text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
|---|---|---|
AP Central
To understand AP Calculus in the 1990s (and the reform movement that shaped what it has become), it is necessary to understand how the AP Program began. In 1952 a group of mathematicians met under the auspices of a pilot program called College Admission with Advanced Standing (CAAS). Their charge was to design a curriculum for a mathematics course that students could take in high school for college credit. The chair of that committee, Heinrich Brinkman of Swarthmore College, held firmly to the position that the course should be nothing less than a full-blown yearlong course in single-variable calculus. Others argued that mathematics could only hope to beef up the existing fourth-year course, and they should look to a sophisticated version of pre-calculus, which was emerging in most colleges as the standard first course anyway. Brinkman's vision prevailed, of course, setting into motion a process that two generations later resulted in AP Calculus playing a major role on the stage of calculus reform.
In 1955 the CAAS program was taken over by the College Board and renamed the Advanced Placement Program. Fewer than 200 students took the first exams in AP Mathematics. In 1999 students from more than 9,000 high schools took a total of 158,468 AP Calculus AB and BC Exams. | 677.169 | 1 |
Dieses Buch gibt es in einer neuen Auflage:Mehr über den Autor
Produktbeschreibungen
SynopsisBuchrückseite
"People who analyze algorithms have double happiness. First of all they experience the sheer beauty of elegant mathematical patterns that surround elegant computational procedures. Then they receive a practical payoff when their theories make it possible to get other jobs done more quickly and more economically.... The appearance of this long-awaited book is therefore most welcome. Its authors are not only worldwide leaders of the field, they also are masters of exposition." --D. E. Knuth
This book provides a thorough introduction to the primary techniques used in the mathematical analysis of algorithms. The authors draw from classical mathematical material, including discrete mathematics, elementary real analysis, and combinatorics, as well as from classical computer science material, including algorithms and data structures. They focus on "average-case" or "probabilistic" analysis, although they also cover the basic mathematical tools required for "worst-case" or "complexity" analysis. Topics include recurrences, generating functions, asymptotics, trees, strings, maps, and an analysis of sorting, tree search, string search, and hashing algorithms.
Despite the large interest in the mathematical analysis of algorithms, basic information on methods and models in widespread use has not been directly accessible for work or study in the field. The authors here address this need, combining a body of material that gives the reader both an appreciation for the challenges of the field and the requisite background for keeping abreast of the new research being done to meet these challenges.
Highlights:
Thorough, self-contained coverage for students and professionals in computer science and mathematics
Focus on mathematical techniques of analysis
Basic preparation for the advanced results covered in Knuth's books and the research literature
Die hilfreichsten KundenrezensionenDie hilfreichsten Kundenrezensionen auf Amazon.com (beta)
Amazon.com:
6 Rezensionen
55 von 55 Kunden fanden die folgende Rezension hilfreich
Clear and concise17. Januar 2001
Von
Amazon Customer
- Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
This is an excellent book on the analysis of algorithms. More specifically, it is a book on the mathematics needed for the analysis of algorithms. Quite a few algorithms are presented and analyzed in great detail, but the emphasis is on the analysis techniques rather than on the algorithms. This is in contrast with Cormen,Leiserson and Rivest, or Sedgewick's own "Algorithms" series which emphasize the algorithms rather than the analysis. If you're looking for a catalog of algorithms along with explanations, you want a different book, but if you want to know how to analyze that bizarre code (which Fred in the next cubicle wrote) and prove that it works well (or doesn't) then this is an excellent choice. The book is aimed at advanced undergrads/graduate students and assumes a certain amount of mathematical sophistication - i.e. calculus, discrete math, probability, etc. On the spectrum from "Mathematical Techniques" through "Analysis of Algorithms" and ending up with "Catalog of Algorithms", I would start with Graham, Knuth and Patashnik "Concrete Mathematics", travel through this book, on to Knuth "The Art of Computer Programming", then to Cormen, Leiserson and Rivest, and finally end up with either Sedgewick's "Algorithms" or Skeina's "Algorithm Design Manual".
21 von 21 Kunden fanden die folgende Rezension hilfreich
Classic textbook in this field1. April 2000
Von
Sen Peng Eu
- Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
Verifizierter Kauf5 von 5 Kunden fanden die folgende Rezension hilfreich
A very readable chapter on generating functions4. Februar 1999
Von
cj2005
- Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe3 von 5 Kunden fanden die folgende Rezension hilfreich
A must have.22. Oktober 2001
Von
Ein Kunde
- Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
I read a lot of books about complexity analysis. And this book is a state of art in the field. Easy to read, and well done.It cover the necessary staff that every new commer to the field should know, can be used as a refference,and it make a good teaching material for graduate student.
