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Welcome to Doug's Blog! This Blog contains a compilation of information created as part of Master's classwork at Arcadia University.
Tuesday, August 3, 2010
Calculators: What are they Good For??
In his paper on Linearity in Calculus, Dan Teague wrestles with the question of knowing one's audience when teaching calculus. In particular, there is a growing dichotomy between those students who have a strong calculator-based precalculus experience and those who do not.
By giving students a visual view of a function, graphing calculators have made it relatively easy to determine whether or not a function is linear. The question with which Teague wrestles is this: How can such an approach strike the proper balance between providing enough rigor for practicing mathematicians and enough information for those students studying calculus as part of their general education?
The approach developed for use at the North Carolina School of Science and Mathematics takes advantage of the use of graphing tools in preparatory calculus courses. Students can use the graphing calculator results as launching point into differential calculus. Thus, this approach lays a good foundation for those students who will take more rigorous mathematical classes.
Should calculators in general, and graphing calculators in particular, be used in such classes? (Interestingly, this is one question I was asked during my job interview for my current teaching position.)
On the con side, if students are using calculators in class, are they truly demonstrating their knowledge of the mathematics being taught? For example, if I am teaching a section on simplifying radical expressions, when I ask students to simplify the square root of 48, I often will get 6.928 as an answer. The correct answer, based on the classroom instruction, is 4√3. The first answer is "correct" according to the calculator, but it does not demonstrate the mathematical knowledge which is being tested. In addition, a calculator will rarely return the negative root solutions to such a question. What is the square root of 49? Many students will say "7", but "-7" is also a valid solution and a calculator will not provide this result. As an instructor, I must become very specific in communicating my instructions and expectations when testing for such knowledge.
On the pro side of using calculators, almost every electronic device which one owns these days has a built in calculator. Cell phone, watches, computers, IPOD's ... calculators are everywhere! So, why not take advantage of them? After all, isn't 6.928 close enough to the square root of 48? 6.928 squared results in 47.997184 ... isn't that close enough to 48? For most of our students and for most of society, the answer is "Yes, this is close enough". And for most practical causes, it truly is close enough.
As a mathematics teacher, I have had to find the balance between acknowledging the technology is readily available and accessible while insisting that my students demonstrate math knowledge. In our state standard exams, students are not allowed to use calculators for the first few math section questions. Once they have completed that section, calculators are allowed to be used. I take a similar approach in my class by allowing calculator use in cases where it does not undermine the fundamental knowledge being tested.
1 comment:
Doug, why do students need 4√3 as opposed to 6.928? In other words, do they appreciate mathematical situations where that form is needed, and have assignments where that form (and not other forms) are necessary for further steps?
For example, they may need to work with vectors with √3 unit lengths, or prove that the answer is irrational (which the calculator can't do). | 677.169 | 1 |
Euclid's Elements by D.E.Joyce
Description: Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional.
Similar booksRotations of Vectors Via Geometric Algebra by James A. Smith - viXra Geometric Algebra (GA) promises to become a universal mathematical language. This document reviews the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve them. (1542 views)
Famous Problems of Elementary Geometry by Felix Klein - Ginn and Co. Professor Pelix Klein presented in this book a discussion of the three famous geometric problems of antiquity -- the duplication of the cube, the trisection of an angle, and the quadrature of the circle, as viewed in the light of modern research. (1583 views) | 677.169 | 1 |
Learning Outcomes
Students will be able to define and apply Gaussian elimination method for solving the systems of linear equations.
Define a homogenous linear system of m equations with n unknowns and identify a sufficient condition for its nontrivial solution.
Students will be able to add and multiply matrices and analyze the properties of Matrix multiplication.
Students will evaluate the determinants of matrices and will apply the Cramer's rule to solve linear systems.
Students will be able to compute the transpose, determinant, and inverse of matrices for a given matrix and prove basic theorems relating to deteminants and matrices.
Students will be able to define subspaces in R-2 and R-3 and inner products; determine the dimension of a subspace and analyze the function that maps two vectors from a vector space to a scalar and prove basic theorems about properties of subspaces.
Students will differentiate between linearly dependent and linearly independent sets of vectors and will be able to find a basis of the subspace; construct orthogonal and orthonormal bases using the Gram-Schmidt Process for a given basis.
Demonstrate the knowledge of constructing the orthogonal diagonalization of a symmetric matrix.
Demonstrate the knowledge of definitions of eigenvalues and eigenvectors and at least of one method to calculate eigenvalues, eigenvectors, and eigenspaces for both matrices and linear transformations.
Students will be able to define linear transformation, transformations from R to R, matrix transformations, one-to-one, kernel, range, rank, nulity and isomotphism, and to solve application problems using the properties of linear mappings: image and kernel. | 677.169 | 1 |
Pages
Thursday, January 19, 2012
Wolfram Launches a New Education Portal
Wolfram Alpha has offered free lesson plans for a couple of years now. Today, Wolfram announced the launch of the new Wolfram Education Portal. The Wolfram Education Portal is an etextbook for Algebra and Calculus. The etextbook includes interactive demonstrations built using Wolfram Mathematica. In the Wolfram Education Portal teachers will have access to lesson plans. While not terribly detailed, the lesson plans do have clear objectives as well as all of the resources a teacher needs to conduct the lesson.
To access all of features of the Wolfram Education Portal you do have to register for a Wolfram account (it's free) and download the Wolfram CDF Player for your computer. Registering and installing the player takes just a couple of minutes.
Applications for Education
The Wolfram Education Portal could be an excellent resource for middle school and high school Algebra and Calculus teachers. The aspect of the Wolfram Education Portal that I find most appealing is the interactive demonstrations accompanying the text. | 677.169 | 1 |
Main Content
Theory of Algorithms
Description:
CS5114: Methods for constructing and analyzing algorithms. Measures of computational complexity, determination of efficient algorithms for a variety of problems such as searching, sorting and pattern matching. Geometric algorithms, mathematical algorithms, and theory of NP-completeness. | 677.169 | 1 |
"The complex social behaviors of ants have been much studied by science, and computer scientists are now finding that these behavior patterns can provide models for solving difficult combinatorial optimization problemsDesigned for high-school students and teachers with an interest in mathematical problem-solving, this stimulating collection includes more than 300 problems that are 'off the beaten path'—i.e., problems that give a new twist to familiar topics or that introduce unfamiliar topics. With few exceptions, their solution requires little more than some knowledge of elementary algebra, though a dash of ingenuity may help second edition extends and improves on the first, already an acclaimed and original treatment of statistical concepts insofar as they impact theoretical physics and form the basis of modern thermodynamicsThis book "can be used to complement courses on differential geometry, Lie groups or probability, differential geometry. It is ideal both as a reference work and as a text for those withing to enter the field | 677.169 | 1 |
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Teaching
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MATHEMATICS 114
Calculus
The module covers introductory Calculus. A key concept in Calculus is that of a limit; thus we begin with a study of functions and limits, providing a foundation for the work that follows. With our knowledge of limits we define the derivative of a function and discover rules and techniques for differentiating. (Derivatives capture mathematically the idea of instantaneous change.) We then apply these skills to help us understand functions, solve optimisation problems and sketch curves. Towards the end of the semester we begin the study of integration, motivated by the problem of finding the area under a graph, finishing the module with basic integration techniques and the Fundamental Theorem of Calculus. (This theorem uncovers the relationship between derivatives and integrals.)
If you miss a test due to illness you need to give Mrs Marais (room 3003 in the Mathematics Building) a valid medical certificate otherwise you will be given 0 for the test.
Please note that lecturers may follow up medical certificates and verify that they are authentic.
If a student misses the class test or too many tests due to illness the lecturers have the option of giving such a student an oral exam in order to calculate their class mark.
Students who do not attend tutorials may be awarded a class mark of less than 40% for this module, regardless of their other marks.
There is no sick test for this module. If you are ill and provide a valid medical certificate, there will be an oral test in the place of the Class Test.
General Information
You will require a basic scientific calculator on occasions during the course.
Calculators may not be used in tests.
Study Guidelines
To succeed in this module it is essential that you work regularly on your own, practising the techniques and applying the results taught in lectures.
Lectures follow the textbooks very closely. Your lecturer will explain the theory in the textbook and work through extra examples. If you read the textbook before you come to class, you will learn a lot more from the lectures.
After class you should make time to read your notes and consolidate what was taught in the lecture by working on exercises in the textbook.
Please consult your lecturer at any time if you are having any trouble with the course material.
Rationale
This module together with Mathematics 144 forms the cornerstone for further study in Mathematics. The module is required by science and commerce students who require a thorough mathematical grounding for further study in mathematics and for their other subjects.
Outcomes
A student who has passed this module should:
Have a solid theoretical and practical grounding in differential and integral calculus.
Understand the concepts of function, limit, derivative and definite and indefinite integral.
Know rules of differentiation and be able to differentiate algebraic and trigonometric functions as well as perform implicit differentiation.
Be able to use differentiation techniques to solve optimisation problems and sketch graphs of functions.
Be able to integrate basic algebraic and trigonometric functions.
Understand methods of proof and reasoning, including mathematical induction, that are used to establish key results in the development of calculus. | 677.169 | 1 |
Applied Mathematics
The subject of mathematics is related to almost all the other subjects. The advancement in the fields of engineering, science, economics statistics etc. are facilitated by use of mathematics. In other words, the application of mathematics helps in development and easier understanding of topics in other subjects. The branch of mathematics that is used for such a purpose is called as applied mathematics. It may be noted that in this branch of math, the important terms and constants used in the other topics also form as parts.
A car leaves an airport at 8 am and runs at an average velocity 45 mph. At 9 am another car leaves the same airport in the same direction and it has to meet the first car before noon. What should be the minimum average velocity with which the second car should run to achieve the necessity?
The situation described in the above physics problem is not very unusual. A person traveling in the first car might have left out something and his friend at the airport may like to reach that article. And by noon, the first person might reach his destination and may not be reachable thereafter. Let us see how applied mathematics helps us to solve. The velocity refers to the rate of change of distance with respect to time. Hence the distance traveled by the first car in the 4 hours (from 8 am to noon) is given by the mathematical equation d1 = v1* t1 = 45*4 = 180 miles, since we know that v1 = 45 mph and t1 = 4 hours. The concept of applied math is same for the second car but now the equation is d2 = v2* t2. But in this case the known values are d2 = 180 miles, since at the point of interception both cars must have traveled the same distance and out of necessity the maximum value of t2 can only be 3 hours (from 9 am to noon). Therefore, 180 = v2*3, which gives the solution as v2 = 60 mph.
Let us study another problem related to physics but which can be considered as an applied mathematics. Two wires of ½ in. diameter are anchored at a ceiling roof as shown in the diagram. These wires are riveted to a hook which is used to hold heavy weights. The wires make angles of 45o and 60o with the ceiling. The wires have an ultimate tensile strength of 16T per sq.in. What could be the maximum weight that can be loaded on the hook?
The concept of this problem is used in material lifting equipment. Ultimate tensile strength of 16T/ sq.in. means the wire can take a load of only 16T for a cross section area of 1 in. Since the diameter of the wires are ½ in. each wire can take only a load of 16(π/4)(1/2)2 = π tons ≈ 3.14 tons. Now mathematically we can draw a vector diagram and find the solution. The same is drawn below.
The wires on the left and right take the loads that are the projection of the main load W in the direction of wires. Let those components be P and Q respectively. As per vector algebra, P = (√2)W/2 and Q = (√3)W/2. Obviously the magnitude of Q is greater and therefore it must be equal to 3.14T. Hence W can be equal to a maximum of 3.14/0.866 = 3.63T approximately.
The concept of matrices is widely used in statistical fields. We will give an introduction to matrices in our next topic.
You need to be a member of The Educator's PLN to add comments!
Unless and until we apply our book knowledge into practical world we found it useless and uninteresting. If we are taught to learn things not just for reading books and solving mathematical problems but also by incorporating these ideas in to real life situations, the interest will develop.
I never analyzed this concept of applied mathematics, It seems really great. Can that be applied to trigonometric problem solving too. I think this has much of application on work and energy concept of physics where a stress and strain calculation for a particular object or a bridge needs to be calculated. Hope I am right!! | 677.169 | 1 |
Description
Get the TOP 25 Education app in the USA!
How this app makes math so much easier:
• This manual shows all important functions of the new graphing calculator TI-84 Plus CE, which are very useful for high school and college. Actual calculator not included!
• The app shows the exact key press sequence for 28 topics of Differential Calculus, Solve Equations, Integral Calculus, Matrices and Tips!
• You can swipe through screenshots of the original graphing calculator which show every step you have to do (220 screenshots in total)!
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• More than 60 math problems with solutions to practice your skills!
Not yet convinced? Here's how you would benefit from this app:
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• To "become warm" CE professional.
• The app is also useful for preparing your examination, because you often don't remember all instructions of the calculator.
• Your teacher will be grateful as well, because he doesn't have to walk around the classroom until everyone's calculator shows the right solution. You only have to take a look at the app and know how to do!
• The app contains the following topics:
DIFFERENTIAL CALCULUS
• Zeros of a Function
• Y-Intercept
• Calculate Tangent
• Intersection of two Functions
• Maxima
• Minima
• Draw Derivation
• Inflection Points
• Y-Calculation
• X-Calculation
• Regression
SOLVE EQUATIONS
• Polynomial
• Solve any Equation
INTEGRAL CALCULUS
• Calculate Integral
• Integral in GRAPH-Menu
• Find Area with Absolute Value
• Area between two Functions
• Integral Function
MATRICES
• Save Matrix
• Delete Matrix
• Put into Row Echelon Form
• Transpose of a Matrix
• Solve Matrix
• Identity Matrix
• Inversion of a Matrix
TIPS & TRICKS
• Insert Functions with Parameter
• Select/ deselect Functions
• Enter Fractions
MATH PROBLEMS
• Practice with over 60 math problems and solutions
• You can practice math problems of a single topic (e.g. Zeros of a Function) or random math problems of Differential Calculus for example.
• As an alternative the app shows you mixed math problems of all topics.
We also have apps for the calculators: TI-Nspire CX (and CAS), TI-84 Plus, TI-84 Plus CE, CASIO fx-9860GlI.
NO ACTUAL GRAPHING CALCULATOR INCLUDED. THE APP IS A MANUAL!
Screenshots
Reviews
Possible refund? Good app, just mistakenly bought this.
5
By Brandonnnnnnnnnnnnn
This is a good app for what it is, yet not the app I was looking for. It is my fault for not reading the title more clearly before purchasing, but is there any possible way I could get a refund? Thank you for taking your time to read.
misleading
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By enma ibarra (student)
i thought i was buying a calculator not a calculator GUIDE. disappointed...
SCAMMMMMM!!!!!
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Do not buy!! I want a refund!!!
SCAM MANUAL , NOT CALCULATOR!!!
1
By Nem.Pro
The developers make the app look like an calculator, but it's NOT, it's just a poor handbook. A poor handbook for 5$ !!! Are you guys serious? The point is the refund progress take a while because you have to fill a form which cost some time, that why some users ignore it and they accept to lose 5$ for this scam. IF YOU ARE A VICTIM OF THIS SCAM APP, PLEASE CONTACT APPLE !!!
Great App
4
By Avery A.
Great app! Easy to use. My only suggestion would be to include more topics that encompass all functions of the calculator on future updates. I do a lot of work with business and statistical calculations. I wish the app included information for how to do these kind of calculations. That is my only suggestion though. Thanks for the great product.
it's a manual!
5
By IKnowHowToRead.IThink.
I bought it thinking it was the calc app. Even thought it say in the description AND the icon. Probably should have noticed haha Besides that, I liked the app it showed me how to do things after I bought the calculator at the store.
Do not buy
1
By Bhjllhresbjkm
Rip off. This is a scam. Do not buy
HORRIBLEEEE
1
By Thaniiamariia
Its a fucjbdhdhk manual no a calculator wasted of money
Horrible
1
By Evfran
It just shows you how to use a calculator. I didn't think they would ever rip you off this bad for instructions on how to use a calculator. Will be calling for refund
DO NOT BUY
1
By Greatest piano app ever
Absolute waste of money this app taught me nothing. Their horrible 3-4 slide tutorials with no text explanation don't teach you a thing. This app should have a descriptive manual on how to use the calculator, but it does not. | 677.169 | 1 |
Description
For Differential Equation and Linear Algebra courses at a sophomore level.Using a unique, student-friendly approach to teaching differential equations, this text encourages students to think both quantitatively and qualitatively when approaching differential equations. Before finding the analytical solution of a differential equation, the text presents the qualitative aspects of the problem--the directional field, the bounded solutions, their range, the presence of constant solutions and so on--to help students use linear algebra to think about the physical systems being modeled. In addition, this text features a fluid integration of linear algebra therefore emphasizing the inter-relatedness of differential equations and linear algebra.show more | 677.169 | 1 |
PDE Textbooks for Undergraduates'm in a similar situation. My prof. recommended Strauss, but I'm broke... I can't find it for less than $60 used. My prof. gives really good notes, but I may get Strauss's book for longevity.
Are there any other books similar to Strauss's? Boyce & Diprima seems to be an ODE book with maybe some PDE. If Strauss is the best (I'm a double major physics and math so I don't mind a lot of talk about sets and open balls), I'll save up for it. | 677.169 | 1 |
Mathematics
R is a statistical computer program used and developed by statisticians around the world. It is probably the leading statistical program, at least among statisticians, and it is freely available. A First Guide to Statistical Computations in R is intended...
Applied Statistics: A Course for Social Sciences will introduce readers to the versatile statistical tools and techniques of estimation of parameters, testing of hypotheses (both in experimental and non-experimental set-ups), the use of multivariate models to answer many research questions in a single on AustralianThis book is an introductory textbook to statistical mechanics that carefully develops new concepts and ideas from basic assumptions and established results. Written for university students in physics and engineering, the prerequisites are mathematics courses on analysis and linear algebra, as well as physics courses...
What does it take to achieve lasting improvement in student numeracy? According to the contributors in this volume, high quality teaching - combined with educational leadership at a school's and system's level - can attain just that. Founded upon evidence and research from...
This book contains a collection of essays on mathematical structures that serve to model the Universe. The contributions discuss such topics as the interplay between mathematics and physics, geometrical structures in physical models, and observational and conceptual aspects of...
Ireland's contribution to modern science is well attested, yet it is not so well known that Ireland, famed for over half a millennium for its saints and scholars, was equally renowned for the scientific endeavor carried out in its monastic...
Perplexingly Easy presents selected letters from the correspondence of Hamilton and Tait during the period from November 1858 to October 1859, critically edited by Wilkins. It offers mathematical historians and others fresh insights into the development... | 677.169 | 1 |
Algebra and Trigonometry presents a finished and multi-layered exploration of algebraic rules. The textual content is acceptable for a customary introductory Algebra & Trigonometry direction, and used to be constructed for use flexibly. The modular strategy and the richness of content material guarantees that the publication meets the desires of numerous courses. Algebra and Trigonometry publications and helps scholars with differing degrees of instruction and event with arithmetic. rules are offered as in actual fact as attainable, and growth to extra complicated understandings with massive reinforcement alongside the best way. A wealth of examples - often numerous dozen in step with bankruptcy - supply distinct, conceptual reasons, so one can construct in scholars a powerful, cumulative origin within the fabric sooner than asking them to use what they have discovered. this can be a full-color textbook built with A. N. Whitehead of their Principia mathematica (1910-1913).
Sheldon Axler's Precalculus focuses in simple terms on subject matters that scholars really need to achieve calculus. due to this, Precalculus is a truly workable dimension although it contains a pupil recommendations manual. The ebook is geared in the direction of classes with intermediate algebra must haves and it doesn't imagine that scholars take into accout any trigonometry.
For instance, consider (pq)3. We begin by using the associative and commutative properties of multiplication to regroup the factors. 3 factors (pq)3 = (pq) · (pq) · (pq) =p·q·p·q·p·q =p·p·p·q·q·q = p3 · q3 3 factors 3 factors In other words, (pq)3 = p3 · q3. the power of a product rule of exponents For any nonzero real number a and natural number n, the negative rule of exponents states that (ab)n = an bn Example 7 Using the Power of a Product Rule Simplify each of the following products as much as possible using the power of a product rule.
2 Exponents and Scientific Notation Try It #7 Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents. _______ 6 7 3 (a b ) Finding the Power of a Quotient To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let's look at the following example. f 14 (e −2f 2)7 = _ e14 Let's rewrite the original problem differently and look at the result. | 677.169 | 1 |
Try asking adults about their math education: They refer to it like some sort of NCAA tournament. Everybody gets eliminated, and it's only a question of how long you can stay in the game. "I couldn't handle algebra" signifies a first-round knockout. "I stopped at multivariable calculus" means "Hey, I didn't win, but I'm proud of making it to the final four."
But there's a new orthodoxy among teachers, an accepted wisdom which says, "Absolutely not." Continue reading → | 677.169 | 1 |
Math 152 Homework Assignments - Spring 2006
About Homework:
Homework assignments are due in class on the days indicated below solutions: Is each step clearly explained? Is the logic sound?
Most of the learning in this course will come from sitting and struggling with these problems, doing lots of examples, and then thinking about what the computations mean. As a result, you MUST work on problem sets by yourself. Office hours for both Kazim and me are an opportunity to discuss problems you are having. | 677.169 | 1 |
INB FOLDABLE & TASK CARDS - Algebra - Calculating Slope
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this file type before downloading and/or purchasing.
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Product Description
This includes a foldable on calculating slope using ALL methods (from a table, two points, an equation, and a graph).
This also comes with task cards. They contain 20 problems for students to calculate slope. This set includes:
* Four task cards with a graph
* Four task cards with a table
* Four task cards with two points
* Four task cards with an equation in standard form
There are two different answer sheets for students. One contains all the answers for students to match. This is a great option if they are just beginning to learn to calculate slope. The other answer sheet does not have the answers for matching. | 677.169 | 1 |
This article allows readers from quite a lot of backgrounds and with restricted technical must haves to discover the dynamical structures and arithmetic usually. The booklet is designed for readers who are looking to proceed exploring arithmetic past linear algebra, yet are usually not prepared for hugely summary fabric.
