive Book Foundation 1 will: * Inspire your students with interactive animations, video clips of students teaching a problem, relevant news clips, real-life maths, informative worked examples, PowerPoint presentations, practice exam questions and more * Adapt to different styles of learning with lots of audio and visual interactive book resources and study aids that launch directly from the Student Book pages onscreen * Personalise your maths ICT teaching by uploading your own resources and web-links directly into the Collins scheme * Integrate ICT into your lessons like a pro (without needing to be one!) with classroom-friendly software and resources which have been developed, written and tested by teachers for teachers * Get everything you need in one box: perfect for individual computers, networks and all VLEs (please note: the VLE version is not interactive, the VLE provides PDFs of the book along with the resources by lesson) * Plan your whole year with a fully flexible Interactive Scheme of Work linked to whiteboard lessons, web-links, external resources and the whole Collins scheme - or simply adapt Collins' suggested Scheme of Work to your teaching. | 677.169 | 1 |
Keep Learning
The majority of reviews on Amazon.com are from parents using the book to home school their children. There are several themes that run through the reviews, and many reviewers comment that the layout of the book is helpful, with well sized fonts and an engaging design. Reviewers also repeatedly mention that the book is easy to understand and use, particularly because of the answer key in the back. Some reviewers feel that the book is too easy and recommend looking through the problems to gauge if it is appropriate for the particular student.
One criticism of the book is that it does not have enough practice problems for students. Additionally, one reviewer states that the pages cannot be easily torn out, making the practice problems somewhat less convenient. One final issue to consider is McGraw-Hill's compatibility with Common Core principles. The book is not fully compatible with all Common Core requirements, which can be a positive or a negative issue depending on personal beliefs, but should be taken into consideration. | 677.169 | 1 |
Showing 1 to 3 of 3
Algebra is a class that you will never stop using. Even if you are not using the Pythagorean Theorem, you are still using the problem solving skills it taught you. This class is wonderfully detailed and tailored to help each student.
Course highlights:
Our class went over most of Algebra II in a crash course on the different chapters. The highlights of taking this course were that the teacher never got frustrated at a student for not understanding or that she would make sure that everyone had a fair chance to pass the class with an exceptional grade if you would put in the work.
Hours per week:
6-8 hours
Advice for students:
Make sure that you go in with a mentality that you will get an A. If you start slacking early on, it can be extreamly difficult to get back on your feet. This class may not be required, but I would definitely recommend it, to whom ever has math left to take.
Course Term:Fall 2016
Professor:Mrs.Noble
Course Tags:Math-heavyMany Small Assignments
May 23, 2016
| Would recommend.
Pretty easy, overall.
Course Overview:
Noble does a wonderful job of making sure everyone understands the information taught in this course.
Course highlights:
I was given a better understanding of what I had learned in previous classes, and was able to learn things that I had not understood in previous classes.
Hours per week:
0-2 hours
Advice for students:
Try not to miss a single assignment, missing even one can be fairly regrettable in the long run.
Course Term:Spring 2016
Professor:Mrs.Noble
Nov 08, 2015
| Would highly recommend.
Pretty easy, overall.
Course Overview:
College Algebra taken during high school is my most recommended class. This class is almost identical to your typical Honors Algebra II class. By basically reviewing and adding a bit more to each concept to the prior year, this course is "a piece of cake."
Course highlights:
The highlights of this course were learning how to break down and get a better understanding of how simple lessons tie into word problems. This course focused on taking what you've already learned and helping you truly understand how everything makes sense.
Hours per week:
0-2 hours
Advice for students:
Although many think that you can't study for a math class, you sure can! The way to study is to ask your teacher for practice problems a few days before your exam and holding yourself accountable for knowing how to do each type of problem. Never forget to complete your homework because it can bring your grade down rocket fast. | 677.169 | 1 |
Investigating Exponential Functions
Students explore exponential function through graphing and investigating patterns of graphs. They graph exponential functions and relate these functions to real world applications of functions. Afterward, they discuss and compare peer graphs | 677.169 | 1 |
Representing Patterns & Evaluating Expressions
Write algebraic expressions, determine patterns, and evaluate expressions in a real-world context. Learners engage in a series of collaborative activities to identify, model, and give variables for real-world patterns. They write algebraic expressions to match each situation and use their graphing calculator to find the nth term in a | 677.169 | 1 |
The study and application of N -body problems has had an
important role in the history of mathematics. In recent years, the
availability of modern computer technology has added to their
significance, since computers can now be used to model material bodies
as atomic and molecular configurations, i.e. as N -body
configurations. more...
Turbulence is the most fundamental and, simultaneously, the most
complex form of fluid flow. However, because an understanding of
turbulence requires an understanding of laminar flow, both are
explored in this book.
Groundwork is laid by careful delineation of the necessary physical,
mathematical, and numerical requirements for the studies... more...
This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic systems. The examples are carefully explained and compiled... more...
In this book the author has tried to apply "a little imagination and thinking" to modelling dynamical phenomena from a classical atomic and molecular point of view. Nonlinearity is emphasized, as are phenomena which are elusive from the continuum mechanics point of view. FORTRAN programs are provided in the Appendices. more...
Computer-Oriented Mathematical Physics describes some mathematical models of classical physical phenomena, particularly the mechanics of particles. This book is composed of 12 chapters, and begins with an introduction to the link between mathematics and physics. The subsequent chapters deal with the concept of gravity, the theoretical foundations... more...
Arithmetic Applied Mathematics deals with concepts of arithmetic applied mathematics and uses a computer, rather than a continuum, approach to the deterministic theories of particle mechanics. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics using only arithmetic. Definitions of energy and... more...
Arithmetic Applied Mathematics deals with the deterministic theories of particle mechanics using a computer approach. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics with the aid only of arithmetic. The computational power of modern digital computers is highlighted, along with simple models... more... | 677.169 | 1 |
most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology. This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The four main chapters present the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find. Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and a full exposition of Steenrod squares and powers. Researchers will also welcome this aspect of the book. | 677.169 | 1 |
Mathematica 9 for Beginners
A course that teaches Mathematica 9 for beginners to the software. It teaches Mathematica for college and high school
3.9This is an example based course aiming to teach Mathematica at an understandable level to students in college. Advanced high school students, or students whose high school teaches Mathematica will also find this course invaluable.
It assumes no understanding of programming languages, although knowledge, even rudimentary, of C/C++/Java is a plus.
This course does NOT teach Mathematica as a programming language.
This course does NOT teach Mathematics in general. Although a textbook on Pure Mathematics will come in handy for reference.
Any high school textbook that teaches college level Pure Mathematics is recommended. College students may use their course textbooks recommended by their Professor.
Curriculum
This video, introduces who I am [Shakil Rafi] and to what Mathematica is.
Here I explain the fact that chances are your college/university will have a site license. What that means is that your college has probably bought a bulk license for Mathematica, and its students can access it for the low, low price of zero, ask your academic advisor.
If your college does not have a site license, you can still get it though, but you will need to verify your studenthood by entering your college email id: joesixpack@somerandomcollege.edu
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Instructor Biography
I am a Math Major at Troy University in Alabama. I am currently a Junior and I have been using software to do my math for me for quite a while. Partly because its easy and partly because my course requires it. I will be instructing you on Mathematica 9. Have a blast ! | 677.169 | 1 |
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Show More differential equations using Jordan normal form). The first three chapters address the basics: matrices, vector spaces, and linear transformations. The next three cover eigenvalues, Euclidean inner products, and Jordan canonical forms, offering possibilities that can be tailored to the instructor's taste and to the length of the course. Bronson's approach to computation is modern and algorithmic, and his theory is clean and straightforward. Throughout, the views of the theory presented are broad and balanced. Key material is highlighted in the text and summarized at the end of each chapter. The book also includes ample exercises with answers and hints. With its inclusion of all the needed features, this text will be a pleasure for professionals, teachers, and students. . Introduces deductive reasoning and helps the student develop a familiarity with mathematical proofs. Gives computational algorithms for fi nding Eigenvalues and Eigenvectors. A balanced approach to computation and theory. Exercise sets ranging from basic drill to theoretical/challenging. Useful and interesting applications not found in other introductory linear algebra | 677.169 | 1 |
Children love sculpting clay or building sand castles, creating objects in three dimensions before they have the motor skills to draw in two dimensions. Similar arguments applied to the study of curves and graphs in high school mathematics would suggest that students' work and calculation with shapes should move sequentially from concrete to more abstract thought. This "Technology Tips" article introduces three-dimensional algebraic surfaces to ninth-grade students, using Surfer®, open-source software that is available at Imaginary ( The immediate visual appeal of algebraic surfaces is apparent, and the developments in technology have made them easily accessible. Teachers found that students' attempts to gain control of the images created in playful exploration drive them to want to understand the underlying mathematics. A bibliography is included. | 677.169 | 1 |
"CliffsQuickReview Trigonometry" provides you with all you need to know to understand the basic concepts of trigonometry -- whether you need a supplement to your textbook and classes or an at-a-glance reference. Trigonometry isn't just measuring angles; it has many applications in the real world, such as in navigation, surveying, construction, and many other branches of science, including mathematics and physics. As you work your way through this review, you'll be ready to tackle such concepts asTrigonometric functions, such as sines and cosinesGraphs and trigonometric identitiesVectors, polar coordinates, and complex numbersInverse functions and equations
You can use "CliffsQuickReview Trigonometry" in any way that fits your personal style for study and review -- you decide what works best with your needs. You can read the book from cover to cover or just look for the information you want and put it back on the shelf for later. Here are just a few ways you can search for topics: Use the free Pocket Guide full of essential informationGet a glimpse of what you'll gain from a chapter by reading through the Chapter Check-In at the beginning of each chapterUse the Chapter Checkout at the end of each chapter to gauge your grasp of the important information you need to knowTest your knowledge more completely in the CQR Review and look for additional sources of information in the CQR Resource CenterUse the glossary to find key terms fast
With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades David A. Kay
Dave Kay is a writer, engineer, and aspiring naturalist and artist, combining professions with the same effectiveness as his favorite business establishment, Acton Muffler, Brake, and Ice Cream (now defunct). Dave has written more than a dozen computer books, by himself or with friends. His titles include various editions of Microsoft Works For Windows For Dummies, WordPerfect For Windows For Dummies, Graphics File Formats, and The Internet Complete Reference.
David A. Kay currently resides in the state of California. David A. Kay was born in 1940.
David A. Kay has published or released items in the following series...
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This book seems to be pretty nice, until you start finding all the super horrible mistakes that are thrown in ever couple of pages. They are all do the this book not being edited, and make it a hassle to read as you have to second guess everything that is written. I got about half way done and the mistakes turned me off so bad I put the book away and don't intend to read from it again. This book could have been a real nice book but the mistakes are just too horrible to look over.
Handy Little Summary Mar 3, 2004
After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."
Great series... Great book. Jun 26, 2001
I'm a (returning :P) university Freshman preparing for the College Board CLEP tests. I was already familiar with the material covered in this book, but needed to refresh my memory. This review turned out to be *exactly* what I needed.
The author's ability to explain the material to the student are just shy of enlightening. The discussions & theorem proofs are written in a very concise, clear style.
I'm a big advocate of the Cliff's QuickReview series. Intended as a course supplement, these books are also *GREAT* for students wanting to refine their skills. Most of them are also very accessible to students with less familiarity on the subject; trying to learn it for the first | 677.169 | 1 |
Synopsis
Oxford GCSE Maths for Edexcel: Higher Teacher's Guide by Chris Green
This book has been specifically written for the new two-tier Edexcel linear GCSE specification for first examination in 2008. The book is targeted at the D to B grade range in the Higher tier GCSE, and it comprises units organised clearly into lesson plans. Each unit offers: BLPrior knowledge identified at the start so teachers are fully prepared to teach the topic BLTeaching objectives identified so it is clear what students need to know BLUseful resources suggested, including links to the interactive CD-ROM, to help teachers plan engaging lessons BLPractical support for the lesson, including mental starters and plenaries BLExercise commentary so that teachers can differentiate effectively even within ability groups BLWorked solutions to past Edexcel questions so that teachers can help students revise This guide forms part of a series of four teacher guides at GCSE, in which the other three books cater for grade ranges G to E, E to C and B to A*. | 677.169 | 1 |
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Introduction To Mathematical Reasoning
Discover Introduction To Mathematical Reasoning book by from an unlimited library of classics and modern bestsellers book. It's packed with amazing content and totally free to try.
Introduction To Mathematical Reasoning
An Introduction to Mathematical Reasoning [Peter J. Eccles] on Amazon.com. *FREE* shipping on qualifying offers. This book eases students into the rigors of In this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku. You will learn the importance of tenacity in approaching mathematical Title: An Introduction to Mathematical Reasoning: numbers, sets and functions Author: Peter J. Eccles Created Date: 10/16/2007 2:00:22 PMGrade 7 » Introduction Print this page. In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional Grade 6 » Introduction Print this page. In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the Mathematical induction is a mathematical proof technique, most commonly used to establish a given statement for all natural numbers, although it can be used to prove Twenty Questions about Mathematical Reasoning : Lynn Arthur Steen, St. Olaf College: The concluding chapter in NCTM's 1999 Yearbook which is devoted to mathematical UW TACOMA Division of Sciences and Mathematics MATHEMATICS – TACOMA Detailed course offerings (Time Schedule) are available for. Winter Quarter 2016Maths Mathematical Reasoning part 1 (Statement and Negation) CBSE class 11 Mathematics XI | 677.169 | 1 |
Mathematics A Level
Course introduction
Mathematics is an important subject that underpins virtually all human endeavours, including: science, technology, business and finance.