0 von 3 Kunden fanden die folgende Rezension hilfreich
Easy to Comprehend30. Oktober 2011
Von
Anon-Ops
- Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe
Verifizierter Kauf
This book is great for the everyday college student, 2 to 3 education required to comprehend thoroughly.Impress your professor with the superfluous knowledge you will get from this well-written book. | 677.169 | 1 |
NS-0315: Elementary Theory of Numbers
Course Information
Instructor Info:
Geremias Polanco Encarnacion
Term:
2013F
Meeting Info:
Tuesday
10:30 AM - 11:50 AM Cole Science Center 316
Thursday
10:30 AM - 11:50 AM Cole Science Center 316
Description:
Number theory is the branch of mathematics that deals with the properties of whole numbers. This is an area in which simplicity and complexity meet in an astonishing way. Therefore, in this course you will be presented with problems that, in most cases, are very easy to state, but whose degrees of difficulty range from very easy to incredibly difficult. We will focus on learning the tools and techniques that are used to attack problems in the field and beyond. By following an inquisitive approach in this exploration of the theory of numbers, we will help sharpen problem solving skills, the basic weapon of a professional mathematician. You will also learn and apply basic principles used in mathematical research. Topics include divisibility, primes and factorization, congruency, arithmetical functions, quadratic reciprocity, primitive roots, Dirichlet's series, and other topics at our discretion and as time permits.
Course Objectives:
There are five main objective in this course:
To develop a conceptual understanding of the Elementary Theory of Numbers.
Learning how to write professional mathematical arguments.
Learning how to produce simple mathematical documents using latex.
Learning how to use computer experiments to make mathematical conjectures and proving them if possible.
Apply the theory learned to other disciplines to solve problems.
Evaluation Criteria:
The privious five objective will be evaluated as follows
Developing conceptual understanding of Number Theory, through
Homework problems and projects.
Classwork
Portfolio
Learning how to write professional arguments
In-class and homework xercises that require proof writing
Portfolio
Producing latex documents
Through problems and assignments that require latex
Portfolio
Using computer experiments to make conjectures
There will be specific assignments througout the course
Applying the theory to other disciplines
Homework sets and projects
Students who complete the following are guaranteed to receive an evaluation:
Portfolio: Every student is required to create a portfolio to be handed on Tuesday December 10. The portfolio will be a collection of items comprised of: all graded homework (including latex-typed exercises), graded quizzes, conjectures, proofs and/or observations from experiments, as well as any projects assigned in class.
Non-classroom work-load Expectation: You are expected to work 6 to 10 hours per week outside class. If you find yourself putting more than 10 hours of work per week, let me know please.
Syllabus: Please, read the attached syllabus below to better understand the description of each aspect of the course. | 677.169 | 1 |
Classes
MTH 148: Functional Math for Elementary Teachers I
This course is the first in a two-course sequence presenting the mathematical concepts and problem-solving techniques necessary for success in a teaching career at the elementary school level. It is not a course solely for math teachers; rather it provides a general mathematical background for teachers of all subjects. Topics include problem-solving, sets, numeration systems, number theory and the whole, integer and rationale number systems | 677.169 | 1 |
Read an Excerpt
Description
A plain-English guide to the basics of trig
From sines and cosines to logarithms, conic sections, and
polynomials, this friendly guide takes the torture out of
trigonometry, explaining basic concepts in plain English, offering
lots of easy-to-grasp example problems, and adding a dash of humor
and fun. It also explains the "why" of trigonometry, using
real-world examples that illustrate the value of trigonometry in a
variety of careers.
Mary Jane Sterling (Peoria, IL) has taught mathematics at Bradley
University in Peoria for more than 20 years. She is also the author
of the highly successful Algebra For Dummies (0-7645-5325-9). | 677.169 | 1 |
This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.Practice makes perfect! Get perfect with a thousand and one practice problems! 1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more... more...
A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing shows how to use a collection of mathematical techniques to solve important problems in applied mathematics and computer science areas. The book discusses fundamental tools in analytical geometry and linear algebra. It covers a wide range of topics,... more...
In 1884, Edwin Abbott Abbott wrote a mathematical adventure set in a two-dimensional plane world, populated by a hierarchical society of regular geometrical figures-who think and speak and have all too human emotions. Since then Flatland has fascinated generations of readers, becoming a perennial science-fiction favorite. By imagining the contact... more... | 677.169 | 1 |
An exploration of conceptual foundations and the practical applications of limits in mathematics, this text offers a concise introduction to the theoretical study of calculus. It analyzes the idea of a generalized limit and explains sequences and functions to those for whom intuition cannot suffice. Many exercises with solutions. 1966 edition. more...
This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject... more...
Self-contained and suitable for undergraduate students, this text offers a working knowledge of calculus and statistics. It assumes only a familiarity with basic analytic geometry, presenting a coordinated study that develops the interrelationships between calculus, probability, and statistics. Starting with the basic concepts of function and probability,... more...