This article offers an creation to the fashionable thought of monetary dynamics, with emphasis on mathematical and computational innovations for modeling dynamic structures. Written to be either rigorous and interesting, the e-book indicates how sound knowing of the underlying conception ends up in potent algorithms for fixing actual global difficulties.
A transparent methodological and philosophical creation to complexity thought as utilized to city and neighborhood platforms is given, including a close sequence of modelling case reports compiled during the last couple of many years. in response to the recent complicated structures pondering, mathematical versions are built which try to simulate the evolution of cities, towns, and areas and the advanced co-evolutionary interplay there is either among and insideWe have x(k) = Ak x0 + Ak−1 + Ak−2 + · · · + A + I b. To simplify this, observe that Ak−1 + Ak−2 + · · · + A + I I − A = I − Ak , and so, provided I − A is invertible, we have x(k) = Ak x0 + (I − Ak )(I − A)−1 b. The case: all |λ| < 1. The case: some |λ| > 1. 9) The formula, of course, is valid provided I − A is invertible, which is equivalent to saying that 1 is not an eigenvalue of A. Now, what happens as k → ∞? If the absolute values of A's eigenvalues are all less than 1 (hence I − A is invertible), then Ak tends to the zero matrix, hence ˜ = (I − A)−1 b.
We start with x(0) a good bit to the right of x ˜. Observe that x(1) is to the left of x ˜, but not nearly as far. Successive iterations take us to alternate sides of x ˜, but getting closer and closer—and ultimately converging—to x ˜. 3. In this case we have a > 1, so the line y = f (x) = ax+b is sloped steeply upward. We start x(0) just slightly greater than x ˜. Observe that x(1) is now to the right of x(0), and then x(2) is farther right, etc. 1. 3: Iterating f (x) = ax + b with a > 1. 4: Iterating f (x) = ax + b with a < −1. | 677.169 | 1 |
Math
Welcome to the Mathematics Department at Moreau Catholic! Our department believes in and delivers a comprehensive as well as applicable foundation of mathematics skills for students in order to prepare them for any of the different postsecondary directions they may choose. We provide a strong and broad curriculum ranging from Basic Algebra to both AP Calculus and AP Statistics. While we will continue to teach basic math concepts and skills that have been part of our curriculum for years, we recognize the need to be aware of the latest trends in mathematics education and implement the most effective ones. An emphasis on a multi-representational approach to functions is made in all of our classes, while also highlighting the importance of group work and hands-on problem solving. In keeping with our commitment to the school-wide initiative of improving the technology skills of our students, we include level-appropriate activities that involve graphing calculators, Geometer's Sketchpad, and statistical software. All of this takes place with the goal of improving the critical thinking, problem solving, and communication skills of every student at Moreau.
Teach to One Math
Moreau Catholic is the first high school and the first Catholic school in the nation to embrace the practice of personalized math instruction via Teach to One. Students in TTO always receive instruction that is tailored to their learning needs. Leveraged technology, in the form of an advanced algorithm, identifies what students should be learning and how they can best learn it.
There are eight methods of instruction in TTO, and each one is intended to identify and enhance how your student best learns. Lessons can be taught via live investigation, in small group, peer-to-peer, coached virtual instruction, one-to-one, virtual practice, independent practice and then applied in task sessions.
Task sessions take place over multiple days and students use a variety of related skills in real-world applications. For example, students might analyze the costs and benefits of purchasing a hybrid car and use the skills they acquire through this work (e.g., multiplication, estimation, calculating gas mileage) to defend a purchasing decision.
This cutting edge program is a strong example of how Moreau is a thought-leader in the educational community and a mission-driven institution determined to meet the needs of all learners.
Department Members
Math Courses
For students interested in more challenging course work, the department offers honors math classes as well as Advanced Placement courses (application required). Note regarding honors placement: Any grade requirements cited are year averages, i.e., the average of both semester grades.
Comment: This course is UC approved. Students will need a scientific calculator.
This course deals with basic algebraic concepts: sets, equations, inequalities, functions, and the operations and applications for real numbers. Methods and applications stressing word problems are presented throughout the course, using linear equations and inequalities, simultaneous and quadratic equations.
Prerequisite: Placement based on performance on the placement test and/or the Algebra Diagnostic Exam.
Students will be required to take a mandatory summer bridge math course. Comment: This course does not meet the UC "c" requirement. Algebra Enrichment will covers and reviews math concepts needed for success in Algebra 1. Additionally, time will be taken to focus on enhancing study skills and effective use of laptops as a learning tool in mathematics. Completion of this course will prepare students for Algebra 1 their sophomore year. This course does not fulfill any mathematics units needed for graduation for students graduating in 2017 or 2018.
Prerequisite: Successful completion of Algebra 1 or 1b with a C- or better in the second semester. Comment: This course is UC approved. Students will need a scientific calculator.
The concepts of congruence, similarity, and the Pythagorean theorem and its applications will be studied, including an introduction to trigonometry. Coordinate geometry will be covered, as well as geometric applications of algebra. There will be some geometric proof.
Prerequisite: Successful completion of Algebra 2 Honors with a B- or better or Completion of Algebra 1 with an A.
This course explores Euclidean Geometry from identifying geometric figures to proving relationships related to these figures. Students will be challenged to think creatively, reason logically and communicate effectively. In addition, students will explore the interdependence between algebra and geometry.
Prerequisite: Completion of one year of Algebra and one year of Geometry with a C- or better in the second semester. Comment: This course is UC approved. Students will need a scientific calculator.
Topics of first year algebra are intensified and expanded with an emphasis on problem solving. The course deals with coordinate systems, systems of linear inequalities and systems of linear equations in three variables, logarithms, complex numbers, quadratic relations and conics.
Prerequisite: One of the following: -Completion of Geometry with an A, if preceded by Algebra 1 -Completion of Honors Geometry with a C- or better, if preceded by Algebra 1 or Algebra 1B Comment: This course is UC approved. Students will need a scientific calculator.
This course will thoroughly cover Algebra 2 with additional problems and exercises of a more difficult type.
Prerequisite: Completion of three years of math with a grade of C- or better in Algebra 2.
Comment: This course is pending UC approval.
In this course, students will be exposed to mathematics through critical thinking and problem solving with selected topics from Geometry, Analysis and Trigonometry, Algebra 2 and Statistics through real-life applications. Throughout this course, students will learn to think with and apply mathematics and will show mastery through culminating projects. Students will also learn to apply previously learned concepts to make decisions, studying financial mathematics concepts and inferential statistics through the lens of their own environment.
Prerequisite: Completion of Algebra 2 with a C- or better average for both semesters. Comment: This course is UC approved. Students will need a scientific calculator.
This course will study relations and functions, matrices, sequences and series, vectors, trigonometric functions and identities, elementary plane analytic geometry, conic sections, functions and their graphs, and exponential and logarithmic functions. Statistics and probability will also be covered.
Prerequisite: One of the following: -Completion of Honors Geometry with a B- or better, if preceded by Freshman Honors Algebra 2 -Completion of Honors Algebra 2 with a B- or better -Completion of Algebra 2 with an A -Completion of Analysis & Trigonometry with a C- or better
Comment: This course is UC approved. Students will need a scientific calculator.
Success in a Calculus course depends upon having acquired a thorough understanding of functions. In this course, considerable emphasis will be given to functions and their graphs. Polynomial, logarithmic, exponential, and trigonometric and circular functions and their inverses will be treated extensively.
Prerequisite: Completion of Honors Pre-Calculus with a B- or better average for both semesters.
Topics covered include limits and continuity, differentiation, applications of the derivative, integration, methods of integration and applications of the integral to surfaces revolution. Practical applications, particularly from Physics, will be emphasized. Students enrolled in this course must take the Advanced Placement test in Calculus AB.
Prerequisite: One of the following: • Completion of Calculus AB with a B- or better or • Completion of Calculus AB with a C- or better and an AP Calculus AB score of 3 or higher
This course covers the topics of a second semester college-level Calculus course. After a review of the content of AP Calculus AB, the courses moves on to cover new types of derivative calculations using Euler's method and L'Hopital's rule. New integration methods are partial fractions, improper integrals, and logistic integration. Polar, parametric, and vector functions, as well as their derivatives and integrals are analyzed. Numerical integration methods are polynomial approximations and Taylor series. Students enrolled in this course must take the Advanced Placement test in Calculus BC.
Prerequisite: One of the following: Completion of Honors Geometry with a C- or better, if preceded by Freshman Algebra 2 Honors -Completion of Algebra 2 with an A -Completion of Algebra 2 Honors with a B- or better -Completion of Analysis & Trigonometry with a C- or better -Completion of Honors Pre-Calculus with a C- or better Comment: This course is UC approved. Students will need a scientific calculator.
The purpose of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students will work on projects involving the hands-on gathering and analysis of real world data. Ideas and computations presented in this course have immediate links and connections with actual events. Computers and calculators will allow students to focus deeply on the concepts involved in statistics. Students enrolled in this course must take the Advanced Placement test in Statistics.
Prerequisite: One of the following: • Completion Calculus BC with a B- or better or • Completion of Calculus BC with a C- or better and an AP Calculus BC score of 3 or higher 9
This course covers topics such as vector and geometry of space, vector functions, space curve derivatives and integrals of vector functions and motion of particle in space. Further, double and triple integrals, volume under a surface with double and triple integrals, as well as partial derivatives, gradient, divergence, and curl will be discussed. The course will also look at forms of derivatives in multi directions, line integral of scalar and vector valued functions, Green theorem, 2-D divergence theorem, surface integrals, Stokes theorem, parameterizing a surface integral, divergence theorem intuition, and proof of regions in 3-D.
Become A Mariner
We love helping people learn more about Moreau. Select a button below to begin the conversation. | 677.169 | 1 |
Regents Exam Recap: August 2012
Here is a brief review of the August 2012 New York State Math Regents exams, as part of my on-going series examining the quality and validity of these tests.
Only the Integrated Algebra and Geometry exams were offered in August 2012, presumably due to financial constraints. Here are a few thoughts regarding some selected questions from those tests.
First, some issues that are probably minor to most, but important to me as a mathematician and a teacher. Here is number 26 on the Geometry exam.
What does it mean to "use" a point in drawing a triangle? Obviously, the author meant that the given points were to be the vertices of the triangle, but why not say that? Technically, I could use a point by drawing a line segment through it; as such, I could draw any kind of triangle using the given points.
Number 3 on the Integrated Algebra exam includes a particular pet peeve of mine.
There is no value of y in the given equation. In fact, that's the entire point of a variable: it can take any value! What the author means is "What value of y makes the following equation true?" , so why isn't that the question? In addition to asking a question that actually has a correct answer, this would also model for students the correct way to think about variables and equations.
Here's another problem from the Integrated Algebra exam, number 36:
There's nothing mathematically wrong with this particular question. However, the Integrated Algebra exam is the first of the high school mathematics exams in New York City. It is meant to be taken after an introductory algebra course, which means that most students will take this test in 9th or 10th grade.
Thanks for stopping by, and for sharing UGA's teaching community forum. And thanks for the kind words! It looks like an interesting site–are you using this with pre-service and in-service math teachers? | 677.169 | 1 |
Also helpful: MyMaths – MyMaths provides a fully interactive online learning resource suitable for all ages and abilities right up to A-Level, and can be accessed 24/7 from school, home, or even when on holiday. MathsBank – Specialists in high-quality, exam-style A-Level mathematics questions. Contains resources you need for all the AS modules and the most common A2 modules. Edudemic – A list of the top 10 Best Free Math Resources on the Web.
Study Help We recommend visiting these websites: Google Scholar – allows you to search peer-reviewed journal articles in your subject Google Books– allows you to search their online book collection and read available chapters. | 677.169 | 1 |
Math
Mathematics in grades 6-8 is a sequential, college preparatory program. It emphasizes the development of math concepts, computational skills, problem solving, and critical thinking. Comprehensive and appropriately challenging, this curriculum is designed to provide students with the math background necessary for subsequent math coursework.
Math 6 (Grade 6) is a continuation of the Progress in Mathematics program used in the Lower School. The continuity of the program helps to ease the Middle School transition and allows the students to expand their mathematical ability. Concepts including numeration, operations, computation, algebra, functions, geometry, measurement, and probability are still presented in a variety of formats to develop higher level critical thinking. Many skills directly foreshadow pre-algebra.
Pre-Algebra (Grades 7-8) reviews the basic computation of real numbers while integrating skills requiring higher levels of thinking. The use of variables throughout prepares for expanded operations required in Algebra I. Algebra-thinking activity labs provide students with opportunities to dig deeper and explore algebraic concepts to build conceptual understanding. This course, normally taught to eighth graders, is also offered to seventh graders who have demonstrated above average quantitative aptitude and skill.
Pre-Algebra A (Grade 7) and Pre-Algebra B (Grade 8) cover the Pre-Algebra curriculum over 2 years.
Algebra ICP (Grades 8-9) is offered to all students who have completed Pre-Algebra. It extends the concept of set theory to include algebraic expressions, algebraic fractions, factoring, solving linear and quadratic equations, as well as solving systems of equations and inequalities. The graphing of linear and quadratic functions is emphasized. The interpretation and solution of verbal problems is incorporated within each skill area. Students are encouraged to develop precise and accurate habits of mathematical expression.
Algebra I Honors (Grades 8-9) is offered primarily to 8th grade students who completed Pre-Algebra in the 7th grade. This is an advanced course; therefore, the pace and rigor of this class will be significantly more challenging than Algebra I CP. Students will study linear, quadratic, absolute value, radical, and rational equations and inequalities, the graphing of linear and quadratic equations and inequalities, solving systems of equations and inequalities, multiplying and factoring polynomials, and simplifying exponential, radical, and rational expressions. Throughout the year, students will work extensively with word problems to develop their critical thinking skills | 677.169 | 1 |
Try these Beginning Algebra with Arithmetic Review Activities with
your students.
Activities provide the focus for each lesson of the text, making it easy for instructors to establish a daily routine of active student involvement. Data is presented (or student-gathered), models are provided (or student-created), and carefully developed questions lead students to a real-world, mathematical connection to the concept.
Answers to Beginning Algebra exercises can be found on the answer pages below. | 677.169 | 1 |
EXTRA OFFICE HOURS BEFORE THE FINAL
EXAMINATION: Thursday, December 10, 4-5 p.m. and
Friday, December 11, 1-3 p.m.
Welcome to the Math 1190 course website. This website is
the primary way for you to get information about our course,
including the homework assignments. It also contains the course
syllabus, and some links to related material on the
web.
Please note:
enrollment in this course is restricted to students in the
Lynch School of Education.
Math 1190-1191 is a course sequence designed for those who plan
to teach mathematics in grades K-8. The emphasis is on building
conceptual understanding of the mathematics present in the
emerging K-8 curriculum and on deepening content knowledge. Number
and number systems through the real number system will be studied;
functions and the structure of algebra will be developed. Problem
solving and reasoning, applications, and making connections will
be featured.
We will focus on mathematics
at the elementary school level. Our goal will be to
develop the pedagogical content knowledge necessary to succeed
in teaching mathematics in elementary school. We will also
learn how to present this material so that it is good
preparation for the topics that come in later grades.
Unfortunately, many teachers find that students in higher grades
have not mastered this material; even BC students who have gone
off to teach in high school have reported it as extremely useful
to them. This course will focus on numbers and arithmetic.
Please note that pedagogical content knowledge related to
geometry is also necessary to be well-prepared to teach.
This is covered in MT1191, and students are strongly advised to
include that class in their schedules for the Spring 2015
semester.
Textbooks: There are two required textbooks for this
course.
1. Mathematics for Elementary Teachers With Activities
(Fourth Edition) by Sybilla Beckmann
2. Knowing and Teaching Elementary Mathematics by
Liping Ma. The Anniversary Edition is in the bookstore, but
a prior edition will also be fine.
Examination Dates: Please note that full information
about the grading policy for this course may be found in the
syllabus (see link above).
The Final Examination
for Math 1190, Section 01
is on Monday, December 15, 2014
at 9:00 a.m. Important: The
date and time of the final examination is fixed by the Registrar
and may not be changed.
SOME USEFUL LINKS
1. The key development in mathematics education of the
last few years has been the adoption of the Common Core
Mathematics Standards by most of the United
States. (You can also find an L.A. Times op-ed I wrote
about this here.)
2.
One
requirement for elementary certification in Massachusetts is
passing the Mathematics Subtest. Here is a practice
test.
4. Here is a review
of the book by Liping Ma that is required for our course.
The review is by Prof. Richard Askey, a very distinguished
mathematics professor who has also been active in mathematics
education for many years.
5. Here
is a provocative article that came out this summer in the New
York Times magazine, entitled "Why do Americans stink at
math?". And here
is a criticism by Tom Loveless of that article. And a thoughtful
response "Americans stink at math but we can fix that" by
Prof. Hung-Hsi Wu which appeared in the San Francisco
Chronicle. And a
criticism of the article by a Japanese educator, Manabu
Watanabe. | 677.169 | 1 |
Instructor (Stephen Boyd):– email – for example, if you don't live in the Bay area, you should email us to let us know when you want the final emailed to you. That's the first announcement. And I guess, even for people in the Bay area, sometimes traffic is a big pain or something and in which case this is an easier option. Second announcement is homework nine – we'll post the solutions Thursday so Thursday evening after homework nine is due. And I think we've now responded to maybe 10, and growing, inquiries. I guess there is a problem involving – the title is something like time compression equalizer, does this strike a bell? Vaguely. You look worn out. No? Okay. It's just early. Okay. All right. So we fielded a bunch a bunch of questions about the convulsion, we didn't put the limits in the sum, in the convulsion, but you're to interpret, I think it's W and C as 0 when you index outside the range. So a bunch of – maybe 10 people pointed this out to us or something like that. An important announcement, sadly, I have to leave tomorrow morning to go to Austin. I don't like doing that, but I have to go. So I'm off to Austin, and that means that Thursday's lecture, which is the last lecture for this class will actually be given this afternoon. And I think it's Skilling Auditorium 415 this afternoon, but whatever the website says that's what it is. And that's on the first page, the announcements page. So that's where. If you were are around this afternoon and want to come, please do come. You should know that it is every professors worst nightmare, maybe second or third worst, but it's way up there on the list that you should give a tape ahead and no one would come. This would cause you to give a lecture to no one. It's never happened, but it doesn't work. So at least, statistically, some of you should come. My guess is someone will come. We've had long discussions about this. Several colleagues have suggested that we should do tape ahead's from wherever we are, sort of like a nova show or something like that. So you could say hi, I'm here in Rio and we're gonna talk about the singular value decomposition or just something like that, but we haven't actually approached SCPD to see if they can pull that off, but I do want to do that sometime. Anyway, this afternoon is a tape ahead. Please come, statically. So as long as some of you come. My guess is that some people will come anyway. All right. Any questions about last time or administrative stuff? Oh, I have to say that one of the problems is because I'm actually in between this lecture and then Thursdays lecture, which is this afternoon, I also have to give a talk at NASA Ames so I'm gonna have to leave my office hours early today around noon. I have to be walking out the door by noon. So I feel quite bad about that. In fact, I'll even be gone when you get your final. That might be a good thing. But I'll be back Saturday morning. I'll be on email and I'll be contact, let's put it that way. And I'll be back Saturday. And we have a couple of Beta testers taking it; I think one in about an hour and a half. So someone is gonna debug it for you. It's already been debugged pretty well. Okay. Any questions? Then we'll continue on reachability. So last time we looked at this idea of just reachability. Reachability is the following state transfer problem. You start from zero and the question is where can you go? So it's a special state transfer problem. You start from zero and you want to hit something like, in states base, at time T. And we said that our sub T is the reachable sub space. This is sub space. If you can hit a point in T, seconds, or epics, you can certainly hit twice the point and it's a sub space, if you can hit one point or another, you can hit the sum. So it's the sub space. And it's a growing family of sub spaces. So we'll know exactly what the family is. Actually, we already know for discrete time. For discrete time it's interesting, but it's just nothing but an application of the material in the course. It's basically this. Our sub T is the range of this matrix, CT; this is the controllability matrix at time T. I think I mentioned last time that this matrix, you will see in other courses. I mean, it comes up in, for example, scientific computing, in which case RT is actually called a [inaudible] sub space. I may have mentioned that last time, but [inaudible] you will see that this matrix doesn't come up in just this context. It comes up in lots of others. So this matrix here and I think we discussed it last time, as you increase T it gets fatter and fatter, in fact, every time you increment time, the matrix gets fatter by the width of B. That's the number of inputs, which is M, is what we're using here. So what happens is you have a matrix, you start with B, that's where you the range of B in one step, then the range of B and AB is where you can get in two steps and that was parched very carefully and I guess I shouldn't have said it so quickly. When I said the range of B and AB, it means the matrix B space AB. So it's the linear combination of columns of B plus columns of AB. That's where you can get the two steps together. Okay. Now we noted by the Cayley Hamilton Theorem, once you get to N steps, A to the N is a linear combination of I, A, A squared up to A and minus one and so the rank of CT or the range, does not increase once you hit above N. So for example, the range of CN and plus 1 is also the range of CN. So it doesn't grow. Okay. Now that means we have a complete analysis of discrete time system where you can get starting from zero in T epics. The answer is just this. You can get to the range of CT for T less than N, and then after that, once you hit N, it's the range of C. And C is just CN. That's called the controllability matrix. And the system is called controllable if CN is onto. So in other words, if it's range is RN. So that's the idea. And so you can say, you get something that's not totally obvious, it's this, you have the following. In the discrete time system any state you can reach in any number of steps, can be reached, in T equals N steps. Now, that doesn't mean that's a good idea. We will see why very shortly, but nevertheless, as a mathematical fact, it says that if you can't reach a state in N steps then you can't reach it ever. So giving you more time to hit the step is not gonna help at all. Okay. In the reachable set, that's the set of points you can hit with no limit on time, is simply the range of C. It's the range of this matrix. Okay. Now a system is called controllable or reachable, now, unfortunately there are people who distinguish between reachable and controllable, sadly, so sometimes controllable means something slightly different, but don't worry about it for now. A system is controllable if you can reach any state in in steps or fewer, and that's if and only if this matrix C is full rank. So that's the condition. And we'll just do a little stupid example here is this. You have XT plus 1 is this matrix zero 1 1 0 X of T plus 1 1 U of T, now, we can just look at this and know immediately what it does. It does absolutely nothing but swap the roles. That's the swap matrix, I mean, if you ask me to describe it in English, that's a swap matrix. It simply swaps X1 and X2. The input, and this is the important part, acts on both states the same way. So the point is there's a symmetry in the system. It's just a stupid simple example. There's a symmetry in the system and it basically says that whatever you can do to one state, and I'm arguing very roughly now, it will do the same thing to the other. So that's a hint right there that there's gonna be some things you can't get to. We'll wait and see what they are. The controllability matrix is B, that's AB, and sure enough, B AB is not on two. It's singular. And the reachable set is all states where X1 is equal to X2. So no matter what you do here, no matter how you wiggle, you will never reach a state that doesn't have the form of a number times the vector 1 1. It just can't happen. And it's obvious here you certainly didn't need controllability analysis to see this here. And to be blunt about it, that's often the case in almost all examples. I mean, sometimes you don't know, you actually have to check, they'll be something, and in fact, not only that, but most lack of controllability comes down to symmetries like this. They can do much more sophisticated in large mechanical systems and things like that or after the fact you'll realize that something symmetric in your actuator configurations is symmetric and of course, you couldn't do something after the fact. We'll see actually there's a much more interesting notion of controllability that we're gonna get to of quantitative work. Okay. Now let's look at general state transfers. So general state transfers, that's a general problem. We're gonna transfer from initial to a final time, from an initial state to a final state, and of course this is the formula that relates the final state to the initial state and of course, this is completely clear, that's simply the dynamics propagating the initial state forward in time. That's nothing else. So this in fact what would happen if you did nothing, if you were zero over the interval? This is the effect, I stacked my inputs in a big M times TF minus T1 plus 1 vector and I multiply it by this controllability matrix here. And this gives you the effect of the input, how it changes your final state. Okay. So what this says is this equation holds, if and only if, I'll take X desired to be the state you want XTF to be, so I take XTF minus this is in the range of that because this is in the range of that and there's your answer. So it actually makes a lot of sense. It's actually quite beautiful. It basically says something like this. If you want to know if you can transfer from an initial state to a desired state, then it's really the same as the reachability problem, what you want to reach is an interesting state. You don't want to reach X desired. You want to reach X desired minus what would happen if your initial state were propagated forward in time. That's what it comes down to. Okay. So this is simple, but it's quite interesting. So I guess another way of saying it is something like this. The U, if you want to transfer from T initial to X of T initial to some X desired, it says don't aim at X desired. What you do is pretend you're starting from zero and aim for this point, which takes into account the drift dynamics. Okay. So that's kind of what you want you want to do. Okay. So general state transfer reduces to reachability problem, and now I believe last time somebody asked the following question. We talked about reachability and your ability to get from one state to another, let's say over some fixed time interval. And the question is if we made the time interval longer, can you get to more points? Certainly if the initial state is zero, that's true. If the initial state is not zero, that's false. It's just wrong. So it is entirely possible in general reachability to be able to hit a state from one initial state in four steps, but then in five steps to be unable to hit it. Okay. That's entirely possible. It does happen and so that's entirely possible. Now, there's a very important special case. Some people think of it as the dual of reachability and sometimes people call this controlling, I mean, if you distinguish between reaching and controlling, that is driving a state to zero. So sometimes the problem of taking a state that's non-zero and finding an input that manipulates the state to zero is called regulation and sometimes it's just called controlling. I can tell you the background there. The basic idea in regulation is that X represents some kind of – your state actually represents what we call X here represents an error. It's an error from some operating conditions. So you have some chemical plant, you have a vehicles, you have whatever you like, X equals zero means you're back in some state that you want to be in, in some target state or bias point in a circuit or trim for an aircraft or something like that and then regulating or controlling means there's been a wind gust or something's happened, you're not in that state and you want to move it back to this standard state which is zero. This equilibrium position, which is zero. So that's why it's called the regulation problem or control problem or something like that. And here you can work out exactly what that is, here it turns out this is just zero so it depends on whether or not, and of course, that's a sub space so I can remove the minus sign here. If I give you a non-zero state, let's just even just check that. So how would we do the following? I give you a system, I give you A and B and I give you a non zero state and I ask, "What is the minimum number of steps required to achieve X of T equals zero?" That's the minimum time control problem or whatever you want to call it. How do you solve that? So this is what you're given. I'm gonna give you A, I'm gonna give you B and I'm gonna give you this, X zero. How do we do it? How do I minimize T for which X of T is zero? Let's handle a simple case. If X zero is zero, then we're already done before we started and the answer is T equals zero in that case. Okay. How can you do it in one step? What do you do?