As a result, an A Level in Maths provides you with a highly regarded qualification that can support you in further study or employment.
On this course you will have a chance to study both pure and applied maths.
The focus of pure maths is very much on problem solving. Many students enjoy the challenge of solving pure maths problems and the satisfaction of finding a precise solution.
Meanwhile, applied maths will give you a taste for how mathematics is used in understanding and modelling the real world.
Why choose Brock for Mathematics A Level?
At Brock we have a team of highly experienced and dedicated teachers who specialise in teaching A Level Maths and Further Maths.
You will learn in dedicated, high quality maths teaching rooms at our new state-of-the-art, IBM-sponsored STEM building.
These rooms include a range of traditional and high-tech teaching technologies to support your learning. Students who need extra help or who want to push themselves further can attend extra workshops and/or use our quiet study area.
Teaching & learning
In addition to traditional classroom learning, we give you extra opportunities for support and enrichment.
Indeed, every year Brock students take part in the regional Senior Team Maths Challenge competition, pitting themselves against teams from other local schools and colleges.
We also facilitate trips further afield, including visits to central London and elsewhere for stimulating talks by leading researchers.
Minimum entry requirements
Five GCSEs at grade A*-C or grade 4 to include:
• Maths grade A (7)
Other GCSEs mostly at A and B grades.
What can I do with this qualification?
A Level Mathematics is a highly regarded qualification that can support you in higher level study or employment.
Furthermore, it is essential for many university degrees such as Physics, Engineering, Computer Science and Architecture, as well as Mathematics itself.
It is also highly desirable for other degree courses such as Chemistry and Economics.
Finally, this qualification shows universities and employers that you have the abstract reasoning and problem solving skills that are required in many professions. | 677.169 | 1 |
Multivariable Calculus in the Lab
A collection of Maple V R3 Worksheets
Used in Math 222 at Cornell, Fall 1994
Some Maple Release 7 Versions (and Related) Worksheets
are now available here.
This collection of worksheets was written for Maple V R3 on the Macs. It was
a first time using these materials, so there are rough edges
to be improved upon.
The Mac interface to Maple is a very nice one, although significantly
different from the Unix xmaple and Windows interfaces. In particular it
supports the pasting of color pictures into worksheets. Also, these worksheets
use multi-line input at times, and hence rely on the Mac option that
only the enter key sends input to Maple.
An introduction to three dimensional graphs. Lots of
examples of contours. Includes some cases where computer
pictures are misleading. Hyperboloids of one and two
sheets with an opportunity to look at sections and what they
tell us about graphs.
This worksheet uses a variety of graphical and numeric methods
to study multi-dimensional limits. It encourages people to continue
thinking about the graphical elements of functions of several variables.
Much attention is placed on examples where limits fail to exist. Looking at the graph of a function over a small domain is perhaps the most naive
strategy introduced. Techniques such as restriction to families of
lines and curves (algebraically, by 2D graphs or by animations) are also used. Subtleties involving loss of precision due to catastrophic cancellation
also arise.
Some examples of tangent lines and planes
graphically. Efficient generation in Maple of the linear approximation
of a function at a point. Use of the transform plots package to geometrically
compare the mapping properties of a function with those of its linear
approximation.
This worksheet looks geometrically at three
different kinds of singularities. A high point is the sketch of an argument at
the end about why the cube root of (x^3-3xy^2) fails to be differentiable
at the origin. A good opportunity to talk about the idea of what it means
for an approximation to be linear.
Instructions for using the transform plots package.
Efficient generation in Maple of the linear approximation at a point of a
transformation from R^2 to R^2. Comparison of the mapping properties of
a transformation with those of its derivative.
This worksheet draws some curves and Frenet frames. It shows how
readily differentiable curves can develop self-intersections and corners. The
Frenet frame package is illustrated and used to draw some Frenet frames for
a helix.
A real "kitchen sink" worksheet. Covers both major
Maple usage issues and mathematical content ones. Most worksheeets
prior to this one were geometric in focus. This one pointed out in passing the
power of the algebraic paarts of the system and attempted to use them
non-trivially. In the usage category here are partial derivatives, vector
differential operators, and the important map function. Three vector identities
are proved; one for scalar triple product and two for the divergence and
curl of cross products. The transformation to spherical coordinates is then
studied and the natural orthonormal moving frame produced. Finally these
are used to compute a formula for gradient in spherical coordinates.
Illustrates the use of the Maple command fsolve for numerically solving algebraic and transcendental equations or systems
of equations. Mostly a "how to do it in Maple" worksheet. Covers one
variable, multi-variable, roots in specified rectangles, and complex
solutions.
The major mathematical focus here is to illuminate
the relationship of curl and divergence to geometric properties of
the flow of a vector field. The worksheet also shows how to compute
numerical solutions to systems of differential equations, use the
draw flows package, and piece the results together into a quicktime movie.
Linear vector fields in the plane are most of the examples here. One
nonlinear vector field is also contrasted with its linearization.
Fieldplot and Numerical ODE's
While specialized programs such as MacMath and DsTool do a better job
at integrating systems of differential equations, there are advantages
to the integrated environment of Maple. This worksheet shows how to
numerically integrate an initial value problem, and graph the solutions
(two coordinates at a time) to a system of differential equations.
It also shows how to use Maple to draw gradient fields and
slope fields of planar vector fields.
A list of vector fields for the
students to quickly explore with MacMath. The students did this for
about 5 or 10 minutes during the lecture. (MacMath is of course much better
at generating phase plane pictures than Maple.)
Uses least squares to fit data in the file "Plane Data 0 "
to a plane z = a x + b y + c. Readily adaptable as a template for
other least squares problems. Also covers the syntax point
of reading data in from an external file.
Explores multivariate Taylor series. Compares graphically
various Taylor polynomials to the original function. Investigates how
errors depend upon the size of the region on which the approximation
is being used. Uses estimates on the size of partial derivatives to
bound errors in Taylor polynomial approximation. (These latter
are obtained from Maple graphs of the partial derivatives.)
This worksheet explores the relationship between symmetric matrices and quadratic forms. It shows how the eigenvalues of such a matrix relate to the geometric character of the graph of the quadratic form. It also discusses in the context of an example how the eigenvectors of the symmetric matrix determine a rotation of coordinates making the quadratic form diagonal.
Extrema for a quadratic form are sought numerically along the intersection of an ellipsoid with a hyperboloid.
Intended to show people how Maple can support a generic Lagrange
multiplier problem numerically.
This worksheet studies geometric behavior near singular points for a mapping of the plane to itself. Relationships with the problem of numerically solving for the inverse are discussed . The collpase of areas near singular points is brought out by the Transform Plots package. And the development of a cusp as the image of a smooth curve is analyzed in detail within the context of an example.
The original routine for displaying regions of integration
in the plane. Can be used as a check on conventional techniques. This version draws Maple curves to describe the region. To improve performance in exchange for lower quality, this is to be replaced by a polygon based version like plot_region_3D.
A package to show how elementary 3D regions
of integration appear. In the interests of better performance, a polyhedral
approximation is displayed. Default is a fairly crude but quick picture.One
can speciify higher resolution if one wants to improve the picture. Once one has generated the graphics structure for the entire volume, one can quickly inspect the "pieces" joining together to assemble it. The package is based on a natural map from the unit cube to an elementary region.
A variety of parameterization and reparameterization examples are presented. Many are quadric surfaces. Singularities of parameterizations are discussed. The worksheet also presents the natural frame (T_u, T_v, N) associated with a parameterization.
Line Integrals
Line integrals of planar vector fields along piecewise polygonal curves
are explored. There are routines both for numerical computation and display
of paths and vector fields. The example of the gradient of the polar
coordinate angle function is discussed, as is its connection with rotation
numbers of curves.
This worksheet does a variety of things related to
Green's theorem for an infinitesimal triangle. Using Taylor series, it shows
how to calculate the line integral of a vector field over a line segment to
second order. Part of the interest here is noting how easily the method generalizes to produce higher order formulas. The result is also used to
show that Green's theorem holds to second order for an infinitesimal triangle.
Because of the naturality of approximately subdividing fairly arbritrary
regions into infinitesimal triangles, an argument is briefly indicated by which one could use this result to give another proof of Green's theorem. | 677.169 | 1 |
Useful Math Software - MathType 6
MathType 6
I've been seeing a lot of maths questions in this forum section and seeing people type in plain text really make it for the answerer to read and understand. So I've downloaded a program called MathType, it's a piece of software that allows you to type any kind of mathematical equation or formula and then paste it into Microsoft Word or upload it here. It's simple and easy to use, no special knowledge is required. | 677.169 | 1 |
Mathematics For class XI
About the
Course
This would be 1 month FREE demo class for the students passing out standard X this year, 2014. The aim is to grow the genuine interest among the students to learning higher Mathematics and Statistics. To create a bridge between the mid school level mathematics to high school level mathematics is the goal to float such training course. The trainer has almost a decade of experience in teaching mathematics from middle school level till university level and he himself is holding M.Sc. in Maths & Computing from IIT and a PhD equivalent degree in Industrial Mathematics. The following are the salient features of this training course.
- Classes would be mostly during the weekend and twice in a week. It may float in the week days as well based on the students' suitability with the time
- Each class would be 1 hour lecture and one hour problem solving session.
- Lot of open book surprised quizzes of 15-20 minutes would be taken off for self-evaluation
- Each individual will get special attention after the class if require
- Students will get the flavor of real life application of higher mathematics
- Minimum assignments and home works would be provided. Each individual will get different problem to think during the week days.
- No burden of hard work, assignments, mental ennui related to the mathematical formulas. It is a fear free training workshop to fall in love with mathematics. Motivation to do maths is the motto.
- Each individual would be guided about his strong and weak areas to improve.
- Students would be encouraged to interact a lot with the instructor not only to ask their doubts but also to challenge in making things more clear
After the one month induction , the interested students will be allocated in batch of maximum size 6 to learn Mathematics and Statistics.
Who
should attend
Students passing out standard X from any board (Karnataka/CBSC/ISCE/IG/International board ) and aiming to go for science and commerce streams are most suitable ones.
Pre-requisites
No pre requisite
What
you need to bring
Pen and paper""
Key
Takeaways
- How to study higher mathematics
- Interest over the subject
- Why is it so important for higher studies
- To solve problem independently
- To develop self confidence
- To prepare for competitive exam of individual choice
Reviews
Recommendations & reviews from previous Customers:
Ishan Prabhakaranattended Mathematics For class XI
Questions and Comments
Thousands of experts Tutors, Trainers & other Professionals are available to answer your questionsDate and Time
Course Id: 16761
About the Trainer
Professional Doctorate in Industrial Statistics(TU/e),M.Sc in Maths and Computing (IIT)
Presently working as a business consultant based in Bangalore, India. More than 7 years of experience in solving various industrial problems using Applied Mathematics and Statistical techniques. Worked in Netherlands and Belgium for 2.5 years on various projects for biggies like Philips Medical System, Dow Chemicals, ASML, PSA-HNN, Bavaria Brewery. Have extensive experience in designing study material and delivering lectures on applying statistics in solving various business problems. | 677.169 | 1 |
...
Show More selected questions, and miscellaneous exercises are presented throughout. This is an invaluable text for students seeking a clear introduction to discrete mathematics, graph theory, combinatorics, number theory and abstract algebra | 677.169 | 1 |
Product Overview
This textbook is an introduction to the concept of factorization and its application to problems in algebra and number theory. With the minimum of prerequisites, the reader is introduced to the notion of rings, fields, prime elements and unique factorization. The author shows how concepts can be applied to a variety of examples such as factorizing polynomials, finding determinants of matrices and Fermats two-squares theorem. Based on an undergraduate course given at the University of Sheffield, Dr Sharpe has included numerous examples which demonstrate how frequently these ideas are useful in concrete, rather than abstract, settings. The book also contains many exercises of varying degrees of difficulty together with hints and solutions. Second and third year undergraduates will find this a readable and enjoyable account of a subject lying at the heart of much of mathematics. *Author: Sharpe, D. W./ Sharpe, David *Binding Type: Paperback *Number of Pages: 124 *Publication Date: 1987/08/28 *Language: English *Dimensions: 5.51 x 8.50 x 0.29 inches | 677.169 | 1 |
Send the Gift of Lifelong Learning!
Algebra IBecause algebra involves a new way of thinking, many students find it especially challenging. Many parents also find it to be the area where they have the most trouble helping their high-school-age children. With 36 half-hour lessons, Algebra I is an entirely new course developed to meet both these concerns, teaching students and parents the concepts and procedures of first-year algebra in an easily accessible way. Indeed, anyone wanting to learn algebra from the beginning or needing a thorough review will find this course an ideal tutor.