"This book is a radical departure from all previous concepts of advanced calculus," declared the Bulletin of the American Mathematics Society, "and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics." Classroom-tested in a Princeton University honors course, it offers students... more...
Comprehensive but concise, this introduction to differential and integral calculus covers all the topics usually included in a first course. The straightforward development places less emphasis on mathematical rigor, and the informal manner of presentation sets students at ease. Many carefully worked-out examples illuminate the text, in addition to... more...
This concise introduction to Lebesgue integration is geared toward advanced undergraduate math majors and may be read by any student possessing some familiarity with real variable theory and elementary calculus. The self-contained treatment features exercises at the end of each chapter that range from simple to difficult. The approach begins with... more...
This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with series, the treatment offers many applications, including those related to the theory of special functions. Numerous problems appear throughout... more...
Two-dimensional calculus is vital to the mastery of the broader field, and this text presents an extensive treatment. Advantages include the thorough integration of linear algebra and development of geometric intuition. 1986 edition. | 677.169 | 1 |
Enter your mobile number or email address below and we'll send you a link to download the free Kindle Reading App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
Computer algebra systems have revolutionized the use of computers within mathematics research, and are currently extending that revolution to the undergraduate mathematics curriculum. But the power of such systems goes beyond simple algebraic or numerical manipulation. In this practical resource Roman Maeder shows how computer-aided mathematics has reached a level where it can support effectively many of the computations in science and engineering. Besides treating traditional computer science topics, he demonstrates how scientists and engineers can use these computer-based tools to do scientific computations. A valuable text for computer science courses for scientists and engineers, this book will also prove useful to Mathematica users at all levels. Covering the latest release of Mathematica, the book includes useful tips and techniques to help even seasoned users.
Editorial Reviews
Review
"This is an excellent introductory textbook in computer science via Mathematica. I recommend it for use in computer science classes for science and engineering students." Computing Reviews
Book Description
Computer algebra systems have revolutionized the use of computers within mathematics research, and are currently extending that revolution to the undergraduate mathematics curriculum. But the power of such systems goes beyond simple algebraic or numerical manipulation. This book shows how computer-aided mathematics has reached a level where it can support effectively many of the computations in science and engineering. In addition to treating traditional computer science topics, an introductory course should show scientists and engineers how these computer-based tools can be used to do scientific computations.A valuable text for computer science courses for scientists and engineers, this book should also prove useful to Mathematica users at all levels. Covering the latest release of Mathematica, the book includes useful tips and techniques to help even seasoned users would be a good book for you if you understand computer science, but don't know anything about Mathematica. Maeder provides examples from physics, differential equations, data sorting, etc, showing how these problem can be solved quickly in mathematica. In the examples, however, Maeder does not always explicitly explain what the Mathematica commands are doing. Every example in the book is downloadable from his website, but he sometimes refers to files under the assumption that you have downloaded them. The most useful parts of the book for me were the section on OOP within mathematica and the section on preparing your programs for distribution.
As a symbolic programming language, and as one that can effectively emulate most programming paradigms, Mathematica is unequaled. In this book, the author makes this abundantly clear as he takes the reader through a sampling of the power of Mathematica, with the target reader being a computer scientist without a knowledge of Mathematica. After a very brief review of computer concepts and architectures in chapter 1, the author begins chapter 2 with an overview of the syntax of Mathematica. The most helpful points in this chapter were: 1. How to implement piecewise-defined functions. 2. Pure functions. 3. The ability of Mathematica to do functional programming via functional operations. 4. Normal expressions and atoms. Chapter 3 is an introduction to iteration and recursion. The author begins the chapter by showing how to use rule-based or recursive programs to construct a program to calculate the greatest common divisor of two integers. He is careful to note however that the use of recursion may be inefficient and so he shows how to convert the program to one that uses loop iteration. The contrast in inefficiency between iterative and recursive programs is illustrated again in the next section which deals with the Collatz problem. An iterative computation of the Collatz sequence is given, and the author encourages the use of loops and not recursion, to obtain efficient programs. The author shows how to use loop invariants to test program correctness for loops. The engineer/physicist reader will appreciate the application of iterative methods to the solution of ordinary differential equations. In chapter 4, the author shows how to build packages in Mathematica, via an example in complex variables.Read more ›
Yes, there are newer books and the mathematica cookbook covers alot of this material. However, the organization of this book I think is better.
This covers the programming paradigms in Mathematica in a very straightforward way. And the author covers some of the standard software constructs like lists, databases, and binary trees. All of which are standard constructs in any programming language, but Mathematica is not C or Pascal, so there are considerations and techniques to using mathematica as a programming language.
The book even covers mathlink, and OOP ( at the end and so-so).