Student:
[Inaudible]
Instructor (Stephen Boyd):It's interesting. What you want to do here is the following. You want to check whether A to the T times X0 is in the range of B up to A T minus 1 B. That's it. I think. Make sense? This is what you need to check and you simply increment T now to check. You try T equals 0, we just did that. You try T equals 1, so you hit AX0; you want to check if that's in the range of this. Okay. Now, if you test this and you get out T equals N and the answer is still no, what do you say?
Student:[Inaudible] Instructor
That is cannot be done. Actually, because of this term, that actually requires a little bit of argument, but that's correct. So that's the basic idea. We have a homework problem that's actually a more, it's actually a more sophisticated version of this. I think. Good. Okay. All right. Okay. Now, again, just applying all the stuff we know, because this is nothing but applied linear algebra. There's nothing interesting here. Let's look at least-norm input for reachability. That's actually much more interesting. So let's assume the system is reachable, although, now that you know about SVD it wouldn't matter if it weren't, but let's assume it is. And let's steer X of 0 to an X desired at time T with inputs user of the UT minus 1. I'll stack them in reverse time. That's just so I can use CT this way. So I stack them in reverse time and I get X desired is this matrix, that's a fat matrix times this is my control, my controls stacked or you could actually call this a control trajectory. That's a good name for that vector. I want to put out one thing about that vector. It runs backwards in time. That's just indexing. I could've run them forward in time, too, but then I would've had to of turn CT around to start A T minus 1B, A T minus 2B….down to B. But everyone writes this as B, AB, A squared B. So time runs backwards in this vector. Okay. Now, in this state C is square or fat and it's full rank so it's on 2 and we want to find the least-norm solution of that. The norm of this by the way is the sum of the squares of the norms of the components. That's true actually for any vector. If I take a big vector and I chunk it up, if I divide it up, any way I like, the sum of the norm squared of the partitioned elements is this norm squared to the original vector. So that's what this is and you just want to get the one that minimizes this. This makes a lot of sense. Some people would call this the minimum energy transfer. That would be one. That's, generally speaking, a lie. It generally has nothing to do with that. It's extremely rare to find a real problem where the actual goal is to minimize the sum of the squares of something. They do come up, but they're very rare. Okay. Well, this is nothing. We know how to do this. So that's called the least-norm or the minimum energy input that affects the given state transfer. And if you write it out in terms of what CT is, you get something very interesting. CT of course is B AB A squared B and so on and when you line that up with C transpose C, you get B transpose on top of A transpose B transpose and so on and when you put all the terms together you get a formula that just looks like that. There it is. So that's the formula. And again, there's nothing here. You're just applying least-norm from week three in the class. That's nothing else. But it's really interesting. First of all, notice that it's just a closed form formula for the minimum energy input that steers you from zero to a desired point in T epics and it just looks like that. And everything's here. The only thing in here is a matrix inverse and you might ask, "Why do you know that that matrix is invertible?" What makes that matrix invertible? This matrix in here is nothing but CT CT transpose. It's a fat matrix multiplied by its transpose. That is non singular if and only if C is full rank. And in that case, it corresponds to controllability. But in the case where it is controllable, C dagger is in fact this whole big thing here. By the way, it's really interesting to see what some of these parts are. Let's see what they are. There's actually one very interesting thing is you see something like this. There's sort of a transpose here and the really interesting part is that its running backwards in time. So we don't have any more time left in the class so I'm not going to go into more detail here, but it's just an interesting observation. By the way, this is related to things like you may have seen in other contexts, in filtering you may have seen single pluses, you may have seen matched filters, which is basically where the optimum receiver is sort of the same as the original signal but running backwards in time. If you've seen that, this is the same thing. It's identical. So this is not exactly sort of unheard of. Okay. Now, this is the minimum input. By the way, these are the things that I showed on the first day, as I recall, you were completely unimpressed. So this is where we're just making inputs to some, I don't know, 16 state mechanical system to take it from one state to another in a certain amount of time. They were pretty impressive. We're just using this formula. Absolutely nothing else. Just this. And all I was doing was varying T to see what the input would look like. To see what it would require to take you to a certain state. This is much more interesting. We can actually work out the energy, the actual two norm squared of this least-norm input. Now, if you work out what that is, I mean in general what the least-norm input is is actually it's going to be a quadratic form. And the quadratic form is very simple. It turns out when all the smoke clears I'll just go through all this. When the smoke clears, it's this. It's a quadratic form. This makes perfect sense that the minimum energy – let me explain what this is. This is the minimum energy, defined as the sum of the squares of the inputs. By the way, this is the minimum energy. So this is the energy if you apply the input to hit that target state if you do the right thing. You are welcomed to use inputs that use more energy than this and many exist. Well, actually, unless C is squared, in which case if you hit it, there's only one way to hit it in that, and oh, I'm sorry, C is squared which means there's a single input and T equals N. If C is square there's only one way to hit it so all inputs are minimum energy. But if square is fat, and real simple, there's lots – you can go on a joyride and burn up a lot of energy and still arrive at X desired. That's it. This is the minimum. It's a quadratic form. And that quadratic form looks like this, and it's actually quite pretty. Inside here it's a sum of positive semi-definite matrixes. Now, I know they're positive semi-definite because each term looks like this. It's A to the tow B times A to the tow B transpose because this part is just that. But whenever you take a matrix and multiply it by its transpose, you get a positive semi-definite matrix. That's what you get. So it's a sum of positive semi-definite matrixes. Well, sums of positive semi-definite matrixes are positive semi-definite. And in fact, you can even say this and as a matrix fact, it's correct. When you increment T you add one more positive semi-definite term to this positive definite matrix once T is bigger than N or at some point and that makes the matrix bigger. And I mean now in the matrix sense. So this is a matrix here, which is getting bigger with T, and I mean in the matrix sense. That means, by the way, the inverse is getting smaller. The inverse is getting smaller. That means that the minimum energy required to hit a target in T seconds, as a function of T can only go down. Well, it could be the same in there. It could be the same. Actually, normally it goes down. All right. So it's actually quite interesting here. It says that we now have a quantitative measure of how controllable a system is or reachable. The reachable is sort of this platonic view that says, "Can you get there at all," and this one is much more subtle. It's less clean but it says basically this. It says oh, I can get to that state, no problem. I can get there, but what it'll do it tells you if for some example, getting there is something that takes a huge amount of input, a very large input is required to get there and for all practical purposes, you can say, "I can't get there." So that's the idea. Then we do beautiful things. I can ask you things like this. I can give a target state and I could say that the energy budget is 10 and I can say, "What is the minimum number of steps required to hit this target and stay within my input energy budget?" I could ask you that question and you could answer it by incrementing T until this goes below 10. One possibility is this will never go below 10. In which case, you announce that, well, you can announce several things. You can announce that is too little energy for me to get there no matter how long you let the journey be. So that's one option there. You can actually solve a lot of very sophisticated problems. So what this does it gives you a quantitative measure of reachability because it tells you how hard it is. It also allows you to say things like, "What points or directions in states base are expensive to hit," and expensive means require a lot of control. Cheap means, you can get there with very little control. And it's actually quite interesting. These are lipoids of course, and they basically show that the set of points in states based are reachable at time T with one unit of energy if that's a one. Actually, let's go through the math first and then I'll say a little bit about how this works. So as I said before, if I have T bigger than S then this matrix, that's a matrix in equality is better than that one because the difference between the two is the sum of a bunch of terms of the form, you know, FX transpose between time S and T. So that's what this happens here. Now, you know that if one matrix is bigger than another, the inverse actually switches them. So the inverse is less than the inverse here. Now we're done because if this matrix is less than that, and anytime you put Z transpose Z here and Z transpose here and Z here, this inequality becomes valid. It's an ordinary scalar in equality and it works. And that says it takes less energy to get somewhere more leisurely. So that's the basic idea. It all makes perfect sense. Now, I should mention something here for general state transfer, the analog is false. Absolutely, or is it? Ewe. Wow, and I put all the intensifier up in front, didn't I. Well, I think it's false. But all of a sudden I had this panic that – I think it's false. Let's just say that. That's what I think. I think it's false. I retract my intensifier at the beginning. It's probably false. There we go. We'll leave it that way. So I think with general state transfer, it's false. Okay. All right. I'm gonna have to think about that one for a minute. I'm pretty sure it's false. Okay. Let's just look at an example. So here's an example. It's a 2 x 2 example because that's the only states based I can draw anyway so here's a 2 x 2 example. And here's some system. It increments like this. There's an input, and I want to hit this target state 1 1. I just made it up. There's no significance to any of this. It's all just made up. And what this shows is the minimum energy required to hit the target point 1 1 as a function of time. And you see a lot of interesting things here. You can see that if you hit it in two samples it costs you an energy of over nine. If you say three, you can get there in almost half the energy. I guess it's half the energy if you double, if you say, instead of two steps, do it in four, and so on and you can see. And it goes down. Now, what's interesting is it appears to be going to an asymptote here, which means that to get to that point, with infinite leisure, it still costs energy. Now, I can explain that. That's actually reasonably easy to explain. If a system is stable – someone have a laptop open. So anyway, no never mind, you don't even need a laptop. Can someone work out the item values of this for me? I need a volunteer. Can you do it? Do you have a pen? So he's working on the item values, which he'll get back to us in a minute. I put him on the spot. We'll let you work on that for a bit and then – it's just because you have to write out a quadratic or something like that. So the conjecture is that this is actually – well, no. Cancel the item value thing. What ai was going to say is if this is stable, then in fact, you have to – if a system is stable and you have to get somewhere, you actually have to fight the dynamics to take it out to some place because if you take your hands off the controls, this is very rough. If you do nothing, the state will just decay back to zero. So you're swimming upstream when you're doing reachability for a system that is stable. Okay. Now, if it's unstable, let's talk about reachability. Let's say a system is violently unstable, so basically, all of the eigenvalues for a discrete time system have magnitude bigger than one. So what that means basically is if you do nothing, the state is gonna grow step by step anyway. Now, let's talk about what happens when I give you more and more time to hit a state. What's gonna happen? If I give you, like, a hundred steps and you have a system that's highly unstable or just unstable. If I give you a hundred steps to hit somewhere, what happens is all you have to do is push X of 0 away from the origin. All you do is you push X away from the origin the tiniest bit and then take your hands off the controls and you let the drift, which is the unstable dynamics, bring the system out to where you want to go. Does this make sense? So you kind of work with the different – there, you're not fighting the stream, it's actually on your side for reachability. Does everybody see what I'm saying? So what that suggests is that for an unstable system, as you give more and more time to hit a target, the energy is gonna go down, in fact, it's gonna go down to zero. So we'll get to that now. It is very hard to hit an isotopic target point like that. It is very easy to hit a target point like that. It's very cheap to hit this one and very expensive to hit that one. So the controllability properties are not isotropic in this case. Okay so let's examine this business of this energy going to zero. That is a sequence of a function of T, that is a sequence of increasing positive definite matrixes. And I mean increasing in the matrix order. That is a sequence of positive definite matrixes, which is getting smaller. Now, a sequence of positive definite matrixes that are getting smaller at each step converges just the way a sequence of non negative numbers that are monotone and decreasing converges. This converges to a matrix. That matrix has a beautiful interpretation. It's called P here, that's actually called the controllability gramian, this matrix. And actually it's the inverse of the gramian, but that doesn't matter what it's called. So this matrix comes up and actually it's beautiful. It's a quadratic form that tells you how hard it is to hit any point in states based with infinite leisure. That's what this matrix tells you. And by the way, if the system is violently unstable, P can be 0. That's extremely interesting. So it takes, basically, 0 energy to hit anywhere in a system that is violently unstable. Let me just do a simple example. Let's take B to I and let's A be 1.01 times the identity. It's a very simple system. U just adds to the input. The dynamics is you just times equal the state at each step by 1.01. So basically it says, "If you do nothing, the state just grows by 1 percent each step." That's all that happens. It's a violently unstable system. All the eigenvalues are outside the unit disc. They're all equal to 1.01 and now it's completely obvious that the longer you take, you name any point you want to hit, and what you do is if you take T samples you go back, I guess, by 1.01, you actually find out what input is required to hit that and you take that point and divide it by 1.01 to the T and that's the U that you've set on the first input. That's a sequence of inputs that just kick it out and then let the dynamics take it there. Those inputs will have, as T gets longer and longer, the energy will go to zero and the – by the way, if P is zero, it does not mean that you can hit any point with zero energy. The only point you can hit with zero energy is the zero state. So when you interpret Z transpose PZ, you'd say that that's the energy required to hit it with infinite leisure. It's really a limit. It says that you can hit it. When this is zero it basically says that you can hit that point, not with zero energy, but with arbitrarily small energy by taking a longer and longer time interval. That's what it really means. Okay. Now, it turns out that if A is stable then this matrix is positive definite. That follows up here. If a matrix is stable, well, what it means, is its power, that's A to the tow are going to zero geometrically. In fact, they go to zero at least as fast as the spectral radius, the largest magnitude and eigenvalue of A to the T. So that means this is a converging series. This thing converges to some positive definite matrix. The inverse of a positive definite makes a positive definite and you have this. So if A is stable, you can't get anywhere for free. But if A is not stable, then you can have a zero null space. Zero null space means just what we were just talking about. You can get to a point in the null space of P using the use of energy as small as you like so that's it. And all you do is just kick if a little bit and let the natural dynamics take you out where you want to go. You have to be careful doing this, obviously that this is way it works. So this is actually used in a lot of things. For example, it's used in a lot of what people call statistically unstable aircrafts, so if you look at various sort of modern fighter aircraft, some of the really bizarre ones will actually have the wings swept forward slightly and it just doesn't look right. It just looks like it's flying backwards actually, and it just doesn't look right, and sure enough, it's not right because it's open loop and unstable. That's what they mean by statically unstable. Most other ones are stable. Commercial ones are, at least so far, stable. I think they're probably gonna stay that way, but who knows. So with forward swept wings or statically unstable aircraft, you might ask why would anyone build an airplane, which basically sitting at a trim position, in some flight condition, is unstable. So let's think about what this means. It means things like your nose goes up and instead of there being a force or moment that pushes your nose down, when your nose goes up, actually, there's an up torque and your nose goes up faster. First of all, why on earth would you ever do this, that is the first question. So and this is just for fun. Someone give me a guess. By the way, I made a guess and it was totally wrong when I talked to someone who knew what they were doing.
Student:
[Inaudible]
Instructor (Stephen Boyd):Yes, that's the idea. You want to get a nice snappy ride. Okay. And you do. You get a very – as you can imagine you do. Right. You pop your elevator down a little bit or whatever it is and your nose is now going to go very fast. So is the idea that you can just do it with a small U so it's efficient? Okay. So what's the objective? Well, I assumed it was – I don't know. I actually finally talked to someone who knew what they were talking about, at least on this topic, and they told me in fact why you do this. The main reason, actually, has nothing to do with efficiency or anything like that. Obviously. You want small control surfaces for smaller radar cross sections. So the reason you want small control surfaces, obviously if you're flying at mock two or something like that, you're not really worried about energy efficiency or anything like that. What you want is a small control surface because control surfaces reflect radar stuff. So that's the real reason. And I actually found out how they work. They have, like, five back up control systems because, let's remember, you flip up, but you better be very careful with this, right, and you flip up with a tiny, very small, little subtle control surface that just goes like that. You flip up, and when you get to where you want, you better have just the right input to make you stabilize there and all that kind of stuff because if you lose it, I guess in this case, it's all over in three seconds. It's in under three seconds that whether the pilot likes it or not that explosive bolts go and you're out. So that's the way it works. And the way it works is I think that there were four redundant control systems. So I guess if the first one fails, the second one is all ready to go, if the fourth one fails, you're out the top whether you push the button or not. And that's the way this is and they actually do this. And actually now there's a move to do this for some chemical processes, too. By the way, there's a name for a chemical process that's statically unstable. What would be the common name for it?
Student:[Inaudible]
Instructor (Stephen Boyd):Yes, it's called an explosive. Yes, that's correct. So I don't know if these things are good or bad or whatever, but that's the – and people are doing it. They just said, no, we operate this process at an unstable equilibrium point because it's more efficient in terms of the overall operation. So that's it. All of these obviously require active control to make sure everything's okay. Right. Everything will become – that's the whole point of an unstable system. Things will become not okay very quickly. There was a question back there.
Student:No.
Instructor (Stephen Boyd):Maybe no? Just stretching. Okay. All right. So. Okay. Let's look at the continuous time case and see how that works. It's a little bit different but there's nothing here you wouldn't expect. And in fact, this allows me to kind of say something that I should've said earlier but that's good. Now I get the excuse to say it. To make a connection between the conditions – there is a question.
Student:[Inaudible]
Instructor (Stephen Boyd):Right.
Student:[Inaudible]
Instructor (Stephen Boyd):Really. It's a homework. I can't do the homework, generally, just like that. I had a discussion once. Some people came to my office and I started explaining something, 10 minutes, dead end. I tried again, dead end. And then after 25 minutes they said, "Do you think it's fair to assign homework that you can't do? And I said, "Yes, absolutely because I said at one point, clearly, I could do it, and at that point, it obviously was trivial then." So all right. So let's answer your question. What was it? I can try, but I'm just – I can't do it. I'm not embarrassed in the slightest, but go on.
Student:[Inaudible]
Instructor (Stephen Boyd):That's a good problem. I wonder who made it up. No, I'm kidding. All right. Okay. So you're given an initial state and you want to steer it, not to the origin, but to within some norm of the origin with what, with a –
Student:Minimum amount of input.
Instructor (Stephen Boyd):– with a minimum amount of input. That's a great problem. Is it continuous time?