Conquer the Challenges of Learning Algebra
Taught by Professor James A. Sellers, an award-winning educator at The Pennsylvania State University, Algebra I incorporates the following valuable features:
Drawing on extensive research, The Great Courses and Dr. Sellers have identified the biggest challenges for high school students in mastering Algebra I, which are specifically addressed in this course.
This course reflects the latest standards and emphases in high school and college algebra taught in the United States.
Algebra I includes a mini-textbook with detailed summaries of each lesson, a multitude of additional problems to supplement those presented in the on-screen lessons, guided instructions for solving the problems, and important formulas and definitions of terms.
Professor Sellers interacts with viewers in a one-on-one manner, carefully explaining every step in the solution to a problem and giving frequent tips, problem-solving strategies, and insights into areas where students have the most trouble.
As Director of Undergraduate Mathematics at Penn State, Professor Sellers appreciates the key role that algebra plays in preparing students for higher education. He understands what entering college students need to have mastered in terms of math preparation to launch themselves successfully on their undergraduate careers, whether they intend to take more math in college or not. Professor Sellers is alert to the math deficiencies of the typical entering high school graduate, and he has developed an effective strategy for putting students confidently on the road to college-level mathematics.
Whatever your age, it is well worth the trouble to master this subject. Algebra is indispensible for those embarking on careers in science, engineering, information technology, and higher mathematics, but it is also a fundamental reasoning tool that shows up in economics, architecture, publishing, graphic arts, public policy, manufacturing, insurance, and many other fields, as well as in a host of at-home activities such as planning a budget, altering a recipe, calculating car mileage, painting a room, planting a garden, building a patio, or comparison shopping.
And for all of its reputation as a grueling rite of passage, algebra is actually an enjoyable and fascinating subject—when taught well.
Algebra without Fear
Professor Sellers takes the fear out of learning algebra by approaching it in a friendly and reassuring spirit. Most students won't have a teacher as unhurried and as attentive to detail as Dr. Sellers, who explains everything clearly and, whenever possible, in more than one way so that the most important concepts sink in.
He starts with a review of fractions, decimals, percents, positive and negative numbers, and numbers raised to various powers, showing how to perform different operations on these values. Then he introduces variables as the building blocks of algebraic expressions, before moving on to the main ideas, terms, techniques, pitfalls, formulas, and strategies for success in tackling Algebra I. Throughout, he presents a carefully crafted series of gradually more challenging problems, building the student's confidence and mastery.
After taking this course, students will be familiar with the terminology and symbolic nature of first-year algebra and will understand how to represent various types of functions (linear, quadratic, rational, and radical) using algebraic rules, tables of data, and graphs. In the process, they will also become acquainted with the types of problems that can be solved using such functions, with a particular eye toward solving various types of equations and inequalities.
Throughout the course, Professor Sellers emphasizes the following skills:
Using multiple techniques to solve problems
Understanding when a given technique can be used
Knowing how to translate word problems into mathematical expressions
Recognizing numerical patterns
Tips for Success
Algebra is a rich and complex subject, in which seemingly insurmountable obstacles can be overcome, often with ease, if one knows how to approach them. Professor Sellers is an experienced guide in this terrain and a treasure trove of practical advice—from the simple (make sure that you master the basics of addition, subtraction, multiplication, and division) to the more demanding (memorize the algebraic formulas that you use most often). Here are some other examples of his tips for success:
Learn the order of operations: These are the rules you follow when performing mathematical operations. You can remember the order with this sentence: Please Excuse My Dear Aunt Sally. The first letter of each word stands for an operation. First, do all work in parentheses; then the exponents; then multiplication and division; finally, do the addition and subtraction.
Know your variables: It's easy to make a mistake when writing an algebraic expression if you don't understand what each variable represents. Choose letters that you can remember; for example, d for distance and t for time. If you have sloppy handwriting, avoid letters that look like numbers (b, l, o, s, and z).
Use graph paper: You'll be surprised at how the grid of lines encourages you to organize your thinking. The columns and rows help you keep your work neat and easy to follow.
Pay attention to signs: Be very careful of positive and negative signs. A misplaced plus or minus sign will give you the wrong answer.
Don't mix units: If you are using seconds and are given a time in minutes, make sure to convert the units so they are all the same.
Simplify: Straighten out the clutter in an equation by putting like terms together. Constants, such as 7, -2, 28, group together, as do terms with the same variable, such as 3x, x, -10x. Then combine the like terms. Often you'll find that the equation practically solves itself.
Balance the equation: When you perform an operation on one side of an equation—such as adding or subtracting a number, or multiplying or dividing the entire side by a quantity—do the exact same thing to the other side. This keeps things in balance.
Above all, check your work! When you have finished a problem, ask yourself, "Does this answer make sense?" Plug your solution into the original equation to see if it does. Checking your work is the number one insurance policy for accurate work—the step that separates good students from superstar students.
By developing habits such as these, you will discover that solving algebra problems becomes a pleasure and not a chore—just as in a sport in which you have mastered the rudiments and are ready to face a competitor. Algebra I gives you the inspirational instruction, repetition, and practice to excel at what for many students is the most dreaded course in high school. Open yourself to the world of opportunity that algebra offers by making the best possible start on this all-important subject.
Hide Full Description
36 lectures
| 30 minutes each
Year Released: 2009
1
An Introduction to the Course
Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations. x
2
Order of Operations
The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now. x
3
Percents, Decimals, and Fractions
Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms. x
4
Variables and Algebraic Expressions
Advance to the next level of problem solving by using variables as the building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables, and operation symbols. Also learn some tricks for translating the language of problems (phrases in English) into the language of math (algebraic expressions). x
5
Operations and Expressions
Discover that by following basic rules on how to treat coefficients and exponents, you can reduce very complicated algebraic expressions to much simpler ones. You start by using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy. x
6
Principles of Graphing in 2 Dimensions
Using graph paper and pencil, begin your exploration of the coordinate plane, also known as the Cartesian plane. Learn how to plot points in the four quadrants of the plane, how to choose a scale for labeling the x and y axes, and how to graph a linear equation. x
7
Solving Linear Equations, Part 1
In this lesson, work through simple one- and two-step linear equations, learning how to isolate the variable by different operations. Professor Sellers also presents a word problem involving a two-step equation and gives tips for how to solve it. x
8
Solving Linear Equations, Part 2
Investigating more complicated examples of linear equations, learn that linear equations fall into three categories. First, the equation might have exactly one solution. Second, it might have no solutions at all. Third, it might be an identity, which means every number is a solution. x
9
Slope of a Line
Explore the concept of slope, which for a given straight line is its rate of change, defined as the rise over run. Learn the formula for calculating slope with coordinates only, and what it means to have a positive, negative, and undefined slope. x
10
Graphing Linear Equations, Part 1
Use what you've learned about slope to graph linear equations in the slope-intercept form, y = mx + b, where m is the slope, and b is the y intercept. Experiment with examples in which you calculate the equation from a graph and from a table of pairs of points. x
11
Graphing Linear Equations, Part 2
A more versatile approach to writing the equation of a line is the point-slope form, in which only two points are required, and neither needs to intercept the y axis. Work through several examples and become comfortable determining the equation using the line and the line using the equation x
12
Parallel and Perpendicular Lines
Apply what you've discovered about equations of lines to two very special types of lines: parallel and perpendicular. Learn how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves. Also try your hand at word problems that feature both types of lines. x
13
Solving Word Problems with Linear Equations
Linear equations reflect the behavior of real-life phenomena. Practice evaluating tables of numbers to determine if they can be represented as linear equations. Conclude with an example about the yearly growth of a tree. Does it increase in size at a linear rate? x
14
Linear Equations for Real-World Data
Investigating more real-world applications of linear equations, derive the formula for converting degrees Celsius to Fahrenheit; determine the boiling point of water in Denver, Colorado; and calculate the speed of a rising balloon and the time for an elevator to descend to the ground floor. x
15
Systems of Linear Equations, Part 1
When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions. x
16
Systems of Linear Equations, Part 2
Expand your tools for solving systems of linear equations by exploring the method of solving by elimination. This technique allows you to eliminate one variable by performing addition, subtraction, or multiplication on both sides of an equation, allowing a straightforward solution for the remaining variable. x
17
Linear Inequalities
Shift gears to consider linear inequalities, which are mathematical expressions featuring a less than sign or a greater than sign instead of an equal sign. Discover that these kinds of problems have some very interesting twists, and they come up frequently in business applications. x
18
An Introduction to Quadratic Polynomials
Transition to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic. Learn how to use the FOIL method (first, outer, inner, last) to multiply linear terms to get a quadratic expression. x
19
Factoring Trinomials
Begin to find solutions for quadratic equations, starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial (a binomial expression consists of two terms, a trinomial of three). Professor Sellers explains the tricks of factoring such expressions, which is a process almost like solving a mystery. x
20
Quadratic Equations—Factoring
In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly. Focus on two such forms: perfect square trinomials and differences of two squares. Learning to recognize these cases makes factoring easy. x
21
Quadratic Equations—The Quadratic Formula
For those cases that defy simple factoring, the quadratic formula provides a powerful technique for solving quadratic equations. Discover that this formidable-looking expression is not as difficult as it appears and is well worth committing to memory. Also learn how to determine if a quadratic equation has no solutions. x
22
Quadratic Equations—Completing the Square
After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched. x
23
Representations of Quadratic Functions
Drawing on your experience solving quadratic functions, analyze the parabolic shapes produced by such functions when represented on a graph. Use your algebraic skills to determine the parabola's vertex, its x and y intercepts, and whether it opens in an upward "cup" or downward in a "cap." x
24
Quadratic Equations in the Real World
Quadratic functions often arise in real-world settings. Explore a number of problems, including calculating the maximum height of a rocket and determining how long an object dropped from a tree takes to reach the ground. Learn that in finding a solution, graphing can often help. x
25
The Pythagorean Theorem
Because it involves terms raised to the second power, the famous Pythagorean theorem, a2 + b2 = c2, is actually a quadratic equation. Discover how techniques you have previously learned for analyzing quadratic functions can be used for solving problems involving right triangles. x
26
Polynomials of Higher Degree
Most of the expressions you've studied in the course so far have been polynomials. Learn what characterizes a polynomial and how to recognize polynomials in both algebraic functions and in graphical form. Professor Sellers defines several terms, including the degree of an equation, the leading coefficient, and the domain. x
27
Operations and Polynomials
Much of what you've learned about linear and quadratic expressions applies to adding, subtracting, multiplying, and dividing polynomials. Discover how the FOIL operation can be extended to multiplying large polynomials, and a version of long division works for dividing one polynomial by another. x
28
Rational Expressions, Part 1
When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials. These functions play an important role in algebra. Learn how to add and subtract rational functions by first finding their common divisor. x
29
Rational Expressions, Part 2
Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions. x
30
Graphing Rational Functions, Part 1
Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves. x
31
Graphing Rational Functions, Part 2
Sketch the graphs of several rational functions by first calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function. In the final exercise, you must simplify the expression in order to extract the needed information. x
32
Radical Expressions
Anytime you see a root symbol—for example, the symbol for a square root—then you're dealing with what mathematicians call a radical. Learn how to simplify radical expressions and perform operations on them, such as multiplication, division, addition, and subtraction, as well as combinations of these operations. x
33
Solving Radical Equations
Discover how to solve equations that contain radical expressions. A key step is isolating the radical term and then squaring both sides. As always, it's important to check the solution by plugging it into the equation to see if it makes sense. This is especially true with radical equations, which can sometimes yield extraneous, or invalid, solutions. x
34
Graphing Radical Functions
In previous lessons, you moved from linear, quadratic, and rational functions to the graphs that display them. Now do the same with radical functions. For these, it's important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol. x
35
Sequences and Pattern Recognition, Part 1
Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence x
36
Sequences and Pattern Recognition, Part 2
Conclude the course by examining more types of number sequences, discovering how rich and enjoyable the mathematics of pattern recognition can be. As in previous lessons, employ your reasoning skills and growing command of algebra to find order—and beauty—where once all was a confusion of numbersDVD Includes:
36 lectures on 6 DVDs
264-page course workbook
Downloadable PDF of the course guidebook
FREE video streaming of the course from our website and mobile apps
What Does The Course Guidebook Include?
Course Guidebook Details:
264-page workbook
Lecture outlines
Practice problems & solutions
Formula listJames A. Sellers, Ph.D.
The Pennsylvania State University
Dr. James A. Sellers is Professor of Mathematics and Director of Undergraduate Mathematics at The Pennsylvania State University. He earned his B.S. in Mathematics from The University of Texas at San Antonio and his Ph.D. in Mathematics from Penn State. In the past few years, Professor Sellers has received the Teresa Cohen Mathematics Service Award from the Penn State Department of Mathematics and the Mathematical Association...
Reviews
Rated 4.9 out of 5 by 70
reviewers.
Rated 5 out of 5 by CuriousJenny Finally... I Understand Algebra 1!