The other side of this is the author provides a clear cut tutorial on how to use mathematica for computation.
The other reviewer really does a great job, but it is 2012 and this is still a very useful resource.
I used Mathematica as a calculator and plotting tool. I switched to IPython/Numpy for my daily work until recently where I needed to model a hidden Markov process. The HiddenMarkovProcess package in Mathematica was handy but I lacked the programming skills to implement the project. This book allowed me to grasp the spirit of Mathematica quickly. Unfortunately the accompanying site is no longer online so I could not access the code of the packages. Nonetheless, this book served me well as a programmer coming from procedural/oo languages. | 677.169 | 1 |
An Illustrated Introduction to Topology and Homotopy explores the beauty of topology and homotopy theory in a direct and engaging manner while illustrating the power of the theory through many, often surprising, applications. This self-contained book takes a visual and rigorous approach that...
An Introduction
The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry.
Following the emergence of his gyroalgebra in 1988, the...
This volume, first published in 2000, presents a classical approach to the foundations and development of the geometry of vector fields, describing vector fields in three-dimensional Euclidean space, triply-orthogonal systems and applications in mechanics. Topics covered include Pfaffian forms,...
Designed for a one-semester course at the junior undergraduate level, Transformational Plane Geometry takes a hands-on, interactive approach to teaching plane geometry. The book is self-contained, defining basic concepts from linear and abstract algebra gradually as needed.
The text adheres to the...
Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.
The book...
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and...
Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations shows how four types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities through their special quasilinear degenerate representations. The authors present a unified...
This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a... | 677.169 | 1 |
Mathematics instructional programs should focus on learning
to reason and construct proofs as part of understanding mathematics
so that all students --
recognize reasoning and proof as essential and powerful parts
of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and proofs;
select and use various types of reasoning and methods of proof
as appropriate.
"Throughout the history of American education, learning
to write proofs has been an important objective of the geometry
curriculum for college-bound students. At the same time, proof
writing has also been perceived as one of the most difficult
topics for students to learn." (Senk, 1985)
"fewer than 15% of high school graduates in the United
States master proof writing." (Senk, 1985)
Externally-based Appeal to an external source
for convincing and persuading Authoritarian rely on textbook, teacher, or more
knowledgeable classmate Ritual correctness judged by the form of the argument
rather than the reasoning Symbolic - rely on symbol manipulation, regardless of
correctness
Empirical Justifications made on the basis of
examples Perceptual rely on how a figure looks (e.g., the
triangle looks equilateral) Examples-based convinced by one or more examples
(e.g., seeing the pattern)
Analytic Mathematical proofs Transformational based on general aspects of a situation,
perceiving underlying structure behind a pattern Axiomatic an ability to work within an axiomatic
system
Balacheff's 4 Stages of Understanding Mathematical Proof
1)Naïve Empiricism
Inductive perspective
Conclusions based on small number of cases
2)Crucial Experiment Question of generalization is considered
Examination of extreme cases
3)Generic Example Arguments based on a class of objects
Highest level prior to deductive proof
4)Thought Experiment Transition from practical to intellectual proofs
Development of deductive proofs
(Balacheff, 1987)
Six Principles of Geometric Proof
1)Implications of Truth Statements are true
if and only if they are true for all cases.
a) A theorem has no exceptions.
b) A counterexample disproves a general statement.
2) Purposes of Proof - The dual role of proof is to convince
and to explain.
a) Proofs are required to establish truth.
b) Proofs can explain.
3)Generality Requirements - A proof must be general.
a) Generality can be achieved by checking all cases.
b) Generality can be achieved by reasoning about general statements.
c) Generality is not achieved by reasoning inductively.
d) Generality is not achieved by checking special cases.
4)Internal Logic Requirements - The validity of
a proof depends on its internal logic. a)Conditional statements contain distinct components.
b)The logical order of statements matters.
c)Ritualistic aspects of proof are irrelevant to its
validity.
5)Logical Equivalences - Statements are
logically equivalent to their contrapositives, but not necessarily
to their converses or inverses.
6)Role of Diagrams - Diagrams that illustrate
statements have benefits and limitations.
a)Diagrams are limited by their specificity.
b)Diagrams may assist with visualization of relationships.
2. Dylan attempted to prove the statement, "When you
add the measures of the interior angles of any triangle, your
answer is always 180º." His work is shown below.
Dylan's Work I measured the angles of all sorts of triangles accurately
and made a table.
A B C Total
34 110 36 180
95 43 42 180
35 72 73 180
10 27 143 180
They all added up to 180º, so the statement is true.
Questionnaire Poor Performance (continued)
2a. Since Dylan checked that the statement is true for both
obtuse and acute triangles, his work shows that the statement
is always true.