Student:[Inaudible]
Instructor (Stephen Boyd):Okay. Fine. All right. So I don't know. Can you solve that? I guess the answer is no. That was a rhetorical question. Let's talk about it. Right. It's safer for me in case I can't solve it. So what happens is you want to – let's fix a time period. Okay. So then it's a linear problem. Right. As to where you can get. So I guess it's sounding, to me, like a bi-objective problem. Am I not wrong? It's sounding to me like one. Right. So the final state is what? Let's just say if you go T seconds, it's T epics, it's A to the T X 0 plus and then something like CT times – I'll call it U, but everyone needs to understand U is really a stack of the times in reverse time. Is that cool? This is actually a sequence of U. The whole trajectory. Right. That's what you got and then what did you want to do? The condition is that this should be less than some number. What was the number I gave?
Student:.1.
Instructor (Stephen Boyd):.1. Good. A nice number. There we go. So we have that. And what did you want to do? You wanted to minimize the norm of U. And then your point is that we never did this, right? Is that your point?
Student:[Inaudible]
Instructor (Stephen Boyd):It seems to be. So we didn't do this. That's true. You can look through the notes and you won't find this anywhere. Any comments?
Student:[Inaudible]
Instructor (Stephen Boyd):What?
Student:[Inaudible]
Instructor (Stephen Boyd):Yes, thank you. Okay. So yeah. We didn't do this. Absolutely true. This is a bi-objective problem. This is a perfect example of how these things go down in practice, right, because basically, you go back and look at like week four, it was all clean. It was, like, "Yes, let's minimize AX minus Y with small x and then we drew beautiful plots and all this kind of stuff, right?" Here, it's clouded by the horrendous notation of the practical application. In this case, the practical notation is steering something from here to there so it doesn't look as clean. But it is the same. So you make a plot here trading off – I don't remember how we did it before, but you would trade off these two things like that and there's an optimal trade off curve here. There we go. I know one thing to do, you could set U equals zero, there, I got one. You could nothing and run up a very small bill here. So how do you solve this? How do you solve this? Anyway, I've already said enough. Are we okay now? So now what happens is you make the trade off curve here and then on this plot what do you look for? I find the point here, which is 0.1 and I go up here and I'm looking for that point and that will solve it, right? Are you convinced?
Student:
Yeah.
Instructor (Stephen Boyd):Okay. So that's it. All right. So it's true. You didn't do that before. But we did things that allowed you do it. So. Okay. Are you happy now? Okay. Good. Okay. Let's continuous time reachability. So how does this work? Well, it's actually in some ways trickier and in some ways it's actually much simpler. It's gonna be interesting, actually. So here's the way it works. Actually, in some ways it's gonna be uninteresting. That's the interesting part about controllability in the continuous time case. Okay. So we X. is AX plus BU and the reachable set of time T is actually now an integral and this, it's parameterized by an infinite dimensional set. It's the set of all possible input trajectories you could apply over the time period zero key. Absolutely infinite dimensional. Okay. Now, it turns out that this sub space is super simple. It's just this. It's actually much simpler than the discrete time case. In a discrete time case you can get weird things like this state you can hit it in five steps, but not four. This state you can hit in seven, but not three. You can get all sorts of weird stuff. I mean, all the weirdness stops. Once you hit N steps, you can hit anything you're ever gonna hit, you can hit. That's starting from zero in the discrete time case. In the continuous time case, it just bumps up to anything you're ever gonna be able to hit, you can hit. You can hit anywhere, you can hit it in one nanosecond, at least according to the model. So it's basically this. You form the matrix B AB up to A and minus 1B, that's the controllability matrix. And it basically says if this matrix is full rank, you can hit this set is all of RN for any positive T. And in continuous time, it says any place you can hit, any point you can reach in any amount of time, you can actually reach infinitely fast. That's what it says. And this makes perfect sense. You have to have your input act over a smaller, and smaller time. And it really couldn't have been otherwise. I mean, it would've been really weird if there was a state here you could reach in three seconds, but not two. That would've been kind of weird because you'd think, "Well, like, what exactly happened?" And in fact, because that's a sub space, it's dimension is an integer, so had this other thing happened, it'd be, like, you know, the dimension of the reachable set would've gone up to equals, you know, T equals 2.237 it would've jumped to three or four. And you think now, "What on earth would allow you, all of a sudden, at some time instance to manipulate the state into some other dimension?" I mean, it makes no sense at all. So in fact, it kind of had to be this way. So this is it. So that's the result. And we'll show it a couple of different ways. Actually, there's a bunch of ways to connect it up here to the discrete time case and see how it works. Now, one way to see that you're always in the range is C is simple. Let's start from zero, E to the TA is a power series, but I could use the Cayley-Hamilton as a back substitute with powers of A starting at N, N plus 1 and so on. I can back substitute powers of smaller powers of A. And I'll end up with this, it says that basically E to the TA is for sure, for any key, it is a polynomial in IA up to A and minus 1 period. [Inaudible] polynomial of A, a degree less than A. Okay. Now, X of T is just this integral, but now I'm gonna plug that in and I get this thing and now I switch the integral and the sum and I get the following. It's the sum from I equals one to N of this. But that is just a number. You could actually work out how these are exactly, but it doesn't really matter for us because that's a number and that's our friend the controllability matrix. So what this says is if you have a continuous time system, no matter you do with the input, and you start from zero, you will never leave the range of the controllability matrix. Ever. Now, we're gonna have to show the converse which is that any point in the range of the controllability matrix can be reached. First we'll cheat a little bit and we'll do that with impulsive input. If we're gonna use impulsive inputs we have to distinguish between zero minus and zero – well, T minus and T plus whenever T is a time when there's an impulse put. So let's just say before the impulse, we'll put zero and we apply an impulse, which is A. It's distributed across the inputs by a constant vector F, that's F1 through FM and it's multiplied by this K differentiated delta function. That's what it is. And here, the laplace transfer of that, is S to KF. The laplace transfer of the state is SI minus A inverse B is S to the KF. I'll do a series expansion on this, I think that's a called a law expansion. Did I say that at the time? I don't think I did. No, I didn't think I did, but that's what it is. I think we used it to do the exponential. So if I expand this, I take out the powers that are going to multiply the S to the K and I get things like this. A bunch of them look like this and let's look at this very, very carefully. When I take the inverse laplace transform these correspond to violent impulses in X of T. This S inverse is gonna be the first one. That's sort of like a step term. This is all the stuff that happens between zero minus and zero plus. This is what happens right after zero plus. It makes perfect sense. It says that if you apply an input differentiated K times, it has an immediate effect on the state and the state is to move it to A to the K B. But now, you know how to transfer the state to anything in the range of C because if I make an input that looks like this, it's a delta function times F 0 up to a delta function differentiator and minus [inaudible] and I multiply this if I apply this, then X of 0 plus is C times this vector and now we're done. Now if it says that at least using impulsive inputs, I can reach anything in zero time using impulsive inputs. That's what this says. So that's the picture there. And the question is can you maneuver the state anywhere starting from X equals zero. Is the system reachable? If not, where can you get it? Well, you can kind of figure out what it is, but to kind do some of the calculations we can actually work out what it is. You work out the controllability matrix. It's A AB A squared B and you get this matrix here and you look at it for a little bit and you'll quickly realize its rank two. All right. Let's move on to a much more important topic, which is least [inaudible] reachability in the continuous case. It's gonna be very similar, except it's gonna be kind of interesting now because it's gonna be that we'll have this possibility of actually affecting a state transfer infinitely fast. And that's gonna come out of this. Let's see how that works. That's your minimum energy input. If you have X. is AX plus BU and you seek an input that steers X of 0 to X desired and minimizes this integral here. Now, this is not anything we did before. In fact, this has got a norm. People would call this, by the way, the two norm – just the norm squared of U. Okay. But this is not anything you've seen before and when this was discrete time, U was sort of a stacked version and it was big, possibility, but it was finite dimensional. That's an integral, were in the infinite dimensional case here. Actually, it's not anything you need to be afraid of. Some of you, depending on the field you're in, will have to deal with infinite dimensional things. I might even just be in continuous time or something like that. My claim is if you actually understand all the material from 263, none of the infinite dimensional stuff has any surprises whatsoever. Absolutely none. I mean, a few details here and there, some technical details, everything we did has an analog. And a simple, elementary one. Now, don't dress it up and make it look very fancy to justify, I don't know, just to make it look fancy, right, but you'll see the concept for example [inaudible] so instead of calling it something symmetric you'll have a self a joint operator. That's the other thing. You're then welcomed to call linear transformation an operator, which sounds fancy by the way. Or some people think of it as fancy. So you can talk about linear operator and you can find out, for example, a symmetric one can be diagonalized. There are some things that get more complicated, but if the operator is what's called compact, then it's gonna be exactly the same. It's gonna look exactly the same. It's gonna be something to the SVD also works, at least for compact operators. I'm just mentioning this because some of you will go on – if you ever have to do that, I mean, it should be avoided of course, dealing with these things, but if you find you've already chosen or are too deep into a field where these [inaudible] dimensional things do appear, don't worry because I claim if you understand 263 you can understand all of that just with some translations. There are a few additional things that come up that you don't – you'll have continuous spectrum and things like that, but otherwise it's fine. Has anyone actually already encountered these things? I think there's a lot of areas in physics where you bump into these things, so okay. All right. This is your first fore ray into that. So let's just discretize the system with an interval T over and. Okay. And later we're gonna let N go to infinity so that's what we're gonna do. So we're actually not gonna look at first over all possible input signals. We're gonna look at input signals that are constant over consecutive periods of length H which is T over N. So that's what we're gonna do. So we're not solving the problem. So we'll let them be constant and we'll just apply our various formulas from various things. It turns out [inaudible] exactly what we had before. Now, it's finite dimensional and this is now the controllability matrix of the discretized system. And remember, these have formulas, like, AD is E to the H A and BD is this integral here. Okay. And the least norm-input – now, this is all finite dimensional so there's no hand waving, nothing. It's week four of the class. The discrete least-norm input is given by this expression here. Now, if I go back and express this in terms of A using these powers of these things, after all, A is an exponential and powers of exponentials is just the same as multiplying the thing by that, you get something kind of interesting. What happens is BD turns into T over NB, so you get the following. That's this expression here. That's this first expression here. As N gets big, that converges to something that looks like that. Now, the sum is nothing but a ream on sum for an integral and the integral is that. Now, you put these together, in other words, you take this thing and then multiply by the inverse of that. Notice that the N conveniently drops out. That just goes away. So does the T for that matter. And I get a formula, and this is in fact different, it's this, it's B transposed times this [inaudible]. By the way, if you compare this to the discrete time case you will see that it is essentially the same, well, you have to change integrals and things like that. Now, what's really cool about this thing is the following. Now that it's completely and horribly marked up and no one can read any of it, but imagining that you could read it, the cool part is this matrix is non singular as long as T is positive. I can make T 10 to the minus nine and this matrix will be non singular. By the way, it's gonna be non singular, but if you integrate something – again, you have to assume some reasonable time scale and things like that, if I integrate something from zero to 10 to minus 9, that integral is gonna be very small. So that says that this inverse is going to be absolutely huge. And so what this says is oh, I can steer the input, I can steer the state from zero to a desired state in any number of steps. Sorry. In any amount of time I can do it very, very quickly, but it's gonna take a huge input. That's what this says. It all makes perfect sense. It all goes together and it makes absolute perfect sense here. Now, in the discrete time case, you might want to know why breaks down and what breaks down is real simple and it's for a simple reason. Let's see if I can say this and not sound like an idiot. The problem in the discrete time case is the time is discrete. This is the problem. Here, time is continuous. I can make it as small as I like. But here, what happens is I'll decrease T. When T equals N, I'm still safe by Cayley-Hamilton, but the minute I drop T below N, then there will be – I can take T down and at some point, this matrix can become non singular, in which the case, the inverse doesn't work. By the way, if I replace the inverse with a dagger, and make that a pseudo inverse, you get something very interestingly related to our famous homework problem. If I put a dagger in here, I'll get something really interesting. I'm gonna get you the least-norm input that will get you as close as you possibly can get to the desired target. Did this make sense? So that's what C dagger will do. And that's not the dagger from lecture four. That's not CC transpose C inverse C. Sorry. C trans – help me with this one. C transposed – whichever it is. C transposed quantity CC transposed inverse. Yes, that was it. It's not that dagger. It's the general dagger that requires the SVD. So that's what happens. Okay. Now, the energy required to hit a state is give by this integral. This integral from zero to T. And the cool thing about the integral is no matter how small T is, Q is positive definite. It's invertible. And I'm not gonna go over a lot of that, but that's sort of the basic idea. Let's see. And I'll just make the connection to the minimum energy [inaudible]. The same story happens. I have an integral, a positive semi-definite matrixes here. If I increase the time T that you're allowed to use to hit a target, this matrix goes up, this one goes down, and that's the quadratic form that gives you the minimum energy so you have the same result again. Okay. Let's quit for today. For those of you who just came in, I think I announced at the beginning of the class there's a tape ahead. It's today. It's today, 4:15, Skilling Auditorium, but as usual, you cannot trust me. Whatever it says on the website is what it really is. And statically, some of you should come because otherwise I'd be put in the terribly awkward position of giving a lecture to no one. It's never happened. Hopefully, this afternoon won't be a first. Okay. We'll quit here. | 677.169 | 1 |
Algebraic reasoning
2017_19_MeganStephens_MEDMATH_SignedCapstone_Spring2017 1 | 677.169 | 1 |
We
will begin with the following quiz as a review from the previous
section covered.
After
the quiz, we will review of some examples using Angle-Angle-Side
Congruence and Hypotenuse Leg Congruence. We may even go over
the worksheet that they did for homework on Section 5.7 covering
these topics.
I will
write the flow diagram on the overhead projector that is at the
beginning of this section 5.8. This will give the kids some review
about how to follow a flow diagram.
Then,
we will do this flow proof.
Then,
we will go over the same proof as a two-column proof to show students
the difference in the two styles. Some of them may prefer one
over the other. I will tell them that it really doesn't matter
to me which way they choose to do the proofs as long as they understand
the basis of a proof. Time permitting, we will go over some more
flow proof examples. I will assign then 2-3 flow proof problems
from the book to do for homework. I will tell them that they can
do the homework problems as two-column proofs if they so desire.
It may
be fun to take the students to the lab and let them work on doing
flow proofs using GSP tomorrow. | 677.169 | 1 |
Math Resource Center
The center assists students in completing math assignments and helps students to develop their quantitative skills. We work hard to support a student's thought process, provide immediate feedback, and help students to clarify misconceptions. We will not solve problems for a student but will offer guidance to help students develop an understanding of a concept. In addition, our goal is to help students gain the necessary confidence in his or her abilities to complete assignments effectively without assistance. Priority will be given to students in Algebra and Geometry courses | 677.169 | 1 |
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Help your students master this difficult but important topic. Step by step illustrated examples plus guided notes will give them the practice thy need to succeed.
Included in the lesson:
✓ A PowerPoint Lesson on Implicit Differentiation with 33 fully animated slides including more than 8 complete problems done step by step. Many of the examples have matching discovery graphs to peak student's interests in problems they cannot graph on their calculators. Also included are tangent line problems.
✓ Student Notes pages to match the main problems in the PowerPoint Lesson
✓ Handout with two problems which can be used as enrichment, homework, group work, or as an assessment.
This resource is designed Calculus 1, Calculus Honors AP, Calculus AB, and AP Calculus BC. It is from Unit 2, Differentiation and Derivatives | 677.169 | 1 |
Mathematics with Computer Science
How do we know that a complicated programme works under a variety of circumstances? How do we know how long it will take to solve a particular problem? Situations like this are at the interface of computer science and mathematics, and form part of this programme. Computer science has both practical and theoretical aspects. In your final year you can choose the area for your individual project from either end of the spectrum, implementing a computer application or analysing some of the algorithms used in a Computer Algebra System for example. | 677.169 | 1 |
Description
This text explores the methods of the projective geometry of the plane. Some knowledge of the elements of metrical and analytical geometry is assumed; a rigorous first chapter serves to prepare readers. Following an introduction to the methods of the symbolic notation, the text advances to a consideration of the theory of one-to-one correspondence. It derives the projective properties of the conic and discusses the representation of these properties by the general equation of the second degree. A study of the relationship between Euclidean and projective geometry concludes the presentation. Numerous illustrative examples appear throughout the text.
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This lucid introductory text offers both an analytic and an axiomatic approach to plane projective geometry. The analytic treatment builds and expands upon students' familiarity with elementary plane analytic geometry and provides a well-motivated approach to projective geometry. Subsequent chapters explore Euclidean and non-Euclidean geometry as specializations of the projective plane, revealing the existence of an infinite number of geometries, each Euclidean in nature but characterized by a different set of distance- and angle-measurement formulas. Outstanding pedagogical features include worked-through examples, introductions and summaries for each topic, and numerous theorems, proofs, and exercises that reinforce each chapter's precepts. Two helpful indexes conclude the text, along with answers to all odd-numbered exercises. In addition to its value to undergraduate students of mathematics, computer science, and secondary mathematics education, this volume provides an excellent reference for computer science professionals.
An ideal text for undergraduate courses in projective geometry, this volume begins on familiar ground. It starts by employing the leading methods of projective geometry as an extension of high school-level studies of geometry and algebra, and proceeds to more advanced topics with an axiomatic approach. An introductory chapter leads to discussions of projective geometry's axiomatic foundations: establishing coordinates in a plane; relations between the basic theorems; higher-dimensional space; and conics. Additional topics include coordinate systems and linear transformations; an abstract consideration of coordinate systems; an analytical treatment of conic sections; coordinates on a conic; pairs of conics; quadric surfaces; and the Jordan canonical form. Numerous figures illuminate the text.
This richly illustrated and clearly written undergraduate textbook captures the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of the space. The authors explore various geometries: affine, projective, inversive, hyperbolic and elliptic. In each case they carefully explain the key results and discuss the relationships between the geometries. New features in this second edition include concise end-of-chapter summaries to aid student revision, a list of further reading and a list of special symbols. The authors have also revised many of the end-of-chapter exercises to make them more challenging and to include some interesting new results. Full solutions to the 200 problems are included in the text, while complete solutions to all of the end-of-chapter exercises are available in a new Instructors' Manual, which can be downloaded from
Based on a historic approach taken by instructors at MIT, this text is geared toward junior and senior undergraduate courses in analytic and projective geometry. Starting with concepts concerning points on a line and lines through a point, it proceeds to the geometry of plane and space, leading up to conics and quadrics developed within the context of metrical, affine, and projective transformations. The algebraic treatment is occasionally exchanged for a synthetic approach, and the connection of the geometrical material with other fields is frequently noted. Prerequisites for this treatment include three semesters of calculus and analytic geometry. Special exercises at the end of the book introduce students to interesting peripheral problems, and solutions are provided.
The basic results and methods of projective and non-Euclidean geometry are indispensable for the geometer, and this book--different in content, methods, and point of view from traditional texts--attempts to emphasize that fact. Results of special theorems are discussed in detail only when they are needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity. Topics include the projective plane, polarities and conic sections, affine geometry, projective metrics, and non-Euclidean and spatial geometry. Numerous figures appear throughout the text, which concludes with a bibliography and index.
Geometry is probably the most accessible branch of mathematics, and can provide an easy route to understanding some of the more complex ideas that mathematics can present. This book is intended to introduce readers to the major geometrical topics taught at undergraduate level, in a manner that is both accessible and rigorous. The author uses world measurement as a synonym for geometry - hence the importance of numbers, coordinates and their manipulation - and has included over 300 exercises, with answers to most of them. The text includes such topics as: - Coordinates - Euclidean plane geometry - Complex numbers - Solid geometry - Conics and quadratic surfaces - Spherical geometry - Quaternions It is suitable for all undergraduate geometry courses, but it is also a useful resource for advanced sixth formers, research mathematicians, and those taking courses in physics, introductory astronomy and other science subjects.
This concise review examines the geometry of the straight line, circle, plane, and sphere as well as their associated configurations, including the triangle and the cylinder. Aimed at university undergraduates, the treatment is also useful for advanced students at the secondary level. The straightforward approach begins with a recapitulation of previous work on the subject, proceeding to explorations of advanced plane geometry, solid geometry with some reference to the geometry of the sphere, and a chapter on the nature of space, including considerations of such properties as congruence, similarity, and symmetry. The text concludes with a brief account of the elementary transformations of projection and inversion. Numerous examples appear throughout the book.
Brief but rigorous, this text is geared toward advanced undergraduates and graduate students. It covers the coordinate system, planes and lines, spheres, homogeneous coordinates, general equations of the second degree, quadric in Cartesian coordinates, and intersection of quadrics. Mathematician, physicist, and astronomer, William H. McCrea conducted research in many areas and is best known for his work on relativity and cosmology. McCrea studied and taught at universities around the world, and this book is based on a series of his lectures.
Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates, and brief introductions to Plücker coordinates and Grassmann coordinates are presented. This book looks carefully at linear, quadratic, cubic and quartic figures in two, three and higher dimensions. It deals at length with the extensions and consequences of basic theorems such as those of Pappus and Desargues. The emphasis throughout is on special configurations that have particularly interesting symmetry properties. The intricate and novel ideas of 'Donald' Coxeter, who is considered one of the great geometers of the twentieth century, are also discussed throughout the text. The book concludes with a useful analysis of finite geometries and a description of some of the remarkable configurations discovered by Coxeter. This book will be appreciated by mathematics students and those wishing to learn more about the subject of geometry. It makes accessible subjects and theorems which are often considered quite complicated and presents them in an easy-to-read and enjoyable manner.