Thank you, Prof. Sellers! I had to pass a college math placement exam for a master's degree in language arts education but hadn't taken math for over 40 years. I am so glad I found this course. In high school, I only half-understood what was covered in math class, and was too embarrassed to ask questions in class. This course has filled in the gaps and clarified so much that was fuzzy and foggy the first time around. Prof. Sellers is so clear and builds on each preceding concept so thoroughly. He makes no assumptions about your understandings and explains everything, if only briefly. The first few lessons were too simple for me but it was helpful to learn the terms again and re-establish my study skills. Prof. Sellers re-defines terms that have been introduced as you go along, thus reinforcing all that has come before. He includes practical real-life applications in each lesson so you understand the point of "learning all this", plus get valuable practice in how to set up word problems. Finally, his personality is so positive that "being with him" is a pleasure. Prof. Sellers seems so patient, encouraging, and friendly. I jokingly refer to him as "my friend" when my family asks me what I am doing ("my friend is explaining the slope intercept formula"), and I realized I no longer brace myself or sit in a hunched posture when doing math, dreading that moment when I will feel hopelessly confused and stupid. Instead, I listen as receptively as I can and if I don't understand something, I just go back and make sure I follow all of Prof. Sellers' steps. Then, bingo... got it! So glad to achieve this level of math literacy and it is all thanks to Prof. Sellers.
January 11, 2017
Rated 4 out of 5 by mikemanMD Algebra 1 Review
Each presentation is done with mastery. It was a fast way to review this subject in preparation for further work.
I enjoyed finding several errors in the workbook and at least one on the slides. It helped me to be sharper. I can see however how errors could be confusing to a high school student. The Teaching company should redo the course book and eliminate the errors and review the slides with the lecture and correct any errors. The slide error had to do with the sign being backwards on the greater than or equal to linear equations.
December 26, 2016
Rated 5 out of 5 by Jump My daughter doesn't need my help anymore!
My daughter is thirteen and in the eighth grade in school. She was having problems with slope-intercept forms of linear equations and I purchased Algebra 1 so I could refine my skills to help her study. We watched one chapter together and after that she said she didn't need me anymore. That was perfect on all levels!!! She scored a 98 on her latest test!
November 22, 2016
Rated 5 out of 5 by johnforce1999 Very solid course
Very good course, with work divided into digestible chunks. The only complaint I have is that the examples given in the video can sometimes be a little too simple, and the workbook will sometimes include certain problems that I wish he had given similar examples to on screen. Despite this, it moves at a very reasonable pace and would work for both high school students and adults wanting a review of algebra.
October 28, 2016 | 677.169 | 1 |
7
ANALYTICAL GEOMETRY
LESSON
Analytical geometry in Gr12 mostly involves circles and tangents to circles. You
will however need all the skills learnt in Gr11 to answer the questions.
Equations of circles.
The general equation for a circle with centre at t
NCEES Principles and Practice of Engineering Examination
CIVIL BREADTH and STRUCTURAL DEPTH Exam Specifications
Effective Beginning with the April 2015 Examinations
The civil exam is a breadth and depth examination. This means that examinees work the brea
Chapter 10
Statically Indeterminate Beams
10.1 Introduction
in this chapter we will analyze the beam in which the number of
reactions exceed the number of independent equations of equilibrium
integration of the differential equation, method of superpositi
This section covers the MIPS instruction set.
1
+ I am going to break down the instructions into two types.
+ a machine instruction which is directly defined in the MIPS architecture and
has a one to one correspondence with a single instruction bit encodi
SVU CE450 Summer 2015
Quiz #2A (Closed Book)
This is a multiple choice exam. Using a no. 2B pencil, place your answer by marking in the appropriate place on the answer sheet.
Name:_
Student ID:_
1.
MIPS contains 32 GPRs. Which one of the following is NOT
Dr.Y.Narasimha Murthy.,Ph.D
yayavaram@yahoo.com
UNIT III DIGITAL SYSTEM DESIGN
Introduction : The concepts of fault modeling ,diagnosis ,testing and fault tolerance of digital
circuits have become very important research topics for logic designers during | 677.169 | 1 |
This text covers all the material required by the Intermediate tier of GCSE Mathematics. The book applies maths to everyday contexts, building on the Foundation tier material. It includes: chapter summaries and revision checklists, which are related to GCSE grades; discovery activities; sample coursework tasks, complete with mark schemes by examiners; exercises, some with part solutions, and answers; and questions taken from "test papers" written by the NEAB Chief Examiner.
"synopsis" may belong to another edition of this title.
From the Back Cover:
'Mathematics For GCSE'
These three books cover all of the material required by the new GCSE syllabuses. They comprise short sections that provide useful short-term goals for pupils and allow teachers to match the books easily to their own course.
• Written specifically for the new GCSE syllabuses. • Lead author, Brian Speed, is the Chief Examiner for NEAB. • Exceptional support for Coursework and 100% terminal-examination Options (NEAB syllabuses A and B). • Continual emphasis on developing mathematical skills. • Full examination-style tests written by the Chief Examiner. • Hundreds of exercises.
Throughout the books, maths is applied to everyday contexts and genuinely interesting situations in the real world.
'General Mathematics For GCSE'
• Supports students as they move from KS3 towards GCSE grades. • Carefully builds on all of the Foundation material. • Full coursework assessment Collins Educational1555057
Book Description Collins Educational156173363403
Book Description Collins Educational 073224627
Book Description Collins Educational 073224627
Book Description Collins Educational 0724627
Book Description Paperback. Book Condition: Good. The book has been read but remains in clean condition. All pages are intact and the cover is intact. Some minor wear to the spine. Bookseller Inventory # GOR003258187956 | 677.169 | 1 |
key concepts, skills to master, a brief discussion of the ideas of the section, and worked-out examples with tips on how to find the solution. | 677.169 | 1 |
Of Special Interest
What is Calculus
Calculus is the study of Rates of Change. Calculus as we know it today was developed in the later half of the seventeenth century by
two mathematicians, Gottfried Leibniz and Isaac Newton. There are two main branches of calculus: Differential Calculus and Integral Calculus.
Differential calculus determines the rate of change of a quantity; integral calculus finds the quantity where the rate of change is known.
Functions are defined by a formula. It may be well you effort to read
Common Algebraic and Calculus Errors for some helpful information.
Differential calculus describes the methods by which, given a function, you can find its associated rate of change function,
called the derivative. The function must describe a constantly changing system, such as the temperature variation over the course of
the day or the velocity of a planet around a star over the course of one rotation. The derivative of those functions would give you the rate
that the temperature changed and the acceleration of the planet, respectively.
Integral calculus is like the opposite of differential calculus. Given the rate of change in a system, you can find the given
values that describe the system's input. In other words, given the derivative, like acceleration, you can use integration to find the original
function, like velocity. Also, you use integration to calculate values such as the area under a curve, the surface area, or the volume of a
solid. Again, this is possible since you begin by approximating an area with a series of rectangles, and make your guess more and more accurate
by studying the limit. The limit, or the number toward which the approximations tend, will give you the precise surface area.
About this page
On the left, under the Suggested Topics is a list of topics typically found in most college or university level Calculus
curriculums. The list is provided to help you determine where you may need tutoring. Your particular topic may not be listed, but that does not
imply that tutoring is unavailable. Just contact me and inquire if I can offer
tutoring for your particular needs. I will promptly respond and you can decide what further action is required.
I have numerous texts on this subject and I am confident that whatever difficulties you are having in Calculus, I can be of assistance.
I wish you the best of luck in your academic success and look forward to any inquiry you may send on how I may be able to help you. | 677.169 | 1 |
... more...
The guide to vector analysis... more...
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 500 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 25 detailed videos featuring math instructors who explain how to solve the most commonly... 1,900 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 30 detailed videos featuring Math instructors who explain how to solve the most commonly... more... 750 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly... more...
The guide conform to the latest... more... | 677.169 | 1 |
A complete high quality Math Curriculum for Grades 1 through 5. FreeMath was developed by conducting a complete analysis of mathtextbooks from a variety of different sources (including mathtextbooks from the highest scoring school district in Florida). From this base of information, we were able to create a detailed list of the specific math skills that should be taught in each elementary...
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Soroban abacus math is the easiest way for children to learn math. Abacus turns math into fun symbols, making math easy for students of all ages. Allowing children to have FUN learning math! MathSecret students experience improvements in not only math, but also other subjects such as science,...
Braille transciption of printed books and music by Valley Braille Service, Inc., your one-stop source for braille. We offer fast, affordable and accurate braille transcription of everything from textbooks to braille menus, braille maps, building markings, and much, much more. We utilize the...
Try a Thinkwell course FREE! Free 2-week trials for any of Thinkwell's math, science, and AP courses for 6th grade through 12th grade homeschool students. Thinkwell wants you to be sure that this is the right course for your family. Not only do we provide a FREE 2-week trial, but we'll also refund your money within 3 business days of purchase, for any reason. And if your children end up in Tutors - Accounting-Tutors.com, Tutors for Accounting Chemistry Tutors - Chemistry-Tutors.com, Tutors for Chemistry Tutors...
Edinformatics has designed a two part -end of year- 8th grade science test that accesses both "Knowledge and Concepts" (Part I), and "Reasoning and Analysis Skills" (part II). Most of the science material used for the test is consistent with current intermediate school textbooks. Several questions require more rigorous mathematics that is contained within the newly initiated Common Core...
View our collections of contest problems from around the world, with tens of thousands of problems from national and international competitions including American Mathematics Competitions, International Math Olympiad, Putnam, Harvard-MIT Math Tournament, Math Prize for Girls, and the USAMTS. We also host problems from national olympiads and IMO Team Selection Tests from countries around Physics Tutors - Physics-Tutors.com, Tutors for Physics Biology Tutors - Biology-Tutors.com, Tutors for Biology Calculus Tutors - Calculus-Tutors.com, Tutors for Calculus Math Tutors - Math-Tutors.com, Tutors for Math Tutors for Toronto Geometry Tutors - Geometry-Tutors.com, Tutors for Geometry Algebra Tutors - Algebra-Tutors.com, Tutors for Algebra Tutors for...
Using John Saxon's Math Books - How homeschool parents can use them - and save money!
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This site provides a series of online textbooks covering electricity and electronics. The information provided is great for both students and hobbyists who are looking to expand their knowledge in this field. Please keep in mind that the textbooks are not complete. You may find missing pages and chapters as you browse. The books are a continuous piece of work, and will be updated over time....
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Montessori Curriculum and lesson guides for classroom and homeschool, A Guide for the Montessori Classroom, Art lesson plan manual ages 3 to 9. The Montessori Method is used with groups of children of varying ages (3 to 6, 6 to 9, and 9 to 12) and it is well suited for the homeschool environment of a single child or group of varying ages. The set of four guides includes day by day lesson... | 677.169 | 1 |
first textbook on mathematical methods focusing on techniques for optical science and engineering, this text is ideal for upper division undergraduate and graduate students in optical physics. Containing detailed sections on the basic theory, the textbook places strong emphasis on connecting the abstract mathematical concepts to the optical systems to which they are applied. It covers many topics which usually only appear in more specialized books, such as Zernike polynomials, wavelet and fractional Fourier transforms, vector spherical harmonics, the z-transform, and the angular spectrum representation. Most chapters end by showing how the techniques covered can be used to solve an optical problem. Essay problems based on research publications and numerous exercises help to further strengthen the connection between the theory and its applications. less | 677.169 | 1 |
Junior Maths Book 1 (So You Really Want to Learn) - Livres de poche
[SR: 3509535], Taschenbuch, [EAN: 9781905735211], Galore Park Publishing Ltd, Galore Park Publishing Ltd, Book, [PU: Galore Park Publishing Ltd], Galore Park Publishing Ltd, This text features a complete set of answers to the questions in "Junior Maths 1". "Junior Maths 1" is the ideal introduction to maths for Key Stage 2 pupils. Suitable for use from Year 3, the first book provides a firm foundation for mathematical study, covering the basics of addition, subtraction, multiplication and division. The course is fully up to date with the new numeracy framework and features mental strategies and introduces formal methods of calculating. Each new concept is clearly explained with examples alongside to demonstrate working. Extensive practice exercises follow to ensure that there is sufficient practice material for every pupil, regardless of ability."The Junior Maths" series is perfect for parents who want to help their children at home but are not certain of the most up to date teaching methods. The course can be used in the classroom by specialist or non-specialist teachers and by home schoolers who want a rigorous and challenging numeracy course for their children. An answer book and a set of downloadable worksheets are also available to accompany "Junior Maths 1"., 56214011, Mathematik, 1320308031, Abbildungen, 56248011, Angewandte Mathematik, 1320307031, Forschung, 56263011, Geometrie & Topologie, 56212011, Geschichte, 56227011, Lernen & Lehren, 56217011, Mathematische Analyse, 56272011, Mathematische Physik, 56218011, Matrizen, 56219011, Messung, 56225011, Nachschlagewerke, 56221011, Populär & Elementar, 56230011, Reine Mathematik, 1320309031, Trigonometrie, 56216011, Unendlichkeit, 56220011, Zahlensysteme, 56047011, Wissenschaft, 54071011, Genres, 52044011, Fremdsprachige Bücher
Hilliard, David
Titre:
Junior Maths Book 1
ISBN:
9781905735211
Presents an introduction to maths for Key Stage 2 pupils. Suitable for use from Year 3, this book provides a foundation for mathematical study, covering the basics of addition, subtraction, multiplication and division. | 677.169 | 1 |
For some topics the following site might help: Khan Academy, offers a library of over 2,700 videos (November 2011) covering topics from arithmetic to physics, finance, and history and 240 practice exercises. For free. | 677.169 | 1 |
This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The material is intended to amplify the examples of Units 81 through 83, titled "Graphical Solution of Differential Equations." Taken as a group, the four units are seen as a general introduction to a number of standard techniques for solving first-order ordinary differential equations. (MP) | 677.169 | 1 |
This course introduces basic algebra concepts and assists in building skills for performing specific mathematical operations and problem solving. Students will solve equations, evaluate algebraic expressions, solve and graph linear equations and linear inequalities, graph lines, and solve systems of linear equations and linear inequalities. These concepts and skills will serve as a foundation for subsequent business coursework. Applications to real-world problems are also explored throughout the course. This course is the first half of the college algebra sequence.