9. Consider the dialogue between Juan and Ling. Juan: In both of our diagrams ABCD is a rectangle and
E is a point on segment AD. The height of BEC is the length of
segment CD. Since the area of a triangle is 1/2(baseoheight),
the area of BEC is half the area of rectangle ABCD. The same
is true in your diagram Ling.
Ling: In my diagram, E is in a different place on segment
AD. So, your argument doesn't apply to my diagram.
Juan's Diagram Ling's Diagram
Questionnaire Moderate Performance (continued)
b. Ling is correct. Since the diagrams are different, the
conclusion that Juan made for his diagram doesn't apply to Ling's
diagram.
Statement 1:
"In a triangle, a line connecting the midpoints of two of
its sides is parallel to the third side."
Argument 1: I drew three different triangles. I labeled each triangle
ABC. In each triangle, D and E are the midpoints of the sides
AB and AC, respectively. I measured the angles in each of the
three different triangles and in each case was congruent to .
Since these angles are corresponding angles (relative to line
, line , and transversal ), is parallel to . Therefore, the statement
is always true. 1. Does argument 1 show that the statement is true for all
triangles? Why or why not?
Proof Quiz Results 3. Conjecture: When I draw a line parallel to a side of a
triangle it creates a new triangle. I checked several examples
and noticed that this smaller triangle is always similar to the
original triangle.
Proof Quiz Results 5.
Given: Quadrilateral KLMN is a parallelogram.
Segments and intersect at P.
N is on line
Prove:
KLP is similar to NQP
Several hints about how this proof may be constructed are
provided below. Please use some of these hints to write a valid
proof that KLP is similar to NQP.
Hints: Recall that proving that triangles are similar requires the
identification of several pairs of congruent angles. Use the
quadrilateral to identify a pair of parallel lines. Use properties
of parallel lines and related angles to identify pairs of congruent
angles.
d. Given: Line l and line m are both cut by
transversal line n. Line n is not perpendicular
to either line l or line m. The alternate exterior
angles are supplementary.
Conclusion:__________________________________
Reason:_____________________________________
Proof Quiz Results 6.
Given: Circle A with radius .
is the perpendicular bisector of .
Point C is on the circle.
Prove: ABC is equilateral.
Several hints about how this proof may be constructed are
provided below. Please use some of these hints to write a valid
proof that ABC is equilateral.
Hints: Recall that proving a triangle to be equilateral requires
showing that several segments are congruent. Use the circle to
find some congruent segments. Using 's relationship to may help
you find a relationship between ADC and BDC.
U.S. Department of Education. National Center for Education
Statistics. (1998) Pursuing Excellence: A Study of U.S. Twelfth-Grade
Mathematics and Science Achievement in International Context.
Washington D.C.: U.S. Government Printing Office. | 677.169 | 1 |
This is one video in the series of lessons on math provided by DeafTEC. Gary Blatto-Vallee, a math and science instructor at the National Technical Institute for the Deaf, guides viewers through a variety of mathematical exercises in this DeafTEC video series. All lessons are fully captioned, signed in ASL, and voiced. In this 7:19 video, Blatto-Vallee uses an electronic whiteboard to show several examples of factoring trinomials with common factors. See the main Math Video Resources page for an introduction to this video series.
The world is becoming more and more quantitative and data focused. Many professions depend on numerical measurements to make decisions in the face of uncertainty. Statisticians use quantitative abilities, statistical knowledge, and communication skills to work on many challenging problems. | 677.169 | 1 |
In this post, cause I believe that limiting the curiosity (or at least waiting for the right time) is important, I want to share a good message to the students from Mary Boas.
To The Student
As you start each topic in this book, you will no doubt wonder and ask "Just why
should I study this subject and what use does it have in applications?" There is a
story about a young mathematics instructor who asked an older professor "What do
you say when students ask about the practical applications of some mathematical
topic?" The experienced professor said "I tell them!" This text tries to follow
that advice. However, you must on your part be reasonable in your request. It
is not possible in one book or course to cover both the mathematical methods
and very many detailed applications of them. You will have to be content with
some information as to the areas of application of each topic and some of the
simpler applications. In your later courses, you will then use these techniques in
more advanced applications. At that point you can concentrate on the physical
application instead of being distracted by learning new mathematical methods.
One point about your study of this material cannot be emphasized too strongly:
To use mathematics effectively in applications, you need not just knowledge but skill.
Skill can be obtained only through practice. You can obtain a certain superficial
knowledge of mathematics by listening to lectures, but you cannot obtain skill this
way. How many students have I heard say "It looks so easy when you do it," or "I
understand it but I can't do the problems!" Such statements show lack of practice
and consequent lack of skill. The only way to develop the skill necessary to use this
material in your later courses is to practice by solving many problems. Always study
with pencil and paper at hand. Don't just read through a solved problem—try to
do it yourself! Then solve some similar ones from the problem set for that section, trying to choose the most appropriate method from the solved examples. See the
Answers to Selected Problems and check your answers to any problems listed there.