This volume became the standard text in the field almost immediately upon its original publication. Renowned for its lucid yet meticulous exposition, it can be appreciated by anyone familiar with high school algebra and geometry. Its arrangement follows the traditional pattern of plane and solid geometry, in which theorems are deduced from axioms and postulates. In this manner, students can follow the development of non-Euclidean geometry in strictly logical order, from a fundamental analysis of the concept of parallelism to such advanced topics as inversion and transformations. Topics include elementary hyperbolic geometry; elliptic geometry; analytic non-Euclidean geometry; representations of non-Euclidean geometry in Euclidean space; and space curvature and the philosophical implications of non-Euclidean geometry. Additional subjects encompass the theory of the radical axes, homothetic centers, and systems of circles; inversion, equations of transformation, and groups of motions; and the classification of conics. Although geared toward undergraduate students, this text treats such important and difficult topics as the relation between parataxy and parallelism, the absolute measure, the pseudosphere, Gauss' proof of the defect-area theorem, geodesic representation, and other advanced subjects. In addition, its 136 problems offer practice in using the forms and methods developed in the | 677.169 | 1 |
Succeed in Maths 11-14 Years by Mike Bell one in a series of books that is both a guide and exercise book for students who want to improve their kwledge and skills in core subjects. Each book contains detailed analysis of all the study relative to the 11-14 year age level and takes into account all levels of ability. | 677.169 | 1 |
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"Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Calculus has widespread applications in science and engineering and is used to solve complicated problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions." | 677.169 | 1 |
Internet @ Schools
Maplesoft has announced the publication of the second edition of The Mathematics Survival Kit - Maple Edition. This ebook is designed to give students the information they need to overcome difficulties encountered while working on homework or assignment problems. It gives students the opportunity to review exactly the concept or technique they are struggling with, learn what they need to know, work through an example, practice as much as they want using randomly generated and automatically graded questions on that exact topic, and then continue with their homework.
The ebook was written by award-winning teacher Jack Weiner and is based on the recent second edition of his printed book, The Mathematics Survival Kit, published by Thomson Nelson. Drawing from his 30 years of teaching experience, Weiner identified topics that both university and high school students find most problematic. The latest release of The Mathematics Survival Kit - Maple Edition contains 25 additional topics, bringing the total to 140. All of the new topics were created by Weiner in direct response to requests from readers of the first edition of the book, including students, teachers, and even parents. New materials range from basic operations such as factoring and fractions, to graph sketching, vectors, and integration.
The new version of The Mathematics Survival Kit - Maple Edition is available from the Maplesoft web store at Students may purchase a specially priced bundle containing Maple 14 Student Edition and the Mathematics Survival Kit - Maple Edition starting at $109 USD. As a stand-alone purchase, the Mathematics Survival Kit is available for $29 USD. | 677.169 | 1 |
Product Description
The concepts covered in this course include postulates, theorems, corollaries, proof, congruence, construction, circles, polygons, triangles, and three-dimensional figures in space. Materials include a textbook and a two-volume teacher's manual (both published by Bob Jones University Press), as well as an exam kit and a parent-teacher guide (each published by Christian Liberty Press). Students are required to complete most exercises in the text as well as all five comprehensive exams. Prerequisite: Saxon Algebra 1 (or equivalent) with a final grade of "B" or better. Saxon Algebra 2 recommended.
1.0 credit.
Grades 10-12 | 677.169 | 1 |
This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity. ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers. For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics. ...Continua Nascondi | 677.169 | 1 |
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MATHEMATICS 324: COMPLEX ANALYSIS
The aim of this module is to introduce fundamental concepts in complex analysis. The following topics are discussed: types of convergence of series of functions, Taylor series and zeroes, differentiation, complex exponential, trigonometric and logarithmic functions, Laurent series and isolated singularities, integration along a path, Cauchy's Theorem and Integral Formula, Liouville's Theorem, the Residue Theorem, applications and other topics.
Complex analysis is a fascinating and highly useful part of Mathematics, and has applications in, among others, Functional Analysis, Number Theory, Approximation Theory, Applied Mathematics and Engineering.
Module information
21539 324 (16) Mathematics 324; language specification: A
Third year, first semester of the Programme in the Mathematical Sciences
Prerequisite pass modules (PP ³ 50): Mathematics 214, 244
Classnotes will be provided
There are several books on the subject in the university library
Lecturer
Dr A Muller, Room 1021, Department of Mathematics
Learning opportunities
The learning material is covered during the lecture periods. During the tutorial periods problems are solved under supervision.
The dates and times of the class test and exam are published on the University's web page. For more information on exam regulations, see the Yearbook of the University, Parts 1 and 5.
Rationale
This module is presented within the Programme in the Mathematical Sciences. It contributes to providing a basic training in Mathematics and is one of the core modules required in order to obtain the B.Sc. degree with a major in Mathematics.
Outcomes
A student who passed this module should be equipped with the knowledge and comprehension of the basic concepts in metric spaces, as well as of complex analysis. Specifically such a student should have the following knowledge and skills:
Can determine Taylor series and circles of convergence of complex functions. Can determine Laurent series and domains of convergence of complex functions.
Can identify and classify the zeroes and isolated singularities of a complex function.
Know and understand the properties of the complex exponential and trigonometric functions. Can determine arguments and complex logarithms.
Can investigate the differentiability and determine the derivative of a complex function using the theory of the Cauchy Riemann equations.
Can integrate complex functions along paths using different techniques.
Can compute residues and use the Residue Theorem to determine complex integrals, as well as improper integrals and/or infinite series. | 677.169 | 1 |
ISBN 13: 9780321536464
MathXL Tutorials on CD for Fundamentals of Precalculus
This interactive tutorial CD-ROM provides algorithmically generated practice exercises that are correlated at the objective level to the exercises in the textbook. Every practice exercise is accompanied by an example and a guided solution designed to involve students in the solution process. Selected exercises may also include a video clip to help students visualize concepts. The software provides helpful feedback for incorrect answers and can generate printed summaries of students' progress
"synopsis" may belong to another edition of this title.
From the Back Cover:
"Fundamentals of Precalculus" is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports students' mathematical needs with a number of tools newly developed for this revision. A student who is well acquainted with the material in this text will have the necessary skills, understanding, and insights required to succeed in calculus.
About the Author:
Mark Dugopolski was born and raised in Menominee, Michigan. He received a degree in mathematics education from Michigan State University and then taught high school mathematics in the Chicago area. While teaching high school, he received a master's degree in mathematics from Northern Illinois University. He then entered a doctoral program in mathematics at the University of Illinois in Champaign, where he earned his doctorate in topology in 1977. He was then appointed to the faculty at Southeastern Louisiana University, where he now holds the position of professor of mathematics. He has taught high school and college mathematics for over 30 years. He is a member of the MAA, the AMS, and the AMATYC. He has written many articles and mathematics textbooks. He has a wife and two daughters. When he is not working, he enjoys hiking, bicycling, jogging, tennis, fishing, and motorcycling | 677.169 | 1 |
Calculus is a limb of maths centred on points of confinement, methods, derivatives, integrals, and unbounded sequence. This subject constitutes a major part of up to date arithmetic training. It has two major limbs, differential maths and necessary analytic, which are identified by the basic theorem of analytic. Maths is the investigation of change, in the same way that geometry is the investigation of shape and variable based maths is the investigation of operations and their requisition to understanding mathematical statements. A course in analytic is a portal to different, more progressed courses in maths dedicated to the investigation of capacities and points of confinement, broadly called scientific investigation.
Calculus has boundless requisitions in science, mass trading, and building and can tackle a large number of situations for which polynomial math apart from everyone else is lacking. | 677.169 | 1 |
The objectives of this course are to
introduce students to elementary set theory,
introduce students to real numbers and mathematical induction,
introduce students to real sequences and series,
introduce students to quadratic equations and polynomials,
introduce students to binomial theorem and expansion,
introduce students to circular measure and trigonometric functions. | 677.169 | 1 |
Welcome! to Free Math Education Online Tutorials, which offers free math tutoring help, math tutorial of good qualifed tutors and team work of high qualified experienced team. If you are interested in improving your math skills of Algebra, Geometry, Trigonometry, Coordinate Geometry etc, then this FREE MATH EDUCATION site is really help you. We also provide math worksheets of these available tutorials/courses.
Wednesday, 8 February 2012
Linear programming problem standard form
For developing a general method of solution of a linear programming problem, we used the standard form. The basics of standard form is
1. Non Negativity Constraints i.e. all the constraints are transformed into equations using the addition and substitution of new non negative constraints and change the equation into standard form. These new varaibles are called as slack or surplus variables and are added if <= sign in constraints and subtract if >= in the constraints.
It you need any help in algebra math topics then free math education online tutorials are really help you. | 677.169 | 1 |
CUSAT Syllabus
Syllabus for CUSAT Entrance TestUNIT 1 : SETS, RELATIONS AND FUNCTIONS:
Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations,functions;. oneone, into and onto functions, composition of functions.
UNIT 2 : COMPLEX NUMBERS AND QUADRATIC EQUATIONS:
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and coefficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3 : MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.
UNIT 9 : INTEGRAL CALCULUS:
Integral as an anti derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities. Evaluation of simple integrals of the type. Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
UNIT 10: DIFFERENTIAL EQUATIONS:
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type:
dy+ p (x) y = q (x)dx
UNIT 11: COORDINATE GEOMETRY:
Cartesian system of rectangular coordinates 10 in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines
Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines.
Circles, conic sections
Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.
UNIT 12: THREE DIMENSIONAL GEOMETRY:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT 13: VECTOR ALGEBRA:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT 3: LAWS OF MOTION
Force and Inertia, Newton's First Law of motion; Momentum, Newton's Second Law of motion; Impulse; Newton's Third Law of motion. Law of conservation of linear momentum and its applications, Equilibrium of concurrent forces. Static and Kinetic friction, laws of friction, rolling friction. Dynamics of uniform circular motion: Centripetal force and its applications.
UNIT 4: WORK, ENERGY AND POWER
Work done by a constant force and a variable force; kinetic and potential energies, workenergy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservative and nonconservative forces; Elastic and inelastic collisions in one and two dimensions.
UNIT 5: ROTATIONAL MOTION
Centre of mass of a two-particle system, Centre of mass of a rigid body; Basic concepts of rotational motion; moment of a force, torque, angular momentum, conservation of angular momentum and its applications; moment of inertia, radius of gyration. Values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems and their applications. Rigid body rotation, equations of rotational motion.
UNIT 6: GRAVITATION
The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Kepler's laws of planetary motion. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a satellite. Geo-stationary satellites.
UNIT 8: THERMODYNAMICS
Thermal equilibrium, zeroth law of thermodynamics, concept of temperature. Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Carnot engine and its efficiency.
UNIT 9: KINETIC THEORY OF GASES
Equation of state of a perfect gas, work doneon compressing a gas.Kinetic theory of gases - assumptions, concept of pressure. Kinetic energy and temperature: rms speed of gas molecules; Degrees of freedom, Law of equipartition of energy,applications to specific heat capacities of gases; Mean free path, Avogadro's number.
UNIT 11: ELECTROSTATICS
Electric charges: Conservation of charge, Coulomb's law-forces between two point charges, forces between multiple charges; superposition principle and continuous charge distribution. Electric field: Electric field due to a point charge, Electric field lines, Electric dipole, Electric field due to a dipole, Torque on a dipole in a uniform electric field. Electric flux, Gauss's law and its applications to find field due to infinitely long uniformly charged straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Electric potential and its calculation for a point charge, electric dipole and system of charges; Equipotential surfaces, Electrical potential energy of a system of two point charges in an electrostatic field. Conductors and insulators, Dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, Energy stored in a capacitor.
UNIT 12: CURRRENT ELECTRICITY
Electric current, Drift velocity, Ohm's law, Electrical resistance, Resistances of different materials, V-I characteristics of Ohmic and nonohmic conductors, Electrical energy and power, Electrical resistivity, Colour code for resistors; Series and parallel combinations of resistors; Temperature dependence of resistance. Electric Cell and its Internal resistance, potential difference and emf of a cell, combination of cells in series and in parallel. Kirchhoff's laws and their applications. Wheatstone bridge, Metre bridge. Potentiometer - principle and its applications.
UNIT 13: MAGNETIC EFFECTS OF CURRENT AND MAGNETISM
Biot - Savart law and its application to current carrying circular loop. Ampere's law and its applications to infinitely long current carrying straight wire and solenoid. Force on a moving charge in uniform magnetic and electric fields. Cyclotron. Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter. Current loop as a magnetic dipole and its magnetic dipole moment. Bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements. Para-, dia- and ferro- magnetic substances. Magnetic susceptibility and permeability, Hysteresis, Electromagnets and permanent magnets.
UNIT 5: CHEMICAL THERMODYNAMICS
Fundamentals of thermodynamics: System and surroundings, extensive and intensive properties, state functions, types of processes. First law of thermodynamics - Concept of work, heat internal energy and enthalpy, heat capacity, molar heat capacity; Hess's law of constant heat summation; Enthalpies of bond dissociation, combustion, formation, atomization, sublimation, phase transition, hydration, ionization and solution. Second law of thermodynamics; Spontaneity of processes; DS of the universe and DG of the system as criteria for spontaneity, Dgo (Standard Gibbs energy change) and equilibrium constant.
UNIT 12: GENERAL PRINCIPLES AND PROCESSES OF ISOLATION OF METALS
Modes of occurrence of elements in nature, minerals, ores; Steps involved in the extraction of metals - concentration, reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13: HYDROGEN
Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen; Physical and chemical properties of water and heavy water; Structure, preparation, reactions and uses of hydrogen peroxide; Hydrogen as a fuel.
UNIT 14: S - BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS) Group - 1 and 2 Elements
General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships. Preparation and properties of some important compounds - sodium carbonate and sodium hydroxide; Industrial uses of lime, limestone, Plaster of Paris and cement; Biological significance of Na, K, Mg and Ca.
UNIT 15: P - BLOCK ELEMENTS Group - 13 to Group 18 Elements
General Introduction: Electronic configuration and general trends in physical and chemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group. Groupwise study of the p – block elements Group – 13
Preparation, properties and uses of boron and aluminium; properties of boric acid, diborane, boron trifluoride, aluminium chloride and alums. Group – 14
Allotropes of carbon, tendency for catenation; Structure & properties of silicates, and zeolites. Group – 15
Properties and uses of nitrogen and phosphorus; Allotrophic forms of phosphorus; Preparation, properties, structure and uses of ammonia, nitric acid, phosphine and phosphorus halides, (PCl3, PCl5); Structures of oxides and oxoacids of phosphorus. Group – 16
Preparation, properties, structures and uses of ozone; Allotropic forms of sulphur; Preparation, properties, structures and uses of sulphuric acid (including its industrial preparation); Structures of oxoacids of sulphur. Group – 17
Preparation, properties and uses of hydrochloric acid; Trends in the acidic nature of hydrogen halides; Structures of Interhalogen compounds and oxides and oxoacids of halogens. Group –18
Occurrence and uses of noble gases; Structures of fluorides and oxides of xenon.
UNIT 25: POLYMERS
General introduction and classification of polymers, general methods of polymerization-addition and condensation, copolymerization; Natural and synthetic rubber and vulcanization; some important polymers with emphasis on their monomers and uses - polythene, nylon, polyester and bakelite. | 677.169 | 1 |
published:29 Sep 2016
views:29927709 May 2017
views:233001
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
published:08 Jul 2013
views:39511920 Apr 2012
views:6757771:27:26
Calculus 1 Lecture 1.1: An Introduction to Limits
Calculus 1 Lecture 1.1: An Introduction to LimitsCalculus - Introduction to Calculus1. What is Calculus (Hindi)
why study differentiation and integration
19:53
Calculus at a Fifth Grade Level
Calculus at a Fifth Grade LevelCalculus: What Is It?What is Calculus Used For? | Jeff Heys | TEDxBozemanBig Picture of Calculus❤︎² Basic Integration... How? (mathbff)24:44
The Birth Of Calculus (1986)
The Birth Of Calculus (1986)21:58
Understand Calculus in 10 Minutes
Understand Calculus in 10 Minutes p...
published: 28 Apr 2017 address is: 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!!
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published: 29 Sep 2016 I in 20 Minutes (The Original) by Thinkwell
1. What is Calculus (Hindi)
why study differentiation and integration
published: 22 Jan 2016 09 May 2017published: 08 Jul 2013 e...
published: 20 Apr 2012 m...
published: 19 May 2011
published: 01 May 2016 ...
published: 22 May 2009Essence of calculus, chapter 1
I want you to feel that you could have invented calculus for yourself, and in this first video of the series, we see how unraveling the nuances of a simple geom... inv... PatreonCalculus - Introduction to Calculus
This video will give you a brief introduction to calculus. It does this by explaining that calculus is the mathematics of change. A couple of examples are pre... jus...Calculus: What Is It?
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wonderi...
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
What is Calculus Used For? | Jeff Heys | TEDxBozeman
This talk describes the motivation for developing mathematical models, including models that are developed to avoid ethically difficult experiments. Three diff...
Introduction to limits
Watch... couldThis video will give you a brief introduction to calculus. It does this by explaining tha...This video shows how calculus is both interesting and useful. Its history, practical uses,...This talk describes the motivation for developing mathematical models, including models th...Calculus is about change. One function tells how quickly another function is changing. Pro...MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integrati...)/Burden Of Grief
The war is over The last battles are gone Swords laying broken My bloodwork is all done I sit down for calming My breath is lessening I�m starting to tremble My sight is clearing My head is weary A dreadful awakening What has driven me Into insanity Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger back on the shelf I look around As I raise from my rest Discover what I�ve done No life I have left My heart is in pieces My soul is laying bare Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy That relegates sexual harassment to a pretty negligible part of the political calculus ... ....
ERBIL, Kurdistan Region (Kurdistan Region) – A Jewish-Kurdish organization in support of the Kurdistan Region's independence, which includes several prominent Jews in Europe and North America, was unveiled on Wednesday ... "There is a kinship between the two peoples, the Jewish one and the Kurdish one, that transcends merely political calculus," the pro-Kurdish group's founder said ... Editing by Sam A. Roy Moore accuser says she was not paid to tell her story ... Here's the thing, though ... That relegates sexual harassment to a pretty negligible part of the political calculus ... ....
GAINESVILLE, Fla. — Seven short minutes. Tick-tock. That's a pot of coffee brewing to completion. It's a melodic tune or two floating from a car radio. Seven minutes are a handful of commercials on television. Seven minutes also might be a lifeline in your golden years ... "People may say, 'Well, doing this little can't mean much.' But it does ... " Researchers aren't surprised that walking is especially critical in the calculus of our health ... Related Articles ... That relegates sexual harassment to a pretty negligible part of the political calculus ... ....
AllahpunditPosted at 8.41 pm on November 21, 2017 Share on Facebook. Share on Twitter. Isn't it interesting that last week, when there was one accuser, the calls for Franken to quit were loudest. Now that there's a second accuser, they've begun to quiet down. You would think it'd be the opposite ... cauterize the wound ... We could probably get left and right together to quantify that calculus too ... Tags ... Share on Facebook ... ....
Facebook Twitter Google + Email CommentPrint. Once ... But eight years with Barack Obama as president of the United States and now, the advent of Trump, have transformed the calculus of the Middle East ... ....
The Senate sex-and-power calculus has come to this. Roy Moore fans say, "Al Franken isn't dropping out. Why should our guy?" Al Franken fans say, "Roy Moore isn't dropping out. Why should our guy?". The language of power has, for some on both sides, displaced the language of right and wrong ...The U.S.Senate must be one of the few remaining workplaces in America that even considers a wrist-slap for such acts | 677.169 | 1 |
How to improve in Algebra
If Algebra is your weak area, here is a comprehensive collection of the exact material you need to make it your strong area.
In order to improve your performance in algebra, you should first of all revise the basics of algebra. You should start from remembering the basic algebraic formulas, and then learn to make equations. This should be followed by methods to solve linear equations, quadratic equations, inequalities and questions based on progressions. You can start from any 8th or 9th standard NCERT books to clear the basic concepts of the same. You can refer the material that you follow for preparing the competitive exams. Do remember, that mastering any chapter or area needs sufficient practice before the exam. You can go through the following articles on algebra on our website/app, which will help you to clear the basics and to gain confidence. | 677.169 | 1 |
Student Print Materials for Algebra, Data Analysis
By Elevated Lab Press
Description
The print materials for students supplement the video lessons of Elevated Math. Over 800 pages of .pdf files are here, including a complete Scope & Sequence, CCSS alignments, a glossary, 40 module tests, and worksheets.
The first two .pdf files for each lesson align with problems in the videos. "Additional Practice" and "Independent Practice" provide more problems to solve.
Each .pdf file is writable. Worksheets can be filled out with the pen/erase tool and the files can be emailed or printed. | 677.169 | 1 |
Can someone reccommend me a book rigerously treating multivariable calculus? I have been through analysis and calc1, and 2 but I want to do calc 3 in a rigorous manner. I have also read the first four chapters of Munkres, if that is worth noting. | 677.169 | 1 |
Essential Mathematics for Economics and Business has become established as one of the leading introductory books on mathematics. It combines a non-rigorous approach to mathematics with applications in economics and business. The fundamental mathematical concepts are explained as simply and as briefly as possible, using a wide selection of worked examples, graphs and real-world applications. | 677.169 | 1 |
PreCalculus Advice
Showing 1 to 1 of 1
Dr. Thomas demonstrates helpful explanations during lecture. As students, we learn at our own pace since we do most work on ALEKS, an online program. Dr. Thomas supplies us with schedules that help us time manage for midterms and the final.
Course highlights:
I learned how to properly time manage as I had to get certain chapters done by certain midterms. I also learned new techniques on strategies to solve problems.
Hours per week:
6-8 hours
Advice for students:
Make sure to work on ALEKS almost every day as it will keep you in stride both for the midterm and problems that Dr. Thomas has students do during lecture. | 677.169 | 1 |
originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists | 677.169 | 1 |
Welcome to 8th
grade Pre-Algebra! This class offers a challenging curriculum focused on linear equations
and functions, systems of equations, geometry, exponents, and data analysis. In
order to create an environment where scholars take responsibility for learning,
I expect all to participate in class discussions, collaborate in groups, create
projects, and complete in-class, online, as well as at-home assignments. Students will be required to critically
think, model work, and defend conclusions verbally, mathematically, and in
sentence form. Organization and effort
are key! I look forward to a successful year together. If you have questions at
any point in the year, feel free to contact me by email or phone.
∞ Be Responsible: with
all required work and materials needed for class
∞ Be Ready: seated and ready
to work when bell rings
∞ Be Truthful &
Trustworthy: Set a positive example for others
∞ Follow all school
rules in handbook
3.Consequences: May include one or more of the
following
∞ Verbal Reminder ∞ Behavior/Academic Contract
∞ Removal from class ∞
Referral to Administration
∞ Conference with
teacher and/or parent ∞ Work
Detail
∞ Detention-teacher
assigned 15 or 30 minutes before or after school only
4.Homework:
∞ Expect
homework approximately three to four days a week. Assignments due at the
beginning of class.