Policies
Faculty and students | 677.169 | 1 |
Algebra Introduction - Expressions, Equations, and Variables
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
0.48 MB
PRODUCT DESCRIPTION
This Algebra Intro pack includes everything you need to introduce algebra to your students!
- PowerPoint: use for your direction instruction. Agenda, essential qusetion, and objectives are all included along with LOTS of student practice slides
- Guided Practice: Memory Match game! Students work in a partner pair to complete the memory game and get the most matches! Great way for them to practice using and applying the new vocabulary and skills you've just taught!
- Independe practice: Students work on a one page exit ticket on their own so you can assess their understanding | 677.169 | 1 |
Interpolation and Regression are fundamental and important calculations in mathematics. Mr. Newton and Mr. Gauss were engaged in-depth with numerical solutions for these problems. Today, there are improved algorithms, that can solve such tasks.
InterReg allows you to do such complex calculations just with some point-and-click. So this program is not only for mathematics and engineers. The applications are many and reach from education over science to productive use in your company: in example a meaningful sales forecast can be done in minutes.
Over this, the software offers some other functionality to do curve sketching, integration and statistics as well. Version 2 calculates with arbitrary precision.
Graph 4.4(2012-06-17) update
Program for plotting graphs of mathematical functions in a coordinate system. This program is for drawing graphs of mathematical functions in a coordinate system. Graphs may be added with different color and line styles. Both standard functions, parameter functions and polar functions are supported. It is possible to evaluate a function at an entered point or tracing the function with the mouse. It is possible to add shadings to functions, and series of points to the coordinate system. Trendlines may be added to point series.
It ...
Random Number Generator Pro 2.07(2012-06-07) update
Generate random numbers. Random numbe...
Falco Calculator 3.8(2012-05-02) new
Falco Calculator is a useful program for education. Falco Calculator is a useful program for education. It allows to calculate entered meanings of variables which are introduced as a formula.
Falco Calculator is usable for school or university. It economies your time and you can decide an exercise quickly....
Magic Box 1.0(2012-04-29) new
Magic Box is a collection of applications. In it you will find... Magic Box is a collection of applications. In it you will find
some popular jokes and puzzles such as magic square,
magic eye, latent image fading dollar and math transformation.
Will the program learn the unknown number? Will the program learn the defined map?
And what happened to that dollar?
Magic box is great fun for you and your friends!...
RekenTest 4.1(2012-04-06) update
Practice arithmetic skills, for kids and professionals RekenTest is freeware educational software to practice arithmetic skills. It supports basic arithmetic operations like addition and subtraction, the muliplication tables and so on, as well as more advanced arithmetic operations like decimals, money problems, percentages and fractions. The software can be used for classes or as a homework helper. It has lots of options to create the lessons you want and lets you organize your classroom with tasks and groups. Availabl...
Multipurpose Calculator - MultiplexCalc 5.4.7(2012-02-07) update
MultiplexCalc is a multipurpose and comprehensive desktop calculator for Windows mathematical functions and constants to satisfy your needs to solve problems ranging from simple elementary algebra to complex equations. Its underling implementation encompass...
Graphically Review Equations:
Equation graph plotter gives engineers and researchers the power to graphically review equations, by putting a large number of equations at their fingertips. Up to ten equations could...
Regression Analysis - CurveFitter 4.5.7(2012-02-07) update
CurveFitter performs regression analysis to estimate values of parameters. CurveFitter program called curve fitting.
Desktop Calculator - DesktopCalc 2.1.7(2012-02-07) update
DesktopCalc is an enhanced, easy-to-use and powerful scientific calculator DesktopCalc is an enhanced, easy-to-use and powerful scientific calculator with an expression editor, printing operation, result history list and integrated help. Desktop calculator gives students, teachers, scientists and engineers the power to find values for even the most complex equation set.
DesktopCalc uses Advanced DAL (Dynamic Algebraic Logic) mechanism to perform all its operation with the built-in 38-digit precision math emulator for high prec...
Regression Analysis - DataFitting 1.7.7(2012-02-07) update
DataFitting performs regression analysis to estimate values of parameters. DataFitting is a powerful statistical analysis program that performs linear and nonlinear regression analysis (i.e. curve fitting). DataFitting determines the values of parameters for an equation, whose form you specify, that cause the equation to best fit a set of data values. DataFitting can handle linear, polynomial, exponential, and general nonlinear functions.
DataFitting performs true nonlinear regression analysis, it does not transform the functi...
Multivariable Calculator - SimplexCalc 4.1.7(2012-02-07) update
SimplexCalc is a multivariable desktop calculator for Windows. SimplexCalc is a multivariable desktop calculator for Windows. It is small and simple to use but with much power and versatility underneath. It can be used as an enhanced elementary, scientific, financial or expression calculator.
In addition to arithmetic operation, more than hundred built-in constants and functions can also be used in your expression.
SimplexCalc also has the unlimited ability to extend itself by using custom (user-defined) variables. ...
Compact Calculator - CompactCalc 4.2.7(2012-02-07) update
CompactCalc is an enhanced scientific calculator for Windows CompactCalc is an enhanced scientific calculator for Windows with an expression editor. It embodies generic floating-point routines, hyperbolic and transcendental routines. Its underling implementation encompasses high precision, sturdiness and multi-functionality. With the brilliant designs and powerful features of CompactCalc, you can bring spectacular results to your calculating routines.
CompactCalc features include the following:
* You can build l... | 677.169 | 1 |
Course Information
Office Hours
MATH 2412 PRECALCULUS: FUNCTIONS AND GRAPHS (4-4-0).This is a course designed to prepare students for MATH 2413 Calculus I. Content includes algebraic, logarithmic, exponential, and trigonometric functions and equations; parametric equations; and the polar coordinate system.
Prerequisites
MATH 1316 with a C or better or equivalent. Another option is an appropriate secondary school course (one semester of trigonometry or precalculus or the equivalent, including trigonometry) and a satisfactory entrance score on the ACC Mathematics Assessment Test. Importance of Prerequisites: This is not a review course. If you do not have current knowledge of the material in our MATH 1314 College Algebra, and MATH 1316, Trigonometry, please ask your instructor about changing to one of these course to better prepare for MATH 2412.
This course is taught in the classroom primarily as a lecture/discussion course.
Use of Graphing Utilities As with any course where either graphing or scientific calculators are used, the calculator will be used as a supportive tool. This course is not about calculator usage, but about precalculus concepts. We will use graphing calculators when their use enhances the understanding of a mathematical idea. Graphing calculators are not required for this course. However, as you progress through the semester you may find it convenient to purchase your own.
Course Rationale
This course is designed to teach students the algebraic and trigonometric modeling concepts needed for scientific/engineering calculus. It is not simply a review of college algebra and trigonometry.
Tests : Four exams throughout the semester. All administered in class.
All tests will be administered in class.
Tests are open answer questions, there won't be multiple choice questions.
Notes/cheat sheets are not allowed. Calculators are allowed.
Homework
The last page of the syllabus has a list of homework problems that you have to work on and turn in every Monday. Sections covered on a given week are automatically due the following Monday.
The assignments will be checked for completion however I will randomly check a few problems for accuracy and your grade will be affected accordingly. Staple your work and make sure to include the section number for each different section.
Late homework will NOT be accepted. The lowest two homework grades will be dropped. Get Help Early
It is your responsibility to check your answers on assigned problems and make sure you understand the concepts. You can always ask homework questions at the beginning of class, come to office hours, go to the learning lab or work with classmates.
Determination of Course Grade
Tests average = 85%
HW average = 15%
Grading Scale:
A: 90 – 100
B: 80 – 89
C: 70 – 79
D: 60 – 69
F: 0 – 59
Missed Exam Policy
If you miss a test, I will let you make it up at the testing center if you provide me with documentation that proves you were unable to come to class on the day of the test (Doctor's note, police report, etc). If your absence is not justified, you will get a zero on the test you missed. Make up tests are slightly harder. Not being ready for a test is not a valid excuse for taking it later at the testing center, please don't ask. No retest allowed.
Attendance/Class Participation
Regular and punctual class attendance is expected of all students. If attendance or compliance with other course policies is unsatisfactory, the instructor may withdraw students from the class.
Students are expected to participate in class by attending, asking questions, and contributing suggestions and ideas.
Students who have excessive absences (4 or more) may be withdrawn. Please be courteous. Disruptive behavior will not be tolerated. Repeat offenders may be withdrawn. All cell phones/pagers must be turned off.
Withdrawal Policy
It is the responsibility of each student to ensure that his or her name is removed from the roll should he or she decide to withdraw from the class. The instructor does, however, reserve the right to drop a student should he or she feel it is necessary. If a student decides to withdraw, he or she should also verify that the withdrawal is submitted before the Final Withdrawal Date. The student is also strongly encouraged to retain their copy of the withdrawal form for their records.
Students who enroll for the third or subsequent time in a course taken since Fall, 2002, may be charged a higher tuition rate, for that course.
State law permits students to withdraw from no more than six courses during their entire undergraduate career at Texas public colleges or universities. With certain exceptions, all course withdrawals automatically count towards this limit. Details regarding this policy can be found in the ACC college catalog.
The withdrawal deadline is Monday, November 26, 2012
Reinstatement Policy:
Once withdrawn from the class, a student can only be reinstated in two situations:
(1) At the student's request, if the student had some documented emergency or tragedy that prevented the student's participation in class for a period of time and had not exceeded six absences or missed any major tests prior to that period of time.
(2) If the withdrawal was made by instructor or college error. In either case the student is responsible for all missed assignments and must complete remaining assignments and tests on schedule.
Incomplete Grade Policy
An instructor may award a grade of "I" (Incomplete) if a student was unable to complete all of the objectives for the passing grade in a course. An incomplete grade cannot be carried beyond the established date in the following semester. The completion date is determined by the instructor but may not be later than the final deadline for withdrawal in the subsequent semester.
Course-Specific Support Services
Sometimes sections of MATH 0185 (1-0-2) are offered. The lab is designed for students currently registered in Precalculus MATH 2412. It offers individualized and group setting to provide additional practice and explanation. This course is not for college-level credit. Repeatable up to two credit hours. Students should check the course schedule for possible offerings of the lab class. ACC main campuses have Learning Labs that offer free first-come first-serve tutoring in mathematics courses. The locations, contact information and hours of availability of the Learning Labs are posted at:
Statement on Scholastic Dishonesty A student attending ACC assumes responsibility for conduct compatible with the mission of the college as an educational institution. Students have the responsibility to submit coursework that is the result of their own thought, research, or self-expression. Students must follow all instructions given by faculty or designated college representatives when taking examinations, placement assessments, tests, quizzes, and evaluations. Actions constituting scholastic dishonesty include, but are not limited to, plagiarism, cheating, fabrication, collusion, and falsifying documents. Penalties for scholastic dishonesty will depend upon the nature of the violation and may range from lowering a grade on one assignment to an "F" in the course and/or expulsion from the college. See the Student Standards of Conduct and Disciplinary Process and other policies at
Statement on Scholastic Dishonesty Penalty Students who violate the rules concerning scholastic dishonesty will be assessed an academic penalty that the instructor determines is in keeping with the seriousness of the offense. This academic penalty may range from a grade penalty on the particular assignment to an overall grade penalty in the course, including possibly an F in the course. ACC's policy can be found in the Student Handbook under Policies and Procedures or on the web at:
Statement on Student Discipline Classroom behavior should support and enhance learning. Behavior that disrupts the learning process will be dealt with appropriately, which may include having the
student leave class for the rest of that day. In serious cases, disruptive behavior may lead to a student being withdrawn from the class. ACC's policy on student
Statement on Students with Disabilities Each ACC campus offers support services for students with documented disabilities. Students with disabilities who need classroom, academic or other accommodations must request them through the Office for Students with Disabilities (OSD). Students are encouraged to request accommodations when they register for courses or at least three weeks before the start of the semester, otherwise the provision of accommodations may be delayed.
Students who have received approval for accommodations from OSD for this course must provide the instructor with the 'Notice of Approved Accommodations' from OSD before accommodations will be provided. Arrangements for academic accommodations can only be made after the instructor receives the 'Notice of Approved Accommodations' from the student.