If you meet an unfamiliar term, look for it in the Index (or in a dictionary if it is
nontechnical).
My students tell me that one of my most frequent comments to them is "You're
working too hard." There is no merit in spending hours producing a solution to
a problem that can be done by a better method in a few minutes. Please ignore
anyone who disparages problem-solving techniques as "tricks" or "shortcuts." You
will find that the more able you are to choose effective methods of solving problems
in your science courses, the easier it will be for you to master new material. But
this means practice, practice, practice! The only way to learn to solve problems is
to solve problems. In this text, you will find both drill problems and harder, more
challenging problems. You should not feel satisfied with your study of a chapter
until you can solve a reasonable number of these problems. Continue reading →
I've been having a problem with my touch screen for a long time. Cause I couldn't find the problem exactly (software or hardware), I didn't send it to any service. And I was trying to fix it but then I realized that it's a hardware problem.
The problem : Touch screen acting up, any side of touch screen is not working
What Can You Try : Firstly, you can calibrate your touch screen. Search on any search engine on how to calibrate your touch screen. For instance, in HTC Desire HD follow the steps:
İf it didn't work try to format your phone. Be sure that you backed up your data before formatting your phone. Check here also if you think that it's a software issue. I've tried most of these but none of them worked for me. Continue reading →
"What Do I Deserve" in this world? This is a question that I've been keeping my head busy with for a long time and hasn't left it without finding an answer. I've been thinking about this recently and have been trying to find answers to it. Actually the answer to this question is quite simple but at the same time also hard. Why simple? Or why hard? Because the answer is "everything" and I cannot own/have everything. Why would I even ask for something like that?
There's a lot of people in this world, and if all these people don't want "everything" they at least want "most of the things". This question has also a simple answer because I deserve this. Because me and you, we are valuable. Because we deserve; seeing places that garnish our dreams, hearing what we want to hear, eating what we want to eat, scenting what we want to scent and doing whatever comes to our mind. I can't tell that we didn't come to this world to have fun, but I can tell that we didn't come here to feel pain…and I'm sure on that. So what is the answer? Why aren't we living the way we want to? Or is someone able or if it isnt able to live like that?
Is it the thing that we werent owning? What goes through our mind? Could it be money?
That time these things are messing with my mind. Do people have to love money? Or do they have to hate it? In conclusion I can say that if I hate money, the hate is because I don't have it. Or if I love money it's because I have money. From this comes a conclusion: the questions "how do I have money?" and "how do I get what I deserve?" are equal questions. And this questioning causes that the most unvaluable thing in this world becomes valuable. | 677.169 | 1 |
MA125 Intermediate Algebra
for F2I Students will develop an understanding of the concepts of algebra through demonstration of principles and individual exercises. Learning is will be based both on reading, lecture, and practical application. Students are encouraged to take advantage of office hours to ensure full understanding of the material.
Learning Outcomes: Core Learning Outcomes
State and use basic terminology and symbols of the discipline appropriately
Solve linear equations and inequalities in one variable and verify solution(s)
Manipulate and simplify exponential expressions
Perform arithmetic on and factor polynomials and solve polynomial equations
Solve "word" problems
Manipulate and simplify rational expressions
Manipulate and simplify radical expressions and translate into/ from exponential form
Class Assessment: Students are responsible for reading all assignments and completing all homework from the text. Students are responsible for obtaining information from the instructor or co-students regarding assignments made during absences. Class assessment will be measured by the performance on five examinations (four tests and a comprehensive final). Their final and mid-terms will cover assigned reading and homework.
Grading:
Attendance: 10%
Homework: 20%
Four Exams: 10% each
Final Exam 30%
Total 100%
Total scores of 90 to 100 % will be considered an A. 80 to 89 % will receive a B, 70 to 79 % will receive a C, 60 to 69 % will receive a D. Any score below 60 % will be failing.
In addition, the final exam MUST be passed to receive any grade higher than a D
Late Submission of Course Materials: Late homework will receive ½ credit without prior coordination with the instructor. Any work submitted more than one class period after assigned due date will receive a zero for that assignment. Students unable to take a test/final exam will coordinate with the instructor prior to the period of the exam for an alternate exam time/place. Failure to do this will result in a zero on the test/ final exam.
Classroom Rules of Conduct: Students should come to class prepared to discuss the subject material scheduled for the day. They are expected to have read the appropriate sections of the textbook and come prepared to ask questions about the topic of the day.
For all homework assignments students must show all steps they use in order to receive full credit for their work.