∞ All homework must be
completed in pencil. Illegible handwriting or completion in pen will result
in re-doing the assignment.
∞ Students must show
all steps in order to receive credit for an assignment.
∞
All problems must be reasonably attempted mathematically or a written
explanation must be provided, addressing the area that is not understood.
∞ Assignment submitted
online must be accompanied by work on paper and turned in for credit.
5.Attendance
and Make Up work:
∞ It is solely the
responsibility of the scholar to check the assignment board, website "Homework
Calendar," and folders after returning from an absence to ensure all work is
received and made up. The teacher is not
responsible for reminding the scholar of this responsibility.
∞ The day you return, all
assignments which were due the first day you were absent must be turned in.
∞ You will have 1 day
per day absence to make up work. (ie. 2 days absent, 2 days to make up work).
∞
Tests and Regular quizzes must be arranged with teacher and be made up outside
of class. Homework quizzes may not be made up.
6.Tardies:
Scholars should be in their seats, prepared to work when the bell rings.
7.Grades: (detailed grading policy to be outlined in a
separate document)
∞ Remind.com:Email and texting communication application
I use to send important updates and reminders regarding upcoming assignments, projects,
quizzes, and tests.
∞ Email: pjmcdonnell@mpsaz.orgThe
best way to contact me is via email.
Phone messages will be returned before or after school. I regularly send emails regarding class
updates to any email provided on the next page.
∞ Online Gradebook:(synergy
portal) Shows missing assignments and grades for assignments and classes. I
update grades several times a week. Contact the front desk if you need
assistance with your parent login or password.
9.Resources:
Students Use Them!
∞ Big Ideas Textbook: Odd
numbered answers are in the back of the book. I highly recommend students check
their answers after each question. However, all steps to support answer must be
shown on paper.
∞ Notes from class composition notebook
(copies of lesson notes are on my website): Review current chapter
lessons each evening for 5-8 minutes.
Before beginning homework, review class notes from that day and refer
back to it if you have questions.
∞ Internet: It is okay to look up concepts in which you
need help! Videos, similar examples etc.
∞ Relatives or Friends:Find a friend or relative who can assist
(not give answers) – Utilize all other resources on your own first.
∞
Tutoring:My before and after school hours are posted
on my Stapley website and updated weekly. | 677.169 | 1 |
Each worksheets is visual, differentiated and fun. Includes a range of useful free teaching resources.
Trigonometry Functions Chart - Great reference for Precalculus Students and Teachers. It includes all the major functions of sin, cos, and tan. It even includes the reciprocal functions of csc, sec, and cot.
Learn about Trigonometry with this clear and comprehensive educational poster. The poster is perfect for being displayed in classrooms, school hallways and at home | 677.169 | 1 |
All of these courses use a combination of traditional mathematics as well as the Cognitive Tutor computer program. The Cognitive Tutor program allows students to work through Algebra concepts at their own pace. It provides addition problems, when needed, to help students achieve mastery.
There is no textbook distributed to the students in these courses. They are, however, required to keep a three-ring binder for all of their lesson notes, example problems, projects, assignments, and assessments. In essence, the students are building their own textbook with these materials.
Grading Procedures
Grades are calculated using the following weighted system:
Tests, Quizzes, Graded Assignments/Projects: 35% of 9 weeks' grade
Homework, Group Work, General Projects: 35% of 9 weeks' grade
Cognitive Tutor Program: 30% of 9 weeks' grade
Homework Policy
Homework is a very integral part of learning mathematics and it is imperative that it is completed on a regular basis. It is the responsibility of the student, when necessary, to ask questions about the homework while working on it or when the class reviews it.
Homework is due at the beginning of the class unless otherwise stated.
Homework is always checked for completeness using a five-point scale. Appropriate work should always accompany homework to receive full credit.
Late homework will receive no more than 3 out of 5 points.
It is the responsibility of the student to get any missing homework due to class absence.
I am a 1994 graduate of Center High School. I began my undergraduate work in Secondary Mathematics at Millersville University and completed it at Robert Morris University. I received my Masters degree in Curriculum and Instruction from Clarion University.
Teaching Career:
I have taught at Hopewell High School since the start of my teaching career in 2003. In addition to the Algebra courses I teach, I am also the Varsity Cheerleading Coach. I also work at Penn State University teaching SAT prep courses in mathematics and a variety of summer camps. | 677.169 | 1 |
What to expect:
Approximately 1/3 of the exam will be calculations and examples, the rematinder will be more theoretical exercises.
You should review the main theorems and techniques from each section. You
will not be required to memorize proofs, but you should have a general idea
of what is involved. Similarly with homework problems.
When studying, try to avoid a "memorizing list of facts" mentality. Instead
focus on "filling in pieces of a puzzle." A useful exercise might be to
figure out for each theorem what previous theorems are used in the proof.
This will show the relationships among the various ideas. It is easier to
remember the ideas if they fit into some framework.
You might also try filling one side of a sheet of paper with all the most
important ideas. Finished? Good. Now start over with a half sheet of paper.
When you have finished with that, try again with a quarter sheet. Keep
going...
Mathematics is a subject that is best learned by doing. You might do well to
try solving some extra problems...
Exercises: You should not rely exclusively on these problems for your
studying, but they can be a big help. | 677.169 | 1 |
The following four investigations showcase various methods to generate and explore parabolas.
Natural Parabolas
Lynda Waters
The purpose of this lesson is to use digital photography of "natural" parabolas to model quadratic functions. Once the digital pictures are downloaded to a computer, they can be sized to fit, exported, and imported onto Geometer's Sketchpad. By placing a coordinate grid onto the picture, the lesson continues with instructions on how to determine the equation of the function that fits the pictured parabola. Pictures of "natural" parabolas will give meaning to the mathematics.
Using Geometer's Sketchpad To Construct Parabolas With Focus & Directrix
Kelley Butler
By definition a parabola is the set of points in a plane that are the same distance from a fixed point - the focus - as from a fixed line - the directrix. In an effort to visually demonstrate the dynamic sketch, step-by-step instructions will be provided using Geometer's Sketchpad. Questions will also be included that are intended to clarify the relationships between the focus, directrix, and graph of any parabola.
An Introduction to Bézier Curves: Generating a Parabola From Three Points
Seth Bundy
Bézier curves are used in computer-aided geometric design. They were initially invented for car design but are commonly used today in font design. The general idea is that curves can be created when we use linear interpolation on line segments defined parametrically. To introduce these ideas to precalculus students, we explore how parabolic curves are created from three points connected by two line segments. We begin by doing string art and showing that the resulting curve is indeed a parabola. We proceed to a review of parametric equations, a foundational concept which will ground our work and provide an algebraic connection. An investigation on Geometer's Sketchpad illustrates how the parametric definition plays out graphically. Finally, some unproven conjectures are posed which will challenge students to utilize prior knowledge in making some discoveries and connections. These explorations will help to enrich this introduction for future use and to smooth the path toward more complicated Bézier curves.
Lab: Using Digital Video to Model Parabolas Parametrically
Connie Savoie
The purpose of this laboratory is to generate data and find the parametric equations to model each situation. The goal is to increase the students' working knowledge of parametric equations. Modeling them in real life will help students visualize what parametric equations really mean.
Students will generate lines and parabolas by rolling a ball across the graph paper while filming with a digital camera. Viewing the path of the ball while using the frame by frame progression allows points to be gathered with t, x and y. Parametric equations can be developed with these points, either using regression on the TI-83/84 or by hand. Further exploration can be made in several directions:
explore the effect of different slopes on the coefficient of the squared term
(in a calculus class) taking the first and second derivatives to explore velocity and acceleration
use trigonometry to find the angles of the slopes
Composing the Plane: Isometries as Graphs and Matrices
Rani Fischer
This unit examines the relationship between the graphs and matrices of isometries and compositions of isometries. Isometries are functions which transform objects in a way that changes neither size nor shape. Students are introduced to isometries: reflections, rotations, translations, and glide reflections with patty paper folding and tracing. From this experience they can construct these images on Geometer's Sketchpad and see a slightly different representation. Finally, students use trial and error first to formulate matrices for reflections over axes and rotations. Translations are shown both by vectors and by writing as 3x3 matrices. Other concepts that arise are line symmetry, symmetry groups, matrix multiplication with and without a calculator, and non-commutativity of matrix multiplication. This unit takes at least two weeks and assumes some familiarity with Geometer's Sketchpad software.
Petunia Has a Plan
Donna Curran, Marta Trimble, Celeste Williams
The working group project is centered around the subject of parabolas. The focus of this particular group is to create a multilevel approach to introducing parabolas which ranges from grade 8 to grade 10. It begins with a story that moves students through the process of forming generalizations and predictions based upon patterns drawn from the information to constructing tables and graphically depicting the information from those tables. In this process, the understanding of parabolas precipitates and is facilitated by questioning from and to the students | 677.169 | 1 |
Cuevas, Gilbert
Mathematics education, representation of mathematical ideas, role of language in the learning and teaching of mathematics.
Gilbert Cuevas is a Professor in the Department of Mathematics at Texas State University. He is also Professor Emeritus of Mathematics Education from the University of Miami. Dr. Cuevas' research focuses on the mathematics education of teachers, instructional strategies for English Language Learners, and the use of mathematical representations. He has published in the Journal for Research in Mathematics Education and other journals such as the Mathematics Teacher and Teaching Children Mathematics. His most recent works include algebra and pre-calculus textbooks as well as Navigating Through Algebra in Grades 3-5, published by the National Council of Teachers of Mathematics, and Integrating Mathematics and Literacy in Secondary Education (co-written with his wife). Dr. Cuevas has given scholarly presentations at regional, national and international conferences on topics of student assessment, language and equity in mathematics education. He has collaborated on research and teacher development projects with colleagues in Australia, Mexico and Central America. In 1990, he received an award from the Mathematical Sciences Education Board of the National Research Council for his contribution to efforts to increase the participation of minority students in mathematics education.
Cuevas, G. (1998). The role of complex mathematical tasks in teacher education. In Mathematical Sciences Education Board (Ed.), High school Mathematics at work, Washington, DC: National Research Council. | 677.169 | 1 |
Calculate this: studying CALCULUS simply received plenty easier!
Written in a step by step layout, this useful advisor starts by way of overlaying the basics--number structures, coordinates, units, and capabilities. you are going to flow directly to limits, derivatives, integrals, and indeterminate varieties. Transcendental capabilities, tools of integration, and purposes of the crucial also are coated. transparent examples, concise motives, and labored difficulties make it effortless to appreciate the fabric, and end-of-chapter quizzes and a last examination aid strengthen key recommendations.
it is a no-brainer! you will get:
Applications of the spinoff and the necessary
Rules of integration
Coverage of incorrect integrals
An rationalization of calculus with logarithmic and exponential features
Details on calculation of labor, averages, arc size, and floor area
easy sufficient for a newbie, yet hard adequate for a sophisticated pupil, Calculus Demystified , moment version, is one ebook you will not are looking to functionality with no!
Instruction manual of Sinc Numerical tools provides a terrific street map for dealing with common numeric difficulties. Reflecting the author's advances with Sinc given that 1995, the textual content so much particularly presents an in depth exposition of the Sinc separation of variables procedure for numerically fixing the total diversity of partial differential equations (PDEs) of curiosity to scientists and engineers.
Speak is on the middle of daily lifestyles, but social scientists have typically handled it as peripheral to human affairs and social constitution. This choice of unique essays deals a brand new and diverse viewpoint that sees speak because the basic framework of social interplay and social associations.
It has frequently been stated that the laser is an answer looking for an issue. The quick improvement of laser expertise over the last dozen years has resulted in the supply of trustworthy, industrially rated laser resources with a wide selection of output features. This, in flip, has led to new laser purposes because the laser turns into a well-known processing and analytical device.
Determine whether there are points of the graph of f corresponding to x = 3, 4, and 1. SOLUTION The y value corresponding to x = 3 is y = f ( 3) = 11/2. Therefore the point ( 3, 11/2) lies on the graph of f. Similarly, f ( 4) = 6 so that ( 4, 6) lies on the graph. However, f is undefined at x = 1, so there is no point on the graph with x coordinate 1. 38 was obtained by plotting several points. Still Struggling Notice that, for each x in the domain of the function, there is one and only one point on the graph---namely the unique point with y value equal to f (x).
10 Determine the equation of the line with slope 3 that passes through the point ( 2, 1) . SOLUTION Let ( x, y) be a variable point on the line. Then we can use that variable point together with ( 2, 1) to calculate the slope: m= y−1 . x−2 15 16 CALCULUS DeMYSTiFieD On the other hand, we are given that the slope is m = 3. We may equate the two expressions for slope to obtain 3= y−1 . x−2 ( ∗) This may be simplified to y = 3x − 5. MATH NOTE The form y = 3x − 5 for the equation of a line is called the slopeintercept form.
13 Sketch the graph of {( x, y) : y = x 2 }. 19). 20. This curve is called a parabola. 14 Sketch the graph of the curve {( x, y) : y = x 3 }. 21). 22. This curve is called a cubic. YOU TRY IT Sketch the graph of the locus |x| = |y|. 15 Sketch the graph of the curve y = x 2 + x − 1. 23). 24. This is another example of a parabola. YOU TRY IT Sketch the locus y2 = x3 + x + 1 on a set of axes. The reader unfamiliar with cartesian geometry and the theory of loci would do well to consult [SCH2]. 7 Trigonometry ...................................................................................................................................... | 677.169 | 1 |
Call 0867227014, 0318115749 or WhatsApp 0825507946 for more detail!
What is Foundational Mathematical Literacy FML
What is Foundational Mathematical Literacy (FML)?
The Foundational Mathematical Literacy is the minimum, generic mathematical literacy that will provide learners with an adequate foundation to cope with the mathematical demands of occupational training and to engage meaningfully in real life situations involving mathematics.
Foundational Mathematical Literacy will provide the foundation for further development of an individual in mathematical literacy contexts and mathematical concepts that may be specific to an occupation or trade.
Learners who have met all the requirements of Foundational Mathematical Literacy are able to solve problems in real contexts by responding to information about mathematical ideas that are presented in a variety of ways. they will be able to solve problems by defining the problem, analysing and making sense of the information provided, planning how to solve the problem, executing their plan, interpreting and evaluating the results, and justifying the method and solution.
in solving problems, individuals will apply skills such as identifying or locating relevant information, ordering, sorting, comparing, counting, estimating, computing, measuring, modelling, interpreting and communicating. Using their mathematical literacy and understanding of numbers, they will make sense of the workplace and the world in which they live.
Foundational Mathematical Literacy consists of:
Number and quantity
Finance
Data and chance
Measurement
Space and shape
patterns and relationships
The curricula describe the learning outcomes, the scope and contexts in | 677.169 | 1 |
Course Description: Math 171 is the first of a three semester beginning calculus sequence, which is taken, for the most part, by math, chemistry, and physics majors. The department expects that students passing Math 171 be able to follow mathematical proofs and handle routine computations, i.e., limits, derivatives, max-min problems, and calculation of definite integrals using the fundamental theorem of calculus. We expect students to be able to state (write) and apply basic definitions and major theorems. These include, but are not limited to, definitions of limit, continuous function, derivative, definite and indefinite integrals, the intermediate value theorem for continuous functions, the mean value theorem, and the fundamental theorem of calculus. Students are also expected to be able supply simple proofs, e.g., some of the limit theorems, some of the rules of differentiation, and applications of the intermediate and mean value theorems. The list is of course endless, but keep in mind the phrase `simple proofs'.
Students should become familiar with the standard notations of logic, set theory, and functions.
Grading.Your grade will be based on weekly quizzes, which are based on the homework and will be given on Fridays, two tests which will
be given out side of class and a final exam. The quizzes will count for 25%, each test for 25% and the final
for 25%. Your letter grade will be assigned this way: 90-100%, A; 80-89%, B; 70-79%, C; 60-69%, D; 59% or less, F.
Make-up policy: Make-ups for missed quizzes and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam or quiz. Otherwise, they forfeit their rights to a make-up.
Scholastic dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules.
Copyright policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.
Americans with Disabilities Act (ADA) Policy Statement.
The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides
comprehensive civil rights protection for persons with disabilities. Among other things, this legislation
requires that all students with disabilities be guaranteed a learning environment that provides for
reasonable accommodation of their disabilities. If you believe you have a disability requiring an
accommodation, please contact Disability Services, in Cain Hall, Room B118, or call 845-1637. For
additional information visit
Tentative Schedule:
This is a projected roadmap of the course. Modifications necessitated by
circumstances are inevitable. Whilst most of the sections below will be
covered in lecture, some might be assigned for reading. | 677.169 | 1 |
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Non-Routine Mean
Non-Routine Mean now available only on CD or by File Download. This in-depth lesson is part of an instructional unit on Pre-Algebra, with 8 in-depth lessons aligned to the Common Core. See the screen shot below, or view our learning objectives. | 677.169 | 1 |
Two mathematical areas provide the content of the course: (1) Geometry and (2) Algebra and Modeling. Mathematical content and pedagogy are fully integrated using contemporary classroom technologies. (Does not fulfill the core requirement.)
Students will be able to understand, process, and interpret statistical information arising in everyday life using real world examples and case studies from a variety of disciplines. Critical thinking and quantitative decision-making skills will be taught. This course focuses on being a consumer of statistics, interpreting research studies, and learning how statistics are used in the real world.
Prerequisites
Introduction to elementary ordinary differential equations with applications to physical processes with emphasis on first and second order equations, systems of linear equations, and Laplace transforms.
Prerequisites
This course introduces the basic concepts and techniques in the study of dynamical systems, including nonlinear ordinary differential equations, difference equations, and systems of equations. Using a wide variety of applications from the physical sciences, we will cover analytical methods such as linear stability, bifurcations, phase plane analysis, limit cycles, Lorenz equations, chaos, iterated maps, period doubling, and fractals.
Prerequisites
An introduction to the study of the integers and related objects. Topics are taken from among the following: divisibility, primes and the Euclidean algorithm, the Euler phifunction, special primes and perfect numbers, congruences mod n, quadratic residues, continued fractions, quadratic forms, Diophantine equations.
Prerequisites
Survey of applied mathematics with an emphasis on modeling problems from science and engineering. Process of formulating the model, solving/simulating, and analyzing/interpreting results. Applications may include: continuous- and discrete-time population models, models of physical and biological phenomena, and statistical models.
3
Prerequisites
This seminar supports students working in local schools as part of the Outreach Excel Program. Students discuss questioning and group work strategies, classroom management, current school mathematics curriculum, and interaction techniques with middle and high school students. This is a Pass/No Pass course and may be repeated for credit. Does not count towards math major.
Practical field experience in selected industries or agencies. Department permission and supervision is required. Students may receive an IP (In Progress) grade until the completion of their internship.3
PrerequisitesPrerequisites
The study of algebraic structures that are like the integers, polynomials, and the rational numbers. The integers and their properties. Groups: examples, properties, and counting theorems. Rings: examples and properties. Fields: roots of polynomials and field extensions.
3
Prerequisites
Unique factorization in special rings. Field theory and the use of groups to understand field extensions: finite fields, Galois theory. Classical construction problems, solution of n-th degree polynomials.
3
Prerequisites
Cryptography is the science of encoding and decoding information for the purpose of secure communication. With an emphasis on utilization of both modern and classical cryptosystems, this course introduces students to the mathematical underpinnings of cryptography.
Practical field experience in selected industries or agencies. Department permission and supervision is required. Students may receive an IP (In Progress) grade until the completion of their internship.
Math 497 is a course for students with strong mathematical preparation. Students will work in teams on a project from an industrial or governmental firm. Student success will depend on realistic industry evaluations such as teamwork, communication, individual initiative, and final products.
3
Prerequisites
3
Prerequisites
Senior standing; 3.0 G.P.A. in the thesis area or good standing in the honors program. | 677.169 | 1 |
CS 779 Splines and Their Use in Computer Graphics
Objectives
Computer-aided design systems originally offered only simple primitives (spheres,
cylinders, polytopes, etc.) as building blocks for the creation of composite
objects. Now such systems provide spline primitives as well for the construction
of free form curves and surfaces. In addition, splines provide an important
class of functions for approximation in areas such as finite element methods
and numerical data fitting.
This course presents a general introduction to spline theory and recent developments
in techniques for representing, manipulating and rendering curves and surfaces
constructed from splines in a graphics environment. Applications of interest
include computer-aided design, synthetic image generation and animation.
References
An Introduction to Splines for Use in Computer Graphics and Computer-Aided
Design, by R. Bartels, J. Beatty and B. Barsky; A Practical Guide to Splines,
by C. de Boor, Springer-Verlag; Curves and Surfaces for Computer Aided Geometric
Design, by G. Farin, Academic Press; Fundamentals of Computer Aided Geometric
Design, by J. Hoschek and D. Lasser; Computer Graphics, by D. Hearn and M.P.
Baker and relevant journal articles.
Outline
Mathematical Background
A review of linear and affine spaces. Introduction to polynomials, polynomial
bases, multi-linear and multi-affine functions. The mathematical description
of curves and surface as parametric mappings. | 677.169 | 1 |
Course Contents:
Linear Systems and Optimization | The Fourier Transform and its Applications
Instructor: Osgood, Brad G
The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and
general principles, and learning to recognize when, why, and how it is used. Together with a great
variety, the subject also has a great coherence, and the hope is students come to appreciate both.
Topics include:
The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.
Osgood is a mathematician by training and applies techniques from analysis and geometry to various engineering problems. He is interested in problems in imaging, pattern recognition, and signal processing. | 677.169 | 1 |
Assessment Specifications
Mode of Assessment
Standards
Format for the assessment
Opportunities for Merit and Excellence will be spread through the examination.
As a result, question parts may not be arranged in order of increasing difficulty.
In order to be awarded Achievement, a candidate may be required to demonstrate evidence within a question part that could also assess a higher level of thinking.