Students with approved accommodations are encouraged to submit the 'Notice of Approved Accommodations' to the instructor at the beginning of the semester because a reasonable amount of time may be needed to prepare and arrange for the accommodations.
Institutions of higher education are conducted for the common good. The common good depends upon a search for truth and upon free expression. In this course, the professor and students shall strive to protect free inquiry and the open exchange of facts, ideas, and opinions. Students are free to take exception to views offered in this course and to reserve judgment about debatable issues. Grades will not be affected by personal views.
With this freedom comes the responsibility of civility and a respect for a diversity of ideas and opinions. This means that students must take turns speaking, listen to others speak without interruption, and refrain from name-calling or other personal attacks.
Student And Instructional Services ACC strives to provide exemplary support to its students and offers a broad variety of opportunities and services. Information on these services and support systems is available at:
For help setting up your ACCeID, ACC Gmail, or ACC Blackboard, see a Learning LabTechnician at any ACC Learning Lab.
Student Rights and Responsibilities
Students at the college have the rights accorded by the U.S. Constitution to freedom of speech, peaceful assembly, petition, and association. These rights carry with them the responsibility to accord the same rights to others in the college community and not to interfere with or disrupt the educational process. Opportunity for students to examine and question pertinent data and assumptions of a given discipline, guided by the evidence of scholarly research, is appropriate in a learning environment. This concept is accompanied by an equally demanding concept of responsibility on the part of the student. As willing partners in learning, students must comply with college rules and procedures.
Safety Statement
Austin Community College is committed to providing a safe and healthy environment for study and work. You are expected to learn and comply with ACC environmental, health and safety procedures and agree to follow ACC safety policies. Additional information on these can be found at Because some health and safety circumstances are beyond our control, we ask that you become familiar with the Emergency Procedures poster and Campus Safety Plan map in each classroom. Additional information about emergency procedures and how to sign up for ACC Emergency Alerts to be notified in the event of a serious emergency can be found at
Please note, you are expected to conduct yourself professionally with respect and courtesy to all. Anyone who thoughtlessly or intentionally jeopardizes the health or safety of another individual will be dismissed from the day's activity, may be withdrawn from the class, and/or barred from attending future activities. You are expected to conduct yourself professionally with respect and courtesy to all. Anyone who thoughtlessly or intentionally jeopardizes the health or safety of another individual will be immediately dismissed from the day's activity, may be withdrawn from the class, and/or barred from attending future activities.
Use of ACC email
All College e-mail communication to students will be sent solely to the student's ACCmail account, with the expectation that such communications will be read in a timely fashion. ACC will send important information and will notify you of any college related emergencies using this account. Students should only expect to receive email communication from their instructor using this account. Likewise, students should use their ACCmail account when communicating with instructors and staff. Instructions for activating an ACCmail account can be found at
Closed Saturday and Sunday
Tel: 512-223-8002
ACC Student ID is required to take a test at the testing center. Do NOT bring cell phones to the Testing Center. Having your cell phone in the testing room, regardless of whether it is on or off, will revoke your testing privileges for the remainder of the semester. ACC Testing Center policies can be found at
Prerequisites for Calculus
There are two calculus sequences at ACC (and at most colleges) -- Business Calculus and Calculus. The prerequisite sequence is different for these. Depending on background, students may start the prerequisite sequence at different places.
Intermediate Algebra (MATD 0390)
Intermediate Algebra (MATD 0390)
College Algebra (MATH 1314)
Math for Bus & Eco or (MATH 1324)
College Algebra
(MATH 1314)
Trigonometry (MATH 1316)
Business Calculus I (MATH 1425)
Precalculus (MATH 2412)
Business Calculus II (MATH 1426)
Calculus I (MATH 2413)
Calculus II (MATH 2414)
Calculus III (MATH 2415)
Where to start: The only way that students may skip courses in a sequence is to begin higher in the sequence, based on current knowledge of material from high school courses.
1. A student who needs a review of high school Algebra II will start in Intermediate Algebra (or below.)
2. A student who completed high school Algebra II, but no higher, and whose assessment test score indicates that he/she remembers that algebra, will start in College Algebra or Math for Business & Economics. A substantially higher assessment test score enables the student to start in Trigonometry.
3. A student who completed some precalculus, elementary analysis, or trigonometry in high school, and whose assessment test score indicates that he/she remembers algebra, is eligible to start higher in the sequence than College Algebra. Check the catalog or the math web page.***
* The material in the Trigonometry course requires that students are quite adept with the skills from high school Algebra II (Intermediate Algebra). Some students will achieve that level of skill in the College Algebra course if their placement score is high enough, while others need an additional semester of work on algebra that is done in two courses, Intermediate Algebra and College Algebra.
** Some students who are very successful in College Algebra are tempted to skip either Trigonometry or Precalculus and enroll in Calculus I.
That is not acceptable. Trigonometry topics are essential to success in Calculus, and while it is true that the topic list for Precalculus has only a few additions from the topic list for College Algebra, the level of sophistication of the presentation and the problems on all topics is greater in Precalculus. That increased sophistication is necessary for an adequate background for the Calculus sequence. ***
Notes about the Business sequence: Texas State University requires Math for Business and Economics and Business Calculus I. Students who will attend the UT College of Business must complete the entire Business Calculus sequence before transferring. For more information, including requirements for UT economics students, see | 677.169 | 1 |
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Showing 1 to 3 of 3
If you have to take it you have to take it, it may be hard but it's not impossible.
Course highlights:
The skills I needed to move on
Hours per week:
6-8 hours
Advice for students:
Pay attention and take notes, and make sure you go over them often
Course Term:Summer 2016
Professor:Stapleton
Course Required?Yes
Course Tags:Math-heavy
Jul 06, 2016
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
Calculus, depending on your major, could be a vital part in what you're doing. Especially for engineering or physics majors! Math is an important thing to know even in basic day-to-day life.
Course highlights:
I learned the more tricky math equations. However, along with that I learned and brushed up on my basic math skills. Having to do math without a calculator also comes in handy when you really want to train your brain to do fast math. It sometimes seems overwhelming but if you keep studying and persevering it will get easier.
Hours per week:
6-8 hours
Advice for students:
The biggest piece of advice I have is to look over your material at least twenty minutes a day. It seems tedious but come test time it is fresh in your mind. Another thing you could do is ask questions. It seems intimidating to ask the teacher for help but that is what they are there for. They love to help you and teach you more.
Course Term:Spring 2016
Professor:Staff
Course Tags:Math-heavyBackground Knowledge ExpectedGo to Office Hours
Nov 10, 2015
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
Dr. Hopkins is a great teacher, she loves to answer your questions and moves at a good pace! The math is basic calc you'll need for a lot of careers so I recommend it.
Course highlights:
Derivatives and integration. Increasing and decreasing functions. How to read a graph and make one with a derivative.
Hours per week:
3-5 hours
Advice for students:
Do your homework and study before a test! The homework is a big part of the learn process for math and if you get these basics down other math classes won't be as hard. | 677.169 | 1 |
Exemplary Practices for Secondary Math Teachers
Chapter 2. The Teaching Assignment
You have your class assignments for the coming school term. You know which subjects you will be teaching. But what do you teach? Algebra, geometry, trigonometry? These generic terms do not tell you what you will be teaching from day to day. You will have to do some planning, but you need some sort of guidelines.
In addition to the standards and curriculum materials published by state agencies, many schools and school districts publish what is often called a scope-and-sequence guide. This at least provides some broad guidelines of what your students should know when they arrive in your room that first day. It will also tell you in general terms what you should be teaching during the year. Ask your chairperson or principal for a copy of the curriculum or course of study guide for each course you will teaching, then check the administrative requirements. Will you be expected to reach a particular place in the algebra curriculum by a specific date? You need this information so that you may pace yourself properly. Will there be a departmental final examination or broader examination at the end of the school year? This can affect your course preparation or pacing. You might try to get copies of previously administered exams to use as a guide.
Examine the curriculum guide carefully. Does it give a day-by-day plan or just a collection of units or topics? Remember, this is a curriculum guide, and you can modify it to fit with your expertise. Some guides give suggestions in detail. For example, the guide might suggest that you teach algebraic factoring in a specific order in an Algebra 1 course:
Find the common factor in the expression 3xy - 5x. The factor common to both terms is x. The factored expression is then x(3y - 5). Then they might have you consider factoring the expression:
5(x + y) - x(x + y),
where the common factor is (x + y), so the factoring gives
(x + y) (5 - x).
Remember, this is really the distributive property of multiplication over addition/subtraction in reverse.
Factoring the difference of two perfect squares, using, for example, x2 - 16. Because both terms are perfect squares, we get (x + 4)(x - 4).
To combine the two previous factoring techniques, the students examine the expression for factoring procedures in the order they have been taught, that is, first for the common factor, then for the difference of two squares. For example, to factor the expression 3x2 - 27, first check for a common factor (here it is 3) to get 3(x2 - 9). Then, because both terms in the parentheses are perfect squares, we get
3(x + 3)(x - 3)
Trinomial factoring can also be simplified by first finding the common factor and then doing the usual trinomial factoring. Consider the trinomial 2x2 + 24x + 54. Factor the expression for the common factor 2: 2(x2 + 12x + 27). Then, factor the trinomial to get 2(x + 9)(x + 3).
In some cases, you may discover that there is no curriculum guide; consider asking a more experienced teacher if you can borrow a lesson plan book from a previous year. This can provide a tremendous amount of material to direct your teaching. Although the content of this teacher's plan book may not fit your teaching style, it at least provides a guide regarding the amount of material that can be presented during a given time span. It also provides a lesson-by-lesson sequence you can examine as you plan your lesson. A word of caution is necessary: Don't directly adopt someone else's lesson plans (tempting as this may be). That will not work. Regard this borrowed plan book simply as a guide to help plan your lessons. You'll see more about lesson planning in Chapter 5.
Above all, don't hesitate to ask for help. Try to find a teacher who can act as a mentor for you for each course you are teaching for the first time. Most experienced teachers will be glad to share their knowledge and experience with you—it is flattering to be asked—as an acknowledgment of mastery!
Is Your Textbook the Curriculum?
Not all curriculum guides go into detail. In fact, it is possible that your school may not even have a curriculum guide. Don't worry. You always have your old friend the textbook to rely on. The textbook is an excellent guide to tell you which topics to teach and in what order. If all else fails, examine the Teacher's Edition of your textbook. It should give you ideas for teaching and many other features to help you. Most textbook publishers provide potentially useful supplementary materials to accompany their textbooks. A word of caution: Do not use these materials just because they are available. Always use your personal judgment so that the instruction is yours and not one provided by prescription. (For more on this, see Chapter 4.)
Your textbook also gives many exercises you can use with your students. Although not a curriculum guide per se, your textbook is definitely a curriculum guide of last resort. Also look at the standards of your local district. The state often generates these because they are ultimately responsible for enforcing the standards in all fields and at all grade levels. Under the No Child Left Behind law, states now require teachers to adhere to the state standards, yet local districts might have some modifications for you to follow. Do not confuse the standards with a curriculum guide. The latter is designed to help you teach the material by providing a suggested order of topics, indicating the depth of your responsibility for covering the material, and providing suggestions for teaching: possible motivational activities, developmental suggestions, and assessment options.
Differentiation Within the Curriculum
Your curriculum may have special sections for teaching gifted or special education students. Yet, you are responsible for providing instruction for all student types: English language learners, "average" students, struggling students, gifted students, as well as special education students. Rest assured your classes will include students from many of these groups. (Note: The Appendix includes more technical information about special education law and the inclusive mathematics classroom.)
The English Language Learner
Teaching mathematics effectively is a daunting task for even the most experienced teacher. However, teaching English language learners is particularly challenging because the teacher must teach both mathematics and English at the same time. The task is more difficult if teachers and students cannot communicate in a common language.
There are several strategies you might employ to create a classroom that is warm, nonthreatening, and rewarding for English language learners. Consider using small groups to allow students who share the same first language to communicate with each other in a relaxed environment. This gives them a chance to ask questions of each other and clarify concepts in both languages while you "manage" the groups. To address the varying levels of understanding a group of English language learners may bring to the classroom, employ a variety of instructional strategies in your classroom not only to keep the classes lively but also to reach more students. Manipulatives enable the English language learner to discover relationships and learn concepts while circumventing the language barrier. You can then ask the student to express the relationship using informal language that does not stress grammatical structure but rather focuses on the mathematical concepts.
Teachers should be sensitive to the frustrations of English language learners and present activities that are both interesting and relevant to the students' lives. English language learners can relate to situations that they are experiencing and are more likely to respond when relevant material is presented. Activities involving sports, music, movies, and games are likely to capture their interest. English language learners can benefit greatly from visual aids, so try to reinforce concepts and skills using charts, graphs, diagrams, and pictures.
Another important factor in the effective instruction of English language learners is the simultaneous acquisition of a mathematics vocabulary and the English language. You should spend some time each day building the English language learners' mathematics vocabulary and making certain that they are well versed in the vocabulary words essential to the day's lesson. You may have students keep a separate vocabulary journal so they can review vocabulary. English language learners may feel more comfortable writing in a journal than speaking up in class. This is natural, and you may wish to first check the journal entries and then call on students to share entries aloud with the class. Knowing that their responses are correct will instill confidence in them and allow them to contribute in a nonthreatening environment.