Test/Date/Material Covered:
Test 1: End of week 2, Chapters 1 & 2
Test 2: End of week 4, Chapters 3 & 4
Test 3: End of week 5 Chapters 5 & 6
Test 4: End of week 7 Chapters 7 & 8
Final Exam: week 8, Chapters 1-9 1
Evaluate 4 out of 4 algebraic expressions
Evaluate 3 out of 4 algebraic expressions
Evaluate 2 out of 4 algebraic expressions
Evaluate 0 or 1 out of 4 algebraic expressions
Synthesis Outcomes 1
Simplify and manipulate 4 out of 4 algebraic expressions
Simplify and manipulate 3 out of 4 algebraic expressions
Simplify and manipulate 2 out of 4 algebraic expressions
Simplify and manipulate 0 or 1 algebraic expressions
Analysis Outcomes 2
Solve and check 4 out of 4 algebraic equations
Solve and check 3 out of 4 algebraic equations
Solve and check 2 out of 4 algebraic equations
Solve and check 0 or 1 out of 4 algebraic equations
Application Outcomes 3
Solve 4 out of 4 practical applications
Solve 3 out of 4 practical applications
Solve 2 out of 4 practical applications
Solve 0 or 1 practical applications
Content of Communication Outcomes 4
Graph 4 out of 4 linear equations or inequalities
Graph 3 out of 4 linear equations or inequalities
Graph 2 out of 4 linear equations or inequalities
Graph 0 or 1 linear equations or inequalities
Technical Skill in Communicating Outcomes 4
Find 4 out of 4 slopes of lines
Find 3 out of 4 slopes of lines
Find 2 out of 4 slopes of lines
Find 0 or 1 slopes of lines
First Literacy Outcomes (Formulas) 1, 2, 3
Use and evaluate 4 out
of 4 formulas
Use and evaluate 3 out
of 4 formulas
Use and evaluate 2 out
of 4 formulas
Use and evaluate 0 or 1 out
of 4 formulas
Second Literacy Outcomes (Order of Operations) 1, 2, 3
Apply order of operations to 4 out of 4 algebraic expressions
Apply order of operations to 3 out of 4 algebraic expressions
Apply order of operations to 2 out of 4 algebraic expressions
Apply order of operations to 0 or 1 out of 4 algebraic expressions
Copyright:
This material is protected by copyright
and can not be reused without author permission. | 677.169 | 1 |
Discrete Math Teaching Resources
Resource Overview
Discrete mathematics is a fundamental building block for any program in Information Sciences and Technology.
This web site provides a set of online resources for teaching and learning many of the concepts in discrete mathematics,
particularly the most fundamental ones.
Topics covered include logic and boolean algebra, elementary number theory, direct proof techniques, sequences
and induction, set theory, combinatorics and probability, functions and relations, grammars, languages and finite
state machines, graphs, and trees.
The resources include static web pages we link to, some static pages we have built, some simple Java applets
to illustate concepts, and some more complex Java applets that we link to.
If you have useful Java applets for teaching concepts in discrete math that you'd like to share, please send
them our way. We'll thank you and put them up on this web site.
Who we are
Development of this web site was partially sponsored by the Penn State Fund for Excellence in Learning and Teaching
(FELT), project "Java-based Teaching of Mathematics in Information Sciences and Technology", supervised
by Frank Ritter and David Mudgett. Andrew R. Freed helped develop it.
Last updated: 16 mar 2012
Suggested Texts
Information Technology - Inside and Outside. D. Cyganski, J.A. Orr, R.F. Vaz, (2001), Prentice
Hall, ISBN 0-13-011496-0 is an interesting book showing many applications of the mathematics in this course. It
is available in the Penn State bookstore. Also available online at
Discrete and Combinatorial Mathematics by R.P. Grimaldi, is another, longer, more detailed textbook.
Bebop to the Boolean Boogie, An Unconventional Guide to Electronics Fundamentals, Components
& Processes, by Clive Maxfield, 1995, is an interesting alternative book on how material in this
course relates to computers and electronics. Call #TK7868.D5M323 in the PSU Library. | 677.169 | 1 |
Christina
I'm a computer science student who isn't exactly thrilled at the prospect of taking discrete math this semester. I've taken up to calc 3 and do not plan to go any further than possibly linear algebra. Otherwise, I'm not really interested in math, nor am I too skilled in it, thus I have found myself here. | 677.169 | 1 |
Elementary Linear Algebra (2nd Edition)
9780131871410
ISBN:
0131871412
Edition: 2 Pub Date: 2007 Publisher: Pearson
Summary: "Elementary Linear Algebra, 2/e" -- Lawrence Spence, Arnold Insel, and Stephen Friedberg Embracing the recommendations of the "Linear Algebra Curriculum Study Group, the authors have written a text that" students will find both accessible and enlightening. Written for a matrix-oriented course, students from a variety of disciplines can expect a greater understanding of the concepts of linear algebra. Starting with ma...trices, vectors, and systems of linear equations, the authors move towards more advanced material, including linear independence, subspaces, and bases. The authors also encourage the use of technology, either computer software (MATLAB) or super-calculators, freeing students from tedious computations so they are better able to focus on the conceptual understanding of linear algebra. Lastly, students will find a variety of applications to engage their interest, demonstrated via economics, traffic flow, anthropology, Google searches, computer graphics, or music to name a few. By leveraging technology and incorporating engaging examples and numerous practice problems and exercises, this text best serves the needs of students attempting to master linear algebra.