Correct answers will generally not be sufficient for showing evidence of the level of thinking required by the standard. This means a candidate who is using a graphing calculator will be required to demonstrate algebraic methods and give the derivate and integrals in calculus.
Candidates are expected to choose their method when solving a problem, although the grade awarded may be affected by the level of thinking applied in solving the problem. Guess-and-check methods are unlikely to show the required thinking.
Candidates must show any working that is asked for in the assessment.
Standards require a range of methods from Explanatory Note 4 to be demonstrated within an assessment.
Candidates will be expected to answer questions that demonstrate an understanding of the mathematical concepts in the solving of problems.
Candidates may be required to understand the use of a letter such as "k" to represent a constant or coefficient. Minor errors will not be penalised unless they directly relate to the methods listed in the standard, e.g. expansion of (x+4)(x-3) to give x2 + x +12 cannot be identified as an algebraic or numerical error and, therefore, cannot be accepted. Rounding in context may be required.
Knowledge of mathematical terms such as indices, exponents, tangents, proportion, and the like is assumed.
The answer from one question part may be required in answering subsequent parts. In this case, consistency of response will be assessed as being correct, provided the solution is not an essential component of the standard and providing the incorrect solution does not result in an easier question to be solved.
Equipment to bring
Candidates will require an approved calculator (preferably a graphing calculator). Candidates who do not have access to graphing calculators will be disadvantaged.
Resources or information provided
Content/context details
Solutions for problems may also require knowledge up to and including Mathematics Curriculum Level 6, and for higher levels of achievement may incorporate content knowledge across different Level 2 Mathematics achievement standards in order to solve a problem.
Questions may be set in a mathematical context.
Questions may require candidates to interpret their solutions in context.
Specific information for individual external achievement standards
Standard
91261
Domain
Algebra
Title
Apply algebraic methods in solving problems
Version
3
Number of credits
4
Further clarification of the achievement standard
For the award of Excellence, candidates may be required to
form and solve exponential equations relating to interest, growth and decay, and suchlike
understand the meaning of rational (fractional) numbers in regards to the roots of equations
Any equations formed by the candidate must be stated in solving a problem.
Candidates must demonstrate algebraic techniques rather than providing only the correct answer.
Given the form of a model, candidates may be required to complete the model using the information given in the context of the question.
Answers should be expressed in their simplest algebraic form.
Standard
91262
Domain
Calculus
Title
Apply calculus methods in solving problems
Version
3
Number of credits
5
Content/context details
Derivatives and anti-derivatives must be shown.
Candidates will be required to use the derivatives and anti-derivatives that they have found.
Candidates may be required to draw the graph of the gradient of a function having been given the graph of the function, or vice versa.
Answers should be expressed in their simplest algebraic form.
Candidates may be required to justify the nature of the maximum or minimum points, e.g. shape of curve, second derivative, or testing points.
At higher levels of achievement, candidates may be required to form their own polynomials.
Candidates may need to be familiar with the terms "local maximum and minimum".
Standard
91267
Domain
Probability
Title
Apply probability methods in solving problems
Version
3
Number of credits
4
Further clarification of the achievement standard
Questions may require knowledge of inverse normal calculations.
Probabilities may be expected to be calculated from one or more tables, written information, or a probability tree.
In describing and comparing distributions, candidates should include reference to the shape of the graph, the centre of the distribution(s), and the spread of the data, from provided statistics or the graph.
Conditional probability questions may be included that can be answered by using informal or intuitive methods. This may include risk or relative risk.
Students should give clear description of "skewness" in their responses, e.g. "skew to the left". | 677.169 | 1 |
A Treatise on the Calculus of Finite Differences(Paperback)
Synopsis
Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published A Treatise on the Calculus of Finite Differences in 1860 as a sequel to his Treatise on Differential Equations (1859). Both books became instant classics that were used as textbooks for many years and eventually became the basis for our contemporary digital computer systems. The book discusses direct theories of finite differences and integration, linear equations, variations of a constant, and equations of partial and mixed differences. Boole also includes exercises for daring students to ponder, and also supplies answers. Long a proponent of positioning logic firmly in the camp of mathematics rather than philosophy, Boole was instrumental in developing a notational system that allowed logical statements to be symbolically represented by algebraic equations. One of history's most insightful mathematicians, Boole is compelling reading for today's student of logic and Boolean thinking | 677.169 | 1 |
415: MATH BOOT CAMP 2
Course Description
A remedial mathematics course designed for those students who need to learn, or re-learn the fundamental concepts of math. The primary emphasis is on algebraic expressions, linear/quadratic equations and applications, polynomials, graphing, and functions. This is a pass/no pass course. Units earned in this course do not count toward the associate degree and/or certain certificate requirements.
Learning Outcomes
Analyze and solve linear equations in one and two variables .
Solve a variety of problems involving applications of linear and quadratic functions .
Determine and implement an appropriate method of solution for these problems .
Graph linear and non-linear relations and utilize the graph in problem solving. | 677.169 | 1 |
California Algebra 1 Ц Student Textbook + Homework Book (2007)
This Algebra 1 program has been specifically written for California - it's everything the teacher needs to deliver successful Math lessons for Grade 8 students. The textbook features a clear layout that will not distract students from the content being taught. It is all written in straightforward, understandable language to aid comprehension and meet the needs of English Language Learners.
Key features: - California Standards are clearly stated and defined in direct language for every Lesson - Each Lesson is supported with carefully selected worked examples - Hints and reminders of key information throughout the Lesson serve as an extension of the classroom - Each Lesson ends with Independent Practice exercises and Round Ups summarizing the whole Lesson - Includes Explorations and Investigations for students to experience Math in real life contexts | 677.169 | 1 |
CALCULOU 2 Advice
Showing 1 to 2 of 2
I recommend this class because this class was an interactive course. Before you came to class, you'd have to watch lecture videos. During the class, the content would be quickly reviewed and then practice problems would be solved within small groups to learn and apply the material reviewed earlier. Afterwards, the solutions would be presented to the class by the students. After class, homework is assigned for more practice and immediate feedback is given. Recitation class is an opportunity to ask more questions when query arises during the homework portion.
Course highlights:
The highlights of this course was being able to apply the material learned from watching lecture videos before class to actual practice and real-world problems, and understanding the steps and the concepts of using each method. I learned methods of integration, power series, and infinite series.
Hours per week:
6-8 hours
Advice for students:
This course is Calculus II and is pretty difficult. However, you can pass this class as long as you put in hard work and dedication into the course, as well as reviewing your Calculus I notes because you're expected to use a lot of it again. Learn the material before going to the lecture class and ask lots of questions if you don't understand the material you are learning. It is also important to do the homework after the lecture so that while the material is fresh in your mind, you'll be able to apply the methods to the homework problems. Also, attending recitation is another opportunity to get your questions answered and learn how to solve the practice problems that wasn't explained during lecture. Be prepared to use critical thinking! | 677.169 | 1 |
Math Resources Index
Abstract
Algebra Online - This site contains many of the definitions
and theorems from the area of mathematics generally called
abstract algebra.
Algebra - An explanation of how the Montessori student learns algebra
while interacting with manipulatives, physical objects, represented
by expressions incorporating the numerals and variables of
mathematics.
Algebra Tutorial - For students and parents, includes lessons, calculators,
and worksheets
Geometry in Action - The Geometry
Junkyard - This page collects various areas in which ideas from
discrete and computational geometry (meaning mainly low-dimensional
Euclidean geometry) meet some real world applications. | 677.169 | 1 |
Ace Mathematics: Gr 7: Teacher's Guide
Description
This publication offers a thorough treatment of mathematics in a straightforward learning programme. Teachers will find the material easy to implement in their classrooms, no matter their resources or level of experience. Simple, plain language and a learshow more | 677.169 | 1 |
Description
Outline These will be some of the topics that I will be covering in this website. This is just a simplified version of each one of these topics that I will be covering more in depth In a separate post.
Learning about derivatives definition of a derivative
Power rule When your x term has a power you simply bring it to the front and then you subtract one from the original power to gain your new power.
Product rule The derivative of a function where two terms are being multiplied can be expressed as the function of the first term times the derivative of the next term plus the derivative of the first term times the function of the second term.
Quotient rule The derivative of a function with terms that are being divided can be expressed by the function of the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.
Lambda Calculus (Part I) - University of Wisconsin–Madison
World Web Math: Calculus Summary
Calculus Summary Calculus has two main parts: ... , and other examples of simple harmonic motion. ... part of Using Maple for ESG Subjects, ... Read more
These presentations are classified and categorized, so you will always find everything clearly laid out and in context.
You are watching Presentation Deceuninck NV presentation right now. We are staying up to date! | 677.169 | 1 |
When you are looking readability and good fortune in studying (or instructing) algebra, this can be the booklet you're looking for. one of many greatest lacking components in math schooling is conversation. This ebook offers that lacking aspect. it's a own coach by way of your aspect, translating the maths into phrases, explaining what issues suggest, providing you with clues to appear for, and telling you ways to unravel difficulties. This advisor makes a speciality of the entire vital subject matters of algebra together with: -Linear Equations -Systems of Linear Equations -Factoring -Trinomials -Quadratic Equations -Complex Rational Expressions -Powers and Radicals This e-book explains this probably advanced topic via certain sections you will not locate in the other examine advisor akin to: -Obscure houses of 0, One and Negatives -The actual Order of Operations -The leading quantity Multiples desk -Is fifty one a major quantity? -GCF vs. liquid crystal display -What Does "Undefined" suggest? -Parallel & Perpendicular strains on a Graph -What Does "Solving by way of" suggest? -The other way to Simplify a Rational Expression -The half everybody Forgets (The final Step of the Quadratic Equation) -Special phrases for detailed circumstances -Prime vs. No resolution -The All-LCD technique -Cross-Multiplying vs. move Cancelling -List of universal Radical Fingerprints -Manipulating & Simplifying Radicals -The Meanings of "Cancelling Out" -What Does "Error" on a Calculator suggest? -Scientific Notation in your Calculator -FMMs (Frequently Made Mistakes). This booklet includes: -Step-by-step directions -Annotated examples -Detailed descriptions -Detailed desk of Contents for fast subject referencing And: -will assist you strategy what you notice and listen to -will inform you tips on how to write and converse the maths -highlights the main regularly made errors -connects key themes that move via various chapters this can be the proper resourceto assist you with homework or organize for an examination. it's going to support any center university, highschool or collage scholar solidify the real basics utilized in simple math, Algebra I, Algebra II, Introductory Algebra, straightforward Algebra, Intermediate Algebra, collage Algebra, Pre-Calculus or even Calculus. via the writer of GRADES, cash, overall healthiness: The e-book each collage pupil should still learn (2010), this is often the publication each math pupil must have. make the most of this ebook to get a clearer realizing of algebra, to enhance your grades... and to benefit why GEMA is the hot PEMDAS! This publication makes an excellent gift for eighth grade, junior excessive and highschool (college certain) graduates.
This compact but thorough textual content zeros in at the elements of the speculation which are rather proper to purposes . It starts off with an outline of Brownian movement and the linked stochastic calculus, together with their courting to partial differential equations. It solves stochastic differential equations by means of numerous equipment and reviews intimately the one-dimensional case.
Approximation conception within the multivariate atmosphere has many functions together with numerical research, wavelet research, sign processing, geographic info structures, computing device aided geometric layout and special effects. This complex creation to multivariate approximation and similar subject matters involves 9 articles written by means of top specialists surveying the various new principles and their purposes.
This is often a longer therapy of the set-theoretic options that have remodeled the research of abelian team and module thought during the last 15 years. a part of the booklet is new paintings which doesn't look in other places in any shape. moreover, a wide physique of fabric which has seemed formerly (in scattered and occasionally inaccessible magazine articles) has been widely remodeled and in lots of instances given new and more suitable proofs.
Additional resources for Algebra in Words: A Guide of Hints, Strategies and Simple Explanations
Sample text
1. If there are any denominators, find the least common denominator and multiply all terms on both sides by the LCD to eliminate all denominators. 2. Simplify: Identify and combine like-terms, if any. Note: #s 1 & 2 are interchangeable. You can eliminate denominators first and combine like-terms next. It is usually easier to "get rid" of fractions first, so you don't have to go through Adding & Subtracting Fractions. 3. " Use addition or subtraction to "move*" the constants (non-variable numbers) to the right of the equal sign and… Use addition or subtraction to "move*" the term(s) containing variables to the left of the equal sign.
You may also get a negative number as the exponent, which is fine, but in the final, simplified form of your answer, you shouldn't leave an exponent negative. If an exponent is negative, move the factor (the base) with that exponent to the opposite part of the fraction and change the sign of the exponent to positive. Remember, any and every factor has an unwritten exponent of "1". Also remember that when exponents add or subtract to equal zero, any base to the power zero equals 1. (Review this in: The Unwritten 1, and: Property Crises of Zeros, Ones and Negatives).
The following are those helpful sub-processes: For 2: Check to see if the number you are factoring is even. Specifically, check to see if the last digit of the number is even. If it is even, then the original number you are factoring is divisible by 2. Then you should divide it by 2 to find the other factor. If it is not even, then the number is not divisible by 2, nor is it divisible by any other even number: 4, 8, 10, 12, etc. Next, test if it is divisible by 3… For 3: Add up the digits in the number you are factoring. | 677.169 | 1 |
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Lesson Overview
Lesson Title: Reimann Sums
Teachers: Mrs. Sara Bonn (Primary)
Brief Description:
Students will explore finding the area under a curve using Left Rectangular, Right Rectangular, and Midpoint Rectangular Approximation Methods.
Topics Introduced:
Reimann Sums
Suggested Grade Levels: 12th Grade
Subjects: Mathematics
Lesson Information
Learning Expectations:
By the end of the lesson, students should be able to approximate the area under various curves using the rectangular approximation methods. They should understand when these approximations are over-estimates versus under-estimates. Finally, they should be left with the idea that ideally, we want to use as many rectangles as possible to give the most accurate approximation.
Plan Of Action:
Typically the first lesson taught when introducing integrals is this one. Students can use their knowledge of geometry to expand the concept of area to approximate the area when speed varies.
Data Set Used:
Students use the area formula for rectangles to approximate the area under a curve.
Materials Needed:
All students should have paper and a pencil to compute their estimates. Besides projecting this lesson, a separate dry erase board or flip chart is helpful, especially when drawing the difference between a left, right and midpoint rectangular approximation.
Preparation Period:
No preparation time is required for this lesson.
Implementation Period:
Students should be able to satisfactorily explore this lesson within 45 minutes.
Unexpected Results:
Several students identified not just the LRAM, RRAM and MRAM options, but also the Trapezoidal approximation. Typically, this method isn't discussed until later.
Lesson Files
Estimating with Finite Sums Students will explore finding the area under a curve using LRAM, RRAM and MRAM. [size: 870912] [date uploaded: Mar 24, 2012, 9:02 pm ] | 677.169 | 1 |
Formal operational thought and mathematicsproblems solving. In the framework of the study of mathematics reasoning processes, a research is presented in order to analyze relationships among cognitive skills reached during the formal operational stage, and mathematics problems solving. 78 boys and girls, aged 16, were assessed with the Logical Thinking Test (TOLT), and with a Mathematics Problems Solving Test. Results in mathematics were compared in function of formal operational thought level achieved. Data suggest that students with a higher level of formal operational thought were those that better solved the mathematics problems. However, just 36% of these were able to solve problems where the proportionality squemes were required. Results suggest that reaching the level of formal reasoning is not enough to apply it in specific mathematics problems. It is also necessary to acquire specific knowledge to carry out a correct mathematics problem solving. | 677.169 | 1 |
Help your student learn to maximize MATLAB as a computing tool to explore traditional Digital Signal Processing (DSP) topics, solve problems and gain insights. An extremely valuable supplementary text, DIGITAL SIGNAL PROCESSING USING MATLAB: A PROBLEM SOLVING COMPANION, 4E greatly expands the range and complexity of problems that students can effectively study in your course. Since DSP applications are primarily algorithms implemented on a DSP processor or software, they require a significant amount of programming. Using interactive software, such as MATLAB, makes it possible to place more emphasis on learning new and difficult concepts than on programming algorithms. This engaging supplemental text introduces interesting practical examples and shows students how to explore useful problems. New, optional online chapters introduce advanced topics, such as optimal filters, linear prediction, and adaptive filters, to further prepare your students for graduate-level success.
Dr. Vinay K. Ingle is an Associate Professor of Electrical and Computer Engineering at Northeastern University. He received his Ph.D. in electrical and computer engineering from Rensselaer Polytechnic Institute in 1981. He has broad research experience and has taught courses on topics including signal and image processing, stochastic processes, and estimation theory. Dr. Ingle has co-authored numerous higher level books including DSP LABORATORY USING THE ADSP-2181 MICROPROCESSOR (Prentice Hall, 1991), DISCRETE SYSTEMS LABORATORY (Brooks-Cole, 2000), STATISTICAL AND ADAPTIVE SIGNAL PROCESSING (Artech House, 2005), and APPLIED DIGITAL SIGNAL PROCESSING (Cambridge University Press, 2011).
Affiliation: University of California, San Diego and Northeastern University Bio: Dr. John Proakis is an Adjunct Professor at the University of California at San Diego and a Professor Emeritus at Northeastern University. He was a faculty member at Northeastern University from 1969 through 1998 and held several academic positions including Professor of Electrical Engineering, Associate Dean of the College of Engineering and Director of the Graduate School of Engineering, and Chairman of the Department of Electrical and Computer Engineering. His professional experience and interests focus in areas of digital communications and digital signal processing. He is co-author of several successful books, including DIGITAL COMMUNICATIONS, 5E (2008), INTRODUCTION TO DIGITAL SIGNAL PROCESSING, 4E (2007); DIGITAL SIGNAL PROCESSING LABORATORY (1991); ADVANCED DIGITAL SIGNAL PROCESSING (1992); DIGITAL PROCESSING OF SPEECH SIGNALS (2000); COMMUNICATION SYSTEMS ENGINEERING, 2E (2002); DIGITAL SIGNAL PROCESSING USING MATLAB V.4, 3E (2010); CONTEMPORARY COMMUNICATION SYSTEMS USING MATLAB, 2E (2004); ALGORITHMS FOR STATISTICAL SIGNAL PROCESSING (2002); FUNDAMENTALS OF COMMUNICATION SYSTEMS (2005).
Review:
"I like the way the authors discuss the solutions to the problems. They provide enough steps in their solution to help the student keep up, but not so many steps that the student doesn't have to think...This book is very accessible compared to many texts. It is written clearly...It has excellent integration with MATLAB. I like the fact that the authors have no trouble introducing functions to complement the MATLAB functions."
"This book has developed a suite of MATLAB functions that can be used to determine standard digital filter structures including the direct form, cascade form, parallel form, and lattice form. The MATLAB scripts are provided with detailed explanations. Numerous examples along with block diagrams and discussions are included to show how these functions are used. Overall, I find the authors' treatment of the subject, including the MATLAB functions and examples, is among the best compared with other textbooks I know of...In terms of the treatment of using MATLAB for DSP teaching at the undergraduate level, counting the number and variety of MATLAB examples and problems, this is the most competitive book on the market in my opinion." | 677.169 | 1 |
Modeling is one of the most effective, commonly used tools in engineering and the applied sciences. In this book, the authors deal with mathematical programming models both linear and nonlinear and across a wide range of practical applications.
Whereas other books concentrate on standard methods of analysis, the authors focus on the power of modeling methods for solving practical problems-clearly showing the connection between physical and mathematical realities-while also describing and exploring the main concepts and tools at work. This highly computational coverage includes: * Discussion and implementation of the GAMS programming system * Unique coverage of compatibility * Illustrative examples that showcase the connection between model and reality * Practical problems covering a wide range of scientific disciplines, as well as hundreds of examples and end-of-chapter exercises * Real-world applications to probability and statistics, electrical engineering, transportation systems, and more
Building and Solving Mathematical Programming Models in Engineering and Science is practically suited for use as a professional reference for mathematicians, engineers, and applied or industrial scientists, while also tutorial and illustrative enough for advanced students in mathematics or engineering
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Review:
"The author covers an amazing amount of ground with clarity and liveliness" --Mathematical Association of America
Book Description Oxford University Press, United Kingdom, 2016 2015 2016. Paperback. Book Condition: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. BTE9780198732822
Book Description OUP Oxford 2015-10250563 | 677.169 | 1 |
Whether your students are learning in a brick-and-mortar school or a homeschool or online, you teachers and parents know how important logic is--but that doesn't make the technical aspects of the subject any easier (in fact the fundamental nature of the subject makes it even more intimidating!). We've painstakingly designed Introductory Logic with that tension in mind: you'll get the benefit of James B. Nance's twenty years of teaching experience, making ingraining the fundamentals of logic in your students as painless (and rewarding!) as possible. Anybody can learn from Introductory Logic. The whole series takes advantage of a brand new, clean, easy-to-read layout, lots of margin notes for key points and further study, a step-by-step modern method, and exercises for every lesson (plus review questions and exercises for every unit). More importantly, anybody can teach Introductory Logic. Here are the features that make the Teacher Edition for Introductory Logic the obvious choice for educators new to logic, no matter where they teach: -A daily lesson schedule for completing Intermediate Logic in a semester or a year-long course. -Answers to all exercises, review questions, review exercises, quizzes, and tests in the order they are taught. -Contains the entire Student Edition text--with the same page numbers as the Student Edition! No more flipping back and forth between answer keys and textbook. -Detailed daily lesson plans for the entire textbook explain each lesson's daily Student Objectives, -Special Notes, step-by-step Teaching Instructions with bolded terms, advice, and more examples, -Assignments for each lesson, -Optional Exercises for further exploration and integration | 677.169 | 1 |
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Linear Circuit Transfer Functions: An introduction to Fast Analytical Techniques teaches readers how to determine transfer functions of linear passive and active circuits by applying Fast Analytical Circuits Techniques. Building on their existing knowledge of classical loop/nodal analysis, the book improves and expands their skills to unveil transfer functions in a swift and efficient manner. Starting with simple examples, the author explains step-by-step how expressing circuits time constants in different configurations leads to writing transfer functions in a compact and insightful way. By learning how to organize numerators and denominators in the fastest possible way, readers will speed-up analysis and predict the frequency response of simple to complex circuits. In some cases, they will be able to derive the final expression by inspection, without writing a line of algebra. Key features: * Emphasizes analysis through employing time constant-based methods discussed in other text books but not widely used or explained. * Develops current techniques on transfer functions, to fast analytical techniques leading to low-entropy transfer functions immediately exploitable for analysis purposes. * Covers calculation techniques pertinent to different fields, electrical, electronics, signal processing etc. * Describes how a technique is applied and demonstrates this through real design examples. * All Mathcad® files used in examples and problems are freely available for download. An ideal reference for electronics or electrical engineering professionals as well as BSEE and MSEE students, this book will help teach them how to: become skilled in the art of determining transfer function by using less algebra and obtaining results in a more effectual way; gain insight into a circuit's operation by understanding how time constants rule dynamic responses; apply Fast Analytical Techniques to simple and complicated circuits, passive or active and be more efficient at solving problems. | 677.169 | 1 |
12: MATH FOR TEACHERS
Course Description
This course is intended for students preparing for a career in elementary school teaching. Emphasis will be on the structure of the real number system, numeration systems, elementary number theory, and problem solving techniques. Technology will be integrated throughout the course. PREREQUISITE: High School Geometry and Math 233 (Intermediate Algebra), or, Math 208 (Plane Geometry) and Math 233 (Intermediate Algebra). All courses must be completed with a grade of 'C' or better. | 677.169 | 1 |
This easy-to-follow book includes terrific tutorials and plenty of exercises and examples that let you learn by doing. It starts by giving you a hands-on orientation to the TI-84 Plus calculator. Then, you'll start exploring key features while you tackle problems just like the ones you'll see in your math and science classes.