Teachers must constantly monitor their teaching habits when working with English language learners. Remember to speak slowly and pause often to allow students to thoroughly comprehend what they are saying. Paraphrase your thoughts using different vocabulary and always write key words on the blackboard. Keep spoken sentences short and build in wait time to allow students to process the information before proceeding with the lesson. By following these guidelines, you are more likely to provide your English language learners rich mathematical experiences.
The Average Student
Working with the average student presents teachers a great opportunity to gauge their own teaching effectiveness. Many teachers of average learners are complacent and forget to challenge these students. Outstanding teaching, however, can transform the average student into an above-average student by engaging the student in interesting and relevant activities that will better reinforce conceptual understanding. In addition, the new mathematics standards require that students have a deeper understanding of mathematics and the ability to apply it to various problem-solving activities. Mathematics instruction is no longer restricted to presenting simple procedural tasks; rather, it has broadened into formulating a process to solve provocative problems using careful analysis and the synthesis of many skills.
Although average students may not be expert mathematicians, each average student is an expert at something. The effective teacher finds a way to include opportunities for the average student to show off personal strengths. By incorporating writing, art, and even sports statistics into your activities, you give each student a chance to shine. The average student's motivation in the mathematics classroom increases with this opportunity to feel confident.
Consider asking average students to pair up with struggling students to provide support in some activities. Being asked to explain a concept to a struggling student can be an effective means of motivating an average student to acquire a fuller grasp of the associated concepts; this often results in a valuable learning exercise for both students. In short, the average student, like all students, should be exposed to a mathematics classroom that is lively, engaging, and rich in content.
The Struggling Student
Teaching is not an exact science. Although teachers plan to reach all students with a single clear explanation, it would be naïve to think that just because you've explained something all students necessarily get it. The realization that some of your students are struggling does not imply that you are failing as a teacher. The real failure comes from refusing to accept the challenge of reaching those struggling students.
Struggling students may need to be retaught, and there are two ways of doing that: Either use the same approach or use a new approach. Simply reteaching using the same approach may not be too inventive, but it can be effective for some students. Mathematics can be considered like a language, and in language learning, repetition can bring benefits. Just rehearing an explanation may help concepts "sink in." However, if you can make the same points in a modified fashion, then there is a good chance that you will avoid student boredom and possibly strike a chord that resonates with the student. Such action can even be the break in the learning frustration that may have begun to set in.
If your re-explanation doesn't work, then it is time to search for a new angle. Try to incorporate visuals, manipulatives, or real-world examples to which struggling students can relate. Your goal might be to enable the struggling student to help himself. If the textbook is beyond that student's reading level, then provide a book that is more appropriate.
If the student is still struggling, consider teaming the student with an average or accelerated student for peer tutoring. The tutoring process benefits the struggling student as well as the student tutor by reinforcing both students' understanding. Working with the parents to devise strategies is another way to effectively address the needs of the struggling student. You might ask parents to monitor their child's study and homework time. This may reveal that simple measures such as increasing study time might improve performance in mathematics. Such parental involvement also sends an important message to the student: There is genuine interest in improved performance. This prominent parent involvement has proved successful in increasing student effort. Without increased effort, improved student achievement might prove evasive.
As you identify struggling students, you can try to prevent future difficulties by anticipating which prerequisite skills each student may lack. An astute teacher will also consider many factors that could impinge on student achievement. Some of the considerations are as follows:
Are there any undetected learning disabilities that may need to be addressed?
Is there a language problem (e.g., for an English language learner)?
Is there adequate support in the home, where many feel that true learning really takes place while doing homework?
Are there any psychological issues that need to be addressed?
Does the student have the proper prerequisites for the course?
Are optimum methods of instruction in use for this student?
The Gifted Student
Teaching the gifted student can be as difficult as teaching the weaker student. The challenge is different, but it is a challenge nevertheless.
You will be able to identify gifted students by their creative talent, curiosity, and ability to achieve at a high level. The gifted student will often come up with an innovative or unusual method for solving problems, reflecting a rare insight into mathematics. You can use these unexpected responses to exhibit to the rest of the class alternative ways to look at the mathematics under discussion. Often, the gifted youngsters take pride in sharing their ideas with the rest of the class.
In some schools, gifted students are moved ahead or accelerated. They may begin their algebra work in 7th grade or earlier and complete geometry by the end of 8th grade. This enables them to continue taking mathematics courses and complete a year of college-level mathematics, such as calculus, while still in high school. In some cases, students may accelerate beyond the capabilities of the high school and be forced to take courses at a nearby college or to take a "vacation" from math. The latter would have deleterious effects on the development of a talented student. A high school may offer advanced (i.e., college-level) courses that do not involve calculus (e.g., probability, number theory, advanced Euclidean geometry, etc.). These are all options to consider.
Within your class, however, you will want to challenge gifted students. You do not want them to be bored by moving along with the class at the regular pace. One way to challenge gifted students is to have them delve more deeply into certain topics. For example, the class may be studying the Pythagorean theorem. You might challenge your gifted students to extend the theorem to non-right triangles. (For an obtuse-angled triangle, a2 + b2 <
c2; for an acute-angled triangle, a2 +
b2 > c2.) Or, you might ask them to generate Pythagorean triples using the following parametric equations:
a = u2 - v2
b = 2uv
c = u2 + v2
Then, have them examine some of the triples that result from these equations:
3, 4, 5
7, 24, 25
5, 12, 13
8, 15, 17
9, 40, 41
Will all primitive Pythagorean triples1
have exactly one leg of even measure? Will the product of the three members of a primitive Pythagorean triple always be a multiple of 60? And so on.
In a geometry class, for example, introduce gifted students to the elements of simple topology. The four-color map problem, the Möbius strip, and the bridges of Königsberg are all topics that will interest gifted students. (See the References and Resources for sources of material on topology.)
There are many questions that can be assigned to the gifted students to interest and intrigue them. Here are some you might consider:
When is 1/x > x?
If a2 = b2, then will a = b?
For what value(s) of x will x2 + 6x + 6 be a negative number?
If x lies between 0 and 1, then can x be less than
x2? Explain your answer.
When does x = 1/x + 1?
The Special Education Student
The special education student comes into your class under guidelines of an entirely different set of rules. In addition to the rules set by your school or school district, these students are governed by the Individuals with Disabilities Education Act (IDEA). (See the Appendix for more information regarding IDEA and your classroom.) If you have a classified special education student in your class, then that student will probably come with an Individual Education Program, or IEP. This has been prepared by the student's previous teacher together with a child study team and the child's parents.
You must follow what appears in this plan; it is a legal document. If a problem arises, you may need to consult with the child study team and discuss modifying the IEP. However, you should always plan to modify your instruction to accommodate the needs of the learning disabled (LD) child. A typical procedure is to have a teacher aid or special education teacher work with the individual student while you are working with the rest of the class. Be certain, however, that the teacher aid has a good understanding of the mathematics taught.
There are other techniques you can use to help these children achieve in your class. For example, you should obtain permission to give the LD child a grade of Pass or Fail rather than a letter or numerical grade. This is often specified in the IEP. You may have to modify what you teach. For instance, when you teach factoring in an algebra class, use simple numbers. Instead of asking the LD child to factor an expression such as 2x2 - x - 6, which factors into (2x + 3)(x - 2), you could ask the child to factor x2 + 5x + 6, which factors into (x + 3)(x + 2).
Remember, students with disabilities are not necessarily slow. They can learn mathematics. Consider reducing the number of problems expected of the child for a class or home assignment. Ask a bright child to work with the LD child and assist with the work. (This might be an excellent way to challenge the gifted child, who might otherwise be bored.) You might consider the following problem:
Ian has $1.35 in nickels and dimes. He has 15 coins altogether. How many nickels and how many dimes does Ian have?
The majority of students in your algebra class would immediately resort to a system of two equations with two variables (where x = the number of dimes, y = the number of nickels):
x + y = 15
10x + 5y = 135
The student working with the special education child might instead encourage a guess at the answer. After all, intelligent guessing and testing is a valid problem-solving strategy and should be encouraged for everyone. Then, the two students could consider how to move on to the pair of equations solved simultaneously as a more efficient method of solution.
Many special education students may not have a good command of the basic arithmetic facts. For example, they may not always recall the multiplication facts. It's usually wise to make a calculator available for the special student. (All students might well have a calculator available all the time.) At the same time, break the various tasks into smaller pieces. Instead of an entire proof of a theorem in geometry, you might have the special education child just do the first part, such as simply proving triangles ABC and DCB congruent (see Figure 2.1). Then, in a second assignment, have the child prove that the line segments AC and BD are parallel by establishing that ∠ACB ≌ ∠DBC.
Figure 2.1.
It is good to have models of geometric figures available. Many LD students need to physically touch geometric shapes to understand them. For example, you might help them grasp the concept of congruence by asking them to place one triangle on the other and make the parts coincide. The student may even have problems associating the symbol for congruence (≌) with the word congruence or the concept of congruence. The word and symbol may have to be placed side by side for several days. Some special education students may forget material within a day or two. For these children, make reteaching and extended drill and practice a regular part of their lessons. Some special education students have trouble organizing numbers on their papers. For example, they might easily misalign numbers in a simple subtraction problem or when adding a column of figures. To help them, always provide graph paper.
How Do You Get Ideas to Teach Something Beyond the Textbook?
There are numerous books that can provide you with ideas appropriate for your class and yet not included in your textbook. These enrichment units, along with student materials, can be found in several books about teaching algebra and geometry (see Posamentier, 2000a, 2000b, 2000c).
Teaching Secondary Mathematics: Techniques and Enrichment Units
(Posamentier, Smith, & Stepelman, 2006) provides enrichment units for all secondary grades, as well as methods of teaching mathematics in the secondary school.
As a new teacher, begin to collect books for a resource library, which will serve you well as you select topics that can be used to enhance your lessons and will enrich your students. A little secret is to find topics and small units you find exciting and you will continue to rejuvenate your professional outlook and become a more enthusiastic teacher. Students appreciate when you take time to show them math "things" not necessarily part of the standard curriculum. You shouldn't think of these short digressions as "wasting time" that could otherwise be used to move ahead in your syllabus; rather, the time spent on these activities will serve you well because they will motivate your students, making them more receptive learners. In that spirit, a list of some of the books that you might obtain as you build your professional library is provided in the References and Resources. | 677.169 | 1 |
WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE) AIMS OF THE SYLLABUS The aims of the syllabus are to test candidates on: (i) (ii) (iii) further conceptual and manipulative skills in Mathematics; an intermediate course of study which bridges the gap between Elementary Mathematics and Higher Mathematics; aspects of mathematics that can meet the needs of potential Mathematicians, Engineers, Scientists and other professionals.
EXAMINATION FORMAT There will be two papers both of which must be taken. PAPER 1: PAPER 2: (Objective) (Essay) 1½ hours (50 marks) 2½ hours (100 marks) This will contain forty multiple-choice questions, testing the areas common to the two alternatives of the syllabus, made up of twenty-four from Pure Mathematics, eight from Statistics and Probability and eight from Vectors and Mechanics. Candidates are expected to attempt all the questions.
PAPER 1 (50 marks)
PAPER 2
-
This will contain two sections – A and B. This will consist of eight compulsory questions that are elementary in type, drawn from the areas common to both alternatives as for Paper 1 with four questions drawn from Pure Mathematics, two from Statistics and Probability and two from Vectors and Mechanics. This will consist of ten questions of greater length and difficulty consisting of three parts as follows: -
SECTION A (48 marks)
SECTION B (52 marks)
-
PART I (PURE MATHEMATICS)
There will be four questions with two drawn from the common areas of the syllabus and one from each alternatives X and Y. There will be three questions with two drawn from common areas of the syllabus and one from alternative X. There will be three questions with two drawn from common areas of the syllabus and one from alternative X.
PART II (STATISTICS AND PROBABILITY) -
PART III (VECTORS AND MECHANICS)
-
Candidates will be expected to answer any four questions with at least one from each part.