About this Book
With so many features and functions, the TI-84 Plus graphing calculator can be a little intimidating. But fear not if you have this book in your hand! In it you'll find terrific tutorials ranging from mastering basic skills to advanced graphing and calculation techniques, along with countless examples and exercises that let you learn by doing.
, Second Edition starts by making you comfortable with the screens, buttons, and special vocabulary you'll use every time you fire up the TI-84 Plus. Then, you'll master key features and techniques while you tackle problems just like the ones you'll see in your math and science classes. You'll even get tips for using the TI-84 Plus on the SAT and ACT math sections!
No advanced knowledge of math or science is required.
What's Inside
Learn hands-on with real examples and exercises Find specific answers fast Compliant with all models of the TI-83 Plus and TI-84 Plus Full coverage of the color-screen TI-84 Plus CE and TI-84 Plus C Silver Edition Christopher Mitchell, PhD. is a research scientist studying distributed systems, the founder of the programming and calculator support site cemetech.net, and the author of Manning's Programming the TI-83 Plus/ TI-84 Plus.
Table of Contents
PART 1 BASICS AND ALGEBRA ON THE TI-84 PLUS
What can your calculator do? Get started with your calculator Basic graphing Variables, matrices, and lists | 677.169 | 1 |
Linear and Integer Programming
(ADM II)
Winter Term 2009/2010
News
Remaining Scheine are now at MA 310.
Photos taken at the Umtrunk are available (as .zip) in
umtrunk/
Content
This course gives an introduction into theory and practice of linear and integer programming. Important algorithms (Fourier-Motzkin, simplex, ellipsoid, and interior point method; cutting planes and branch&bound), numerical aspects of these methods, as well as the theoretical background (Farkas Lemma, LP duality and optimality criteria, polyhedral theory, polyhedral combinatorics) will be described and elucidated. Moreover, application areas will be mentioned and modelling issues discussed.
The development of linear programming is – in my opinion – the most important contribution of the mathematics of the 20th century to the solution of practical problems arising in industry and commerce. The subsequent progress in the applicability of integer and combinatorial optimization begins, at present, to even surpass this impact. Today, the utilization of linear and integer programming abounds. Almost every product available on the market has some linear or integer programming "inside". Solutions obtained by IP&LP-methods impact our daily life. | 677.169 | 1 |
Mathematics (MATH) 271
Linear Algebra II (Revision 5)
Mathematics Diagnostic Assessment. This online test contains 70 questions that will help you assess your mathematical skills. Based on your score we will recommend which Athabasca University mathematics course you are likely ready to take successfully.
Important Links
Overview
Mathematics 271: Linear Algebra II continues the study of linear algebra from Mathematics 270. Mathematics 271 is suggested for students in the science programs. The course covers intermediate topics of linear algebra such as general vector spaces, eigenvalues and eigenvectors, inner product spaces, diagonalization and quadratic forms, and general linear transformations and applications of linear algebra.
Outline
Unit 1: General Vector Spaces
Some of the topics covered in this unit are vector spaces; subspaces; linear independence; bases and dimension; change of basis; row, column and null spaces; rank and nullity; matrix transformations; and applications to computer graphics in 3D.
Unit 2: Eigenvalues and Eigenvectors
Some of the topics covered in this unit are eigenvalues and eigenvectors; matrix diagonalization; and applications to genetics.
Unit 3: Inner Product Spaces
Some of the topics covered in this unit are inner product spaces; orthogonality; Gram-Schmidt process; QR-decomposition; and method of least squares.
Unit 4: Diagonalization and Quadratic Forms
Some of the topics covered in this unit are orthogonal matrices; orthogonal diagonalization; symmetric matrices; and applications of quadratic forms to conics.
Unit 5: General Linear Transformations
Some of the topics covered in this unit are general linear transformations; composition and inverse of linear transformations; isomorphism; similarity; and applications to cryptography.
Evaluation
To receive credit for MATH 271, you must achieve a composite course grade of at least "D" (50 percent) and a grade of at least 50 percent on the final examination. The weighting of the composite grade is as follows:
5 Tutor-marked Exercises (4% each)
Final Exam
Total
20%
80%
100%
To learn more about assignments and examinations, please refer to Athabasca University's online Calendar. | 677.169 | 1 |
Dave's Moodle
Content: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.
However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a `pure' subject like group theory can be part of the daily armoury of the `applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.
In this module we will cover relatively simple examples, first order equations
,
linear second order equations
and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.
We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.
The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.
Aims: To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phase-plane analysis.
Objectives: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour; and to do the same for simple difference equations.
Status for Mathematics students: List B for third years. If numbers permit second and fourth years may take this module as an unusual option, but confirmation will only be given at the start of Term 2.
Assessment: 10% from weekley seminars, 40% from assignment, 50% two hour exam in June
Prerequisites: None
Introduction This module gives you the opportunity to engage in mathematical problem solving and to develop problem solving skills through reflecting on a set of heuristics. You will work both individually and in groups on mathematical problems, drawing out the strategies you use and comparing them with other approaches.
General aims This module will enable you to develop your problem solving skills; use explicit strategies for beginning, working on and reflecting on mathematical problems; draw together mathematical and reasoning techniques to explore open ended problems; use and develop schema of heuristics for problem solving.
This module provides an underpinning for subsequent mathematical modules. It should provide you with the confidence to tackle unfamiliar problems, think through solutions and present rigorous and convincing arguments for your conjectures. While only small amounts of mathematical content will be used in this course which will extend directly into other courses, the skills developed should have wide ranging applicability.
Intended Outcomes
Learning objectives
The intended outcomes are that by the end of the module you should be able to:
Use an explicit problem solving scheme to control your approach to mathematical problems
Most weeks the Thursday slot will be used for the weekly (assessed) problem session, but this will not be the case every week. You are expected to attend all three timetabled hours.
Assessment Details
A flat 10% given for 'serious attempts' at problems during the course. Each week, you will be assigned a problem for the seminar. At then end of the seminar, you should present a 'rubric' of your work on that problem so far. If you submit at least 7 rubrics, deemed to be 'serious attempts', you will get 10%.
One problem-solving assignment (40%) (deemed to be the equivalent of 2000 words) due by noon on Monday 20th March 2017 by electronic upload (pdf).
This module investigates how solutions to systems of ODEs (in particular) change as parameters are smoothly varied resulting in smooth changes to steady states (bifurcations), sudden changes (catastrophes) and how inherent symmetry in the system can also be exploited. The module will be application driven with suitable reference to the historical significance of the material in relation to the Mathematics Institute (chiefly through the work of Christopher Zeeman and later Ian Stewart). It will be most suitable for third year BSc. students with an interest in modelling and applications of mathematics to the real world relying only on core modules from previous years as prerequisites and concentrating more on the application of theories rather than rigorous proof. | 677.169 | 1 |
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About this product
Description
Description
Explores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.
Author Biography
Hermann Weyl held the chair of mathematics at Zyrich Technische Hochschule from 1913 to 1930; from 1930 to 1933 he held the chair of mathematics at the University of Gottingen; and from 1933 until he retired in 1952 he was a Permanent Member of the Institute for Advanced Study in Princeton. | 677.169 | 1 |
Math
Instructors: Mr. Scott Ahrendsen and Mr. Ryan Steines
Algebra I 1825A / 1825B Full Year Course Grades 8-10
Algebra I has a scope far wider than traditional algebra books, highlighting applications, using statistics and geometry to develop the algebra of linear equations and inequalities, and including probability concepts in conjunction with algebraic fractions. Applications motivate virtually all lessons. Considerable attention is given to graphing. Manipulation with rational algebraic expressions is delayed until later courses.
UCSMP Geometry, diverging from the order of topics in most geometry texts, presents coordinates, transformations, measurement formulas, and three-dimensional figures earlier in the year. To teach students how to write proofs and construct other mathematical arguments more effectively, the course lays a foundation of prerequisite understanding step by step. Again, applications abound throughout.
Transition Algebra courses review and extend algebra and geometry concepts for students who have already taken both Algebra I and Geometry. Transition Algebra review and extend topics such as properties and operations of real numbers, evaluation of rational algebraic expressions; solutions and graphs of first degree equations and inequalities; translation of word problems in equations; operations with and factoring of polynomials; simple quadratics; properties of plane and solid figures; rules of congruence and similarity; coordinate geometry including lines, segments, and circles in the coordinate plane; and angle measurement in triangles including trigonometric ratios.
This course emphasizes facility with algebraic expressions and forms, especially linear and quadratic forms, powers, and roots, and functions based on these concepts. Students study logarithmic, trigonometric, polynomial, and other special functions as tools for modeling real-world situations. The course applies geometrical ideas learned in the previous years, including transformations and measurement formulas.
This course covers foundations of real analysis, analytic geometry, sequences, series, limits, exponential and logarithmic functions, deferential and integral calculus (an introduction only). Recommended taken by all students planning on a 4 year college or technical school. It is highly recommended that the student have for their own permanent use a TI-83 calculator.
This course uses technology, group work, and individual study. This class will study functions, parametric equations and polar equations. The student will do mathematical modeling of the real world. We will work at limits, the derivative and the integral and discover what they really are. It is highly recommended that the student have for their own permanent use a TI-83 calculator. The expectation is for the student to take the AP exam in the spring | 677.169 | 1 |
Basic Linear Algebra by Andrew Baker
Description: Linear Algebra is one of the most important basic areas in Mathematics, having at least as great an impact as Calculus, and indeed it provides a significant part of the machinery required to generalise Calculus to vector-valued functions of many variables. These notes were originally written for a course at the University of Glasgow in the years 2006-7. They cover basic ideas and techniques of Linear Algebra that are applicable in many subjects including the physical and chemical sciences, statistics as well as other parts of mathematics.
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Always ask yourself.. why? Here is a simplified case. You travel on the street and you see a girl carrying a cage. Inside the coop at hand are 4 infinitesimal game birds. Suddenly the cage's door opens and one of the ducks escapes. You can revolve this occurrence into a bare pure mathematics equation:
3 - 1 = 2
See what I mean? Now that's the prototypical step! After that it will be undemanding to revolve be highly structured situations into algebra concepts. Your intellect will before long turn familiar with beside the hypothesis. And shortly you will be able to twirl your entire energy into algebra! | 677.169 | 1 |
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How is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school? Can understanding how to do one kind of division help you with understanding the other kind? What are some examples from real life in which you might use polynomial division?
factoring diamond method use this skill in real life? | 677.169 | 1 |
This NON-calculator unit is designed for math department chairs, specialists, coaches, administrators, lead teachers, and classroom teachers. It contains IMPORTANT information you need to cover Virginia SOL A.9 and AII.11 (Z-Scores, Standard
The following activity was designed to assist students with solving equations, listing all algebraic steps, justifying steps using properties, and modeling equations using Algebra Tiles. These topics will be assessed on the Virginia Standards of
12 question assessment on Virginia SOL A.9 and AII.11. This includes terms like "balance point" for the mean and "how many standard deviations away from the mean" for the z-score. It provides you with the multiple representations of what your
This assessment can be used as a quiz or an activity during stations. It provides students with questions that are formatted in a comparative analysis design.
This activity is perfect for Virginia SOL A.10 comparing data from more than one
Z-Score, Variance, Standard Deviation, Mean Absolute Deviation, Box-and-Whisker review
Calculator Steps for TI 84+ are included for this objective!
This lesson has examples of it all. It's a guided lesson that is ready to present to your class.
As a math instructional specialist, my department has achieved a 40 percentage point growth over the past year, under my direct leadership...
As a former teacher, my students have scored above the state of Virginia's overall average on the algebra one and two End of Course EOC exams for the past eight consecutive years. | 677.169 | 1 |
Business Mathematics and Statistics course unit focuses on an integrated treatment of basic mathematics and statistics techniques, with an emphasis on application in business and economics. This course unit requires a minimal knowledge in mathematics and aims to give the students the basic knowledge of each topic. Further course unit stresses the use of arithmetic, algebra, calculus and statistics in solving problems in business related areas. The course also includes topics from statistics such as collection and representation of data, characteristics of a distribution, basic probability concepts and probability distributions, regression analysis and time series analysis. This course unit stresses logical reasoning and problem solving skills. The importance of this course is to provide students with a sound foundation in mathematics and statistics that is required for future courses in area such as economics, finance, operations research and operations management. At the end of this course unit students are expected to have a basic knowledge of principles of mathematics and statistics that are required to solve business related problems. | 677.169 | 1 |
ISBN-10: 0130457973
ISBN-13: 9780130457974Combining a careful selection of topics with coverage of theirgenuineapplications in computer science, this book, more than any other in this field, is clearly and concisely written, presenting the basic ideas of discrete mathematical structures in a manner that is understandable.Limiting its scope and depth of topics to those that readers can actually utilize, this book covers first the fundamentals, then follows with logic, counting, relations and digraphs, functions, order relations and structures, trees, graph theory, semigroups and groups, languages and finite-state machines, and groups and coding.With its comprehensive appendices and index, this book can be an excellent reference work for mathematicians and those in the field of computer | 677.169 | 1 |
Saturday, May 24, 2008
With the spread of the powerhouse MATLAB software into nearly every area of math, science, and engineering, it is important to have a strong introduction to using the software. Updated for version 7.0, MATLAB® Primer, Seventh Edition offers such an introduction as well as a "pocketbook" reference for everyday users of the software. It offers an intuitive language for expressing problems and solutions both numerically and graphically. The latest edition in this best-selling series, MATLAB® Primer, Seventh Edition incorporates a number of enhancements such as changes to the desktop, new features for developing M-files, the JIT accelerator, and an easier way of importing Java classes. In addition to the features new to version 7.0, this book includes: · A new section on M-Lint, the new debugger for M-files · A new chapter on calling Java from MATLAB and using Java objects inside the MATLAB workspace · A new chapter on calling Fortran from MATLAB · A new chapter on solving equations: symbolic and numeric polynomials, nonlinear equations, and differential equations · A new chapter on cell publishing, which replaces the "notebook" feature and allows the creation of Word, LaTeX, PowerPoint, and HTML documents with executable MATLAB commands and their outputs · Expanded Graphics coverage-including the 3D parametrically defined seashells on the front and back covers Whether you are new to MATLAB, new to version 7.0, or simply in need of a hands-on, to-the-point reference, MATLAB® Primer provides the tools you need in a conveniently sized, economically priced pocketbook | 677.169 | 1 |
Advanced Mathematical Decision Making Version: 2011-2012
Strand: Number and Operations
Students will extend the understanding of proportional reasoning, ratios, rates, and percents by applying them to various settings to include business, media, and consumerism.
Element: MAMDMN1.a
Use proportional reasoning to solve problems involving ratios.
Element: MAMDMN1.b
Understand and use averages, weighted averages, and indices.
Element: MAMDMN1.c
Solve problems involving large quantities that are not easily measured.
Element: MAMDMN1.c
Solve problems involving large quantities that are not easily measured.
Element: MAMDMN1.d
Understand how identification numbers, such as UPCs, are created and verified.
Strand: Algebra
MAMDMA1
Students will use vectors and matrices to organize and describe problem situations.
Element: MAMDMA1.a
Represent situations and solve problems using vectors in areas such as transportation, computer graphics, and the physics of force and motion.
Element: MAMDMA1.b
Represent geometric transformations and solve problems using matrices in fields such as computer animations.
MSMDMA2
Students will use a variety of network models to organize data in quantitative situations, make informed decisions, and solve problems.
a. Solve problems represented by a vertex-edge graph, and find critical paths, | 677.169 | 1 |
Hey dudes, I have just completed one week of my high school , and am getting a bit worried about my algebra with power home work. I just don't seem to grasp the topics. How can one expect me to do my homework then? Please help me.
Can you give more details about the problem? I can help you if you explain what exactly you are looking for. Recently I came across a very handy product that helps in solving math problems easily . You can get help on any topic related to algebra with power , so I recommend trying it out.
I remember I faced similar difficulties with complex fractions, equivalent fractions and factoring expressions. This Algebrator is rightly a great piece of math software program. This would simply give step by step solution to any math problem that I copied from workbook on clicking on Solve. I have been able to use the program through several College Algebra, Remedial Algebra and College Algebra. I seriously recommend the program.
Algebrator is a easy to use product and is surely worth a try. You will also find many exciting stuff there. I use it as reference software for my math problems and can say that it has made learning math much more enjoyable. | 677.169 | 1 |
Assignments
Resources
Miscellaneous
Project Ideas
Dr Nicolai Vorobjov
This is Dr Vorobjov's 2003 list of projects. It will be updated
with a new list when one is provided. In the meantime, take it as
a guide to likely interests.
Program for Solving Two-Person Games
The aim is to design an application for finding optimal strategies in
finite two-person games. Such games are called bimatrix. They include all
possible situations in which two parties have finite number of possible
ways to act (strategies) and the outcomes lead to payoffs depending on
the pair of chosen strategies. Chess is an example, though hardly
tractable, stone-paper-scissors is another, very simple to solve. The
application should be able to accept any pair of pay-off matrices as an
input and to output optimal mixed strategies. For games usually defined
in the positional form an appropriate universal representation of the
input should be designed. The project includes the stage of learning the
background game-theoretical material including the existing algorithms
for solving bimatrix games.
Pre-requisite knowledge: Some modest knowledge of linear algebra
Indicative reading: any introductory text on game theory
Program for Teaching Game Theory
The aim of the project is to design a program which will help to
explain and illustrate the basic concepts of the theory of games. When
the number of strategies of one of the players in a matrix game is small
(say, two or three) then the optimal strategies can be found by drawing
certain two- or three-dimensional pictures. The program will draw such
pictures and explain interactively some elements of the theory.
Pre-requisite knowledge: Some modest knowledge of linear algebra
Indicative reading: any introductory text on game theory
Computer Linear Algebra System
The aim is to design a basic linear algebra system as a very
simplified analogy (and a part) of usual computer algebra systems, like
REDUCE, Maple, etc. It should include algorithms for matrix
multiplication (trivial, as well as advanced and more efficient, like
Strassen's), for solving systems of linear equations (Gauss, and based on
fast matrix multiplication), computing determinants, etc.
The system should have a convenient interactive interface with an
ability to extend by adding new commands. Existing systems could be used
as examples, but all programming is supposed to be original.
The aim is to design a graphical application for robot motion planning
on the plane in the presence of polygonal obstacles. In one version of
the problem the robot is considered as a point and we are looking for a
shortest piece-wise linear path connecting "start" and "finish".
In another version the robot is a polygon, and the aim is to decide
whether it is possible for him to pass through a corridor of a complex
form, and if yes, to plan the move.
Pre-requisite knowledge: some basic knowledge of computer graphics and algorithms
Domino Partners
The game of domino is usually played between two teams of two players
in each. Since it might be hard to find enough partners willing to play,
it's desirable to have a computer program which can model one, two or
three players, breaking them in teams, as required. The project consists
of designing such a program together with a graphic interface to show
moves. | 677.169 | 1 |
Main page Content
Mathematics
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Page Content
Mathematics incorporates the skills of numeracy, rote learning of procedures, problem solving and critical thinking; research shows that students who learn mathematics achieve better overall results in education. This is why in Queensland Mathematics is a compulsory subject for all students from year 7 to year 12
Year 7 – 9
At Dalby State High School the Maths team use the Australian Curriculum along with the text resource of "Essential Mathematics for the Australian Curriculum" and digital resources including IXL Maths to present our Mathematics program. In years 7 to 9 classes are not streamed, instead our teachers cater for the individual needs of students within mixed ability classes.
Year 10
In year 10 we offer three different Mathematics subjects, each designed for the different future pathways available to our students. "Short Course in Numeracy" is designed for students who work hard but still struggle with mathematical concepts, it teaches real world applications of maths and recognises them as numerate citizens. "Year 10 Core Maths" follows on from our year 7 to 9 program, it combines real world and abstract mathematics and is assessed by assignments and exams. This subject leads into "Mathematics A" in senior. Finally "Year 10 Extension Maths" challenges our students who enjoy or are good at Mathematics, it combines complex mathematical reasoning and prepares our students to go on to study "Mathematics B" and "Mathematics C" in Senior.
Senior Subjects
At Dalby State High School we offer face to face classes for all the senior Mathematics subjects; Prevocational Maths, Mathematics A, Mathematics B and Mathematics C. More information can be found on each of these subjects individually in the fact sheets provided in "related links".
Tutoring
The Mathematics department run a regular maths tutoring session in the support space on Tuesday mornings from 8 am until 9 am. Individual teachers also run tutorials at lunchtimes during the week. Contact individual teachers for more information.
If you have any inquires about Mathematics at Dalby State High School please contact Mrs Kelly Moody (HOD). | 677.169 | 1 |
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