From Olusegun Fapohunda of
233
WEST AFRICAN SENIOR SCHOOL CERTIFICATE EXAMINATION FURTHER MATHEMATICS/MATHEMATICS (ELECTIVE) Electronic calculators of the silent, cordless and non-programmable type may be used in these papers. Only the calculator should be used; supplementary material such as instruction leaflets, notes on programming must in no circumstances be taken into the examination hall. Calculators with paper type output must not be used. No allowance will be made for the failure of a calculator in the examination. A silent, cordless and non-programmable calculator is defined as follows: (a) It must not have audio or noisy keys or be operated in such a way as to disturb other candidates; It must have its own self-contained batteries (rechargeable or dry) and not always be dependent on a mains supply; It must not have the facility for magnetic card input or plug-in modules of programme instructions. DETAILED SYLLABUS In addition to the following topics, harder questions may be set on the General Mathematics/ Mathematics (Core) syllabus. In the column for CONTENTS, more detailed information on the topics to be tested is given while the limits imposed on the topics are stated under NOTES. NOTE: Alternative X shall be for Further Mathematics candidates since the topics therein are peculiar to Further Mathematics. Alternative Y shall be for Mathematics (Elective) candidates since the topics therein are peculiar to Mathematics (Elective).understanding the recommended homework problems is necessary for mastering the course material. The beginning of each class period will usually be used to discuss homework problems from the previous lecture. The course schedule at the end of the syllabus includes the recommended homework problems for each section of the course textbook that will be covered. Quizzes: Short announced and unannounced quizzes will frequently be given in class. They will be based on the recommended... | 677.169 | 1 |
This book demonstrates scientific computing by presenting twelve computational projects in several disciplines including Fluid Mechanics, Thermal Science, Computer Aided Design, Signal Processing and more. Each follows typical steps of scientific computing, from physical and mathematical description, to numerical formulation and programming and critical discussion of results. The text teaches practical methods not usually available in basic textbooks: numerical checking of accuracy, choice of boundary conditions, effective solving of linear systems, comparison to exact solutions and more. The final section of each project contains the solutions to proposed exercises and guides the reader in using the MATLAB scripts available | 677.169 | 1 |
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2007 Pearson Education Asia Concept of a matrix. Special types of matrices. Matrix addition and scalar multiplication operations. Express a system as a single matrix equation using matrix multiplication. Matrix reduction to solve a linear system. Theory of homogeneous systems. Inverse matrix. Use a matrix to analyze the production of sectors of an economy. Chapter 6: Matrix Algebra Chapter Objectives
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.1 Matrices A matrix consisting of m horizontal rows and n vertical columns is called an m×n matrix or a matrix of size m×n. For the entry a ij, we call i the row subscript and j the column subscript.
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.2 Matrix Addition and Scalar Multiplication Example 3 – Demand Vectors for an Economy Demand for the consumers is For the industries is What is the total demand for consumers and the industries? Solution: Total:
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.3 Matrix Multiplication Example 13 – Matrix Form of a System Using Matrix Multiplication Write the system in matrix form by using matrix multiplication. Solution: If then the single matrix equation is
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Elementary Row Operations 1.Interchanging two rows of a matrix 2.Multiplying a row of a matrix by a nonzero number 3.Adding a multiple of one row of a matrix to a different row of that matrix
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.4 Solving Systems by Reducing Matrices Properties of a Reduced Matrix All zero-rows at the bottom. For each nonzero-row, leading entry is 1 and the rest zeros. Leading entry in each row is to the right of the leading entry in any row above it.
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.5 Solving Systems by Reducing Matrices (Continue) The system is called a homogeneous system if c 1 = c 2 = … = c m = 0. The system is non-homogeneous if at least one of the cs is not equal to 0. Concept for number of solutions: 1.k < n infinite solutions 2.k = n unique solution
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.5 Solving Systems by Reducing Matrices (Continue) Example 3 – Number of Solutions of a Homogeneous System Determine whether the system has a unique solution or infinitely many solutions. Solution: 2 equations (k), homogeneous system, 3 unknowns (n). The system has infinitely many solutions.
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.6 Inverses Example 1 – Inverse of a Matrix When matrix CA = I, C is an inverse of A and A is invertible. Let and. Determine whether C is an inverse of A. Solution: Thus, matrix C is an inverse of A.
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.6 Inverses Example 3 – Determining the Invertibility of a Matrix Determine if is invertible. Solution: We have Matrix A is invertible where Method to Find the Inverse of a Matrix When matrix is reduced,, -If R = I, A is invertible and A 1 = B. -If R I, A is not invertible.
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.6 Inverses Example 5 – Using the Inverse to Solve a System Solve the system by finding the inverse of the coefficient matrix. Solution: We have For inverse, The solution is given by X = A 1 B:
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2007 Pearson Education Asia Chapter 6: Matrix Algebra 6.7 Leontiefs Input-Output Analysis Example 1 – Input-Output Analysis Entries are called input–output coefficients. Use matrices to show inputs and outputs. Given the input–output matrix, suppose final demand changes to be 77 for A, 154 for B, and 231 for C. Find the output matrix for the economy. (The entries are in millions of dollars.) | 677.169 | 1 |
This website is intended to provide extra learning resources in algebra for middle school and high school students. The...
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This website is intended to provide extra learning resources in algebra for middle school and high school students. The approach is to teach math concepts in basic terms using examples and diagrams, if ChiliMath - Algebra Lessons to your Bookmark Collection or Course ePortfolio
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Description: Written for VELS, this exciting new Maths series takes a new approach to learning.Maths Xpress 7 for VELS Level 5 Student Resource Book eBookPLUS has received a fantastic review in Vinculum, Volume 46, Term 4 2009 (a Victorian Maths Publication) by Dr John Gough from Deakin University.Dr Gough has given praise to this title saying:'I do not think I have seen anything that offers this kind of personalised, customizable flexibility, for teachers or students.'AND'I can only end by suggesting that any interested reader explore the free 7-day 'demo', to not just 'feel the width' but 'experience the quality'.' See the Title Information file attached.Maths Xpress is based on research about the positive impact of deep learning strategies and the importance of developing mathematically literate students who not only understand the concepts and procedures of mathematics but can also apply them in a non-routine problem-solving context. Maths Xpress 7 for VELS Level 5 Student Resource Book includes a variety of features to promote a deep conceptual understanding of key Mathematical concepts and encourage students to express their understanding and apply it to non-routine problems. Maths Xpress 7 for VELS Level 5 Student Resource Book eBookPLUS is an electronic version of the textbook available onli | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
From signed numbers to story problems — calculate equations with ease
Practice
100s of problems!
Hundreds of practice exercises and helpful explanations
Explanations mirror teaching methods and classroom protocols
Focused, modular content presented in step-by-step lessons
Practice on hundreds of Algebra I problems
Review key concepts and formulas
Get complete answer explanations for all problems
Synopsis
Algebra I Workbook For Dummies, 2nd Edition, tracks to a typical high school Algebra class with hundreds of practice problems to guarantee understanding and retention—now with 25 percent new and revised content to ensure it meets the needs of students and parents today. Updates include:
Standard For Dummies materials that match the current standard and design
About the Author
Mary Jane Sterling is the author of Algebra I For Dummies, 2nd Edition, Trigonometry For Dummies, Algebra II For Dummies, Math Word Problems For Dummies, Business Math For Dummies, and Linear Algebra For Dummies. She taught junior high and high school math for many years before beginning her current 30-years-and-counting tenure at Bradley University in Peoria, Illinois. Mary Jane especially enjoys working with future teachers and trying out new technology. | 677.169 | 1 |
Review of Systems of Equations
Pupils work to complete various algebraic equations using different methods. Given a word problem, students create a poster identifying the various parts of the equation as well as the solution to the problem. Posters are presented in class. | 677.169 | 1 |
Measurement
Content Standard
Performance Indicator
IMP
Students understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring.
Analyze how changes in the measurement of one or more attributes of an object relate to other measurements (e.g., "In doubling the volume of a cube, what happens to the length of the sides?").
Explain rate of change as a quotient of two different measures (e.g., velocity = change in displacement/change in time).
Use degree measures in problem situations.
Determine precision, accuracy and measurement errors; identify sources and magnitudes of possible errors in a measurement setting; describe how errors can propagate within computations; and determine how much imprecision is reasonable in various measurements.
Experimentally determine and use formulas for the volume of a sphere, cylinder, and cone.
Apply limit concepts to develop concepts of area under a curve and instantaneous rate of change.
Combine measurements using multiplication or rations to produce measures such as force, work, velocity, | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
11.61 MB | 42 pages
PRODUCT DESCRIPTION
Included in this zip folder are 4 SMART Notebook files. 1 is an introductory lesson, 1 is a student reference and 2 are assignments. A brief description of each:
The Introductory File is a 37 slide presentation. The student is introduced to odd and even functions. Odd functions are defined as rotational symmetry about the origin. Even functions are defined as symmetric over the y axis. Values from an odd and even function are plotted to begin the concept of f(x)=f(-x) for even functions and f(-x)=-f(x) for odd functions. 16 formative assessment questions are included in this file.
The student reference is information discussed in the intro. It is on 1 page for easy printing.
Assignment #1 is a 24 question multiple choice assignment. The student asked to identify rotational symmetry, fold symmetry and no symmetry of functions. Also, the student is asked to identify an odd or even function from a graph and a table of values. It is on 1 page for easy printing and coded for SMART Response.
Assignment #2 is a 24 question multiple choice assignment. The student is asked to identify graphs and tables of values as odd, even or neither. The student is asked to predict a value of an odd or even function given a value. It is on 1 page for easy printing and coded for SMART Response | 677.169 | 1 |
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Book Description
Do you get cold sweats just thinking about teaching your teen high school math? Did your last exposure to trigonometry leave you covered in hives? If so, you are not alone! Every homeschool parent "loses it" at some point during high school math.
Lee Binz, The HomeScholar, can help guide the way! Lee's practical advice and gentle encouragement will take your math anxiety down a few notches and give you the confidence to push through the pain so your child can achieve math success.
In this book, you will learn how to teach high school math with the correct:
-Sequence,
-Curriculum,
-Attitude, and
-Speed.
You will discover curriculum options, learn how to keep great math records, and get beyond mere good intentions to actually get the job done in your homeschool. There's even a special section on how to teach any particularly nerdy kids that might be living under your roof. | 677.169 | 1 |
So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality. Albert Einstein (1879-1955) The subject of differential equations is large, diverse and useful. Differential equations can be studied on their own or they can be studied by different scientists, whether they are physicists, engineers, biologists or economists. Many physical and even abstract systems can be explained by transposition into this mathematical concept.
2.1. Creating useful models using differential equations The broad area of applied mathematics usually consists in: * Formulation of a mathematical model to describe a physical situation * Precise statement and analysis of an appropriate mathematical model * Approximate numerical calculation of important physical quantities * Comparison of physical quantities with experimental data to check the validity of the model. Although tasks are never clear enough, physicists and engineers handle parts 1 and 4 (formulation of mathematical model and comparison of physical quantities), while parts 2 and 3 (precise statement and approximate numerical calculation) are directed to mathematicians. In order for these four steps to be followed in the most efficient manner, when a mathematical model is formulated, one must take into account both the lack of precision when describing a physical situation and the inability to analyze the mathematical model which may be forthcoming. Since nature is very complex and changes may occur unexpectedly, the mathematician cannot argue that a solution exists and it is unique because the physical situation seems to prove so, since by the time the problem reaches him, the description is no longer accurate. It becomes an approximate model. Therefore, the mathematician...
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...FIRST-ORDER
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...6 Systems Represented by Differential and Difference Equations
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I YEAR B.Tech
By
Mr. Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.
SYLLABUS OF MATHEMATICAL METHODS (as per JNTU Hyderabad)
Name of the Unit
Unit-I
Solution of Linear
systems
Unit-II
Eigen values and
Eigen vectors
Name of the Topic
Matrices and Linear system of equations: Elementary row | 677.169 | 1 |
Instructor Class Description
Functions, Models, and Quantitative Reasoning
Explores the concept of a mathematical function and its applications. Explores real world examples and problems to enable students to create mathematical models that help them understand the world in which they live. Each idea will be represented symbolically, numerically, graphically, and verbally. Prerequisite: minimum grade of 2.5 in B CUSP 122, a score of 145-153 on the MPT-AS assessment test, or a score of 151 or higher on the MPT-GS assessment test. Offered: AWSp.
Class description
Overview of Course: We will explore the concept of a mathematical function and its applications. Functions are the key to how mathematical models are built. Various mathematical models will be created through the usage of real world examples . This course is designed to prepare students for calculus I and serves as a prerequisite for BCUSP124. Upon successful completion of the course, students are expected to build solid skills in algebra, trigonometry, logarithms, exponentials, composition of functions, and graphing.
Student learning goals
linear,exponential, logarithmic and trigonometric functions
brief introduction to limits
General method of instruction
Recommended preparation
Class assignments and grading
Homework assignments will be assigned online on WileyPlus. Instructions and registration key will be available the first day of class Alla Genkin
Date: 12/09/2010
Office of the Registrar
For problems and questions about this web page contact icd@u.washington.edu,
otherwise contact the instructor or department directly.
Modified:April 23, 2014 | 677.169 | 1 |
QUANTITATIVE / FORMAL REASONING COURSES
Williams students should be adept at reasoning mathematically and abstractly. The ability to apply a formal method to reach conclusions, to use numbers
comfortably, and to employ the research tools necessary to analyze data lessen barriers to carrying out professional and economic roles. Prior to their senior year,
all students must satisfactorily complete a Quantitative/Formal Reasoning (QFR) course-those marked with a "(Q)." Students requiring extra assistance (as
assessed during First Days) are normally placed into Mathematics 100/101, which is to be taken before fulfilling the QFR requirement.
The hallmarks of a QFR course are the representation of facts in a language of mathematical symbols and the use of formal rules to obtain a determinate answer.
Primary evaluation in these courses is based on multistep mathematical, statistical, or logical inference (as opposed to descriptive answers). | 677.169 | 1 |
Math 212: Path Integrals
In this integral instructional activity, students identify a vector field and explore line integrals. Students use the Fundamental Theorem for path integrals to solve problems. This two-page instructional activity contains three | 677.169 | 1 |
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