Lawrence E. Spence is the author of Elementary Linear Algebra (2nd Edition), published 2007 under ISBN 9780131871410 and 0131871412. Three hundred eighty three Elementary Linear Algebra (2nd Edition) textbooks are available for sale on ValoreBooks.com, seventy used from the cheapest price of $83.76, or buy new starting at $1440136001001102-3-0-12 Orders ship the same or next business day. Exp [more]
ALTERNATE EDITION: Instructor Edition: Same as student edition but has free copy markings. Has minor wear and/or markings. SKU:9780136001102 shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less] | 677.169 | 1 |
Intermediate Algebra: Connecting Concepts through Applications
9780534496364
ISBN:
0534496369
Edition: 1 Pub Date: 2011 Publisher: Brooks Cole
Summary: INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate so...lutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and skills.
Clark, Mark is the author of Intermediate Algebra: Connecting Concepts through Applications, published 2011 under ISBN 9780534496364 and 0534496369. Four hundred sixty six Intermediate Algebra: Connecting Concepts through Applications textbooks are available for sale on ValoreBooks.com, ninety three used from the cheapest price of $39.13, or buy new starting at $110.59edited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less] | 677.169 | 1 |
overview of the central ideas in calculus and gives examples of how calculus is used to translate many real-world phenomena into mathematical functions. Beginning with an explanation of the two major parts of calculus - differentiation and integration - Gudmund R Iversen illustrates how calculus is used in statistics: to distinguish between the mean and the median; to derive the least squares formulas for regression co-efficients; to find values of parameters from theoretical distributions; and to find a statistical -value when using one of the continuous test variables such as the t-variable.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":18.05,"ASIN":"0803971109","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":17.5,"ASIN":"0803920520","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":15.96,"ASIN":"0803958757","moqNum":1,"isPreorder":0}],"shippingId":"0803971109::3kj1dBzsLEL0002jRhzr5mw%2Fe3%2F%2FPdVqjy%2F68dtU%2BCyZlHmwkDFqIU%2FSHiiS7YgSXKUV4O%2FvYXABHYozbTF%2BAsvReusYlPmjVQsi7D4NcXQ%3D,0803920520::AApT%2FexqNdXYQcXzph4OHj6ntFH8CS3aWY1vG%2B7CxyjrtG%2FnxwKk5rvN%2BNTANpqBOzB3CeE1XT4ANurS9mVgZIgAG97aEcJ%2FgybaopI54l0%3D,0803958757::AApT%2FexqNdXYQcXzph4OHqbG%2BBdvDmCl5yiZfDhL2Tl9h0HWmuCIdCmSoe5TezsLheLlKlAh%2FvJ3U23n0Ppni9CX5b4ebjnORm4n%2BD3wynudmund Iversen explains the concepts, formulas and applications of calculus to social scientists and highlights their applications to analysis of research data. Four successive chapters review prerequisite concepts, discuss differential calculus and its usefulness in modeling change, introduce integral calculus and its applications to reasoning about probabilities, and illustrates these concepts with examples from research.
The book is clear, concise, and covers the necessary material. I recommend it as a good review of calculus for the mathematically unenthused.
I purchased this text as guide for a refresher course on calc on statistics.com. The book is very concise and easy to read, though without the course notes I am not sure I would have really advanced my understanding of the subject - I needed the additional example problems to really see how things worked and relearn what had been forgotten.
If looking for a primer and quick review of important calculus methods, this is a solid text. But insufficient for newbies or those interested in a more rigorous approach. Only recommended for those who may have been away from Calculus for a while, and just need a refresher on the basics of derivatives and integration. | 677.169 | 1 |
An IT based activity for beginners to understand the functions of a small scale business operation. The case considers...
see more
An IT based activity for beginners to understand the functions of a small scale business operation. The case considers break-even points and cash flow management. The case is made up of three activities,, includes downloadable spreadsheets, and contains a worksheet for analyzing costs, and a teacher's guide. A tutor version of the case is also available for use online. Materials are designed for United Kingdom students. Some of the material by this author was drawn from Spreadsheets and Mathematics for I.T. by Andrew Rothery, John Murray Publishers, London, 1991 Transforming Course Design - Business Math to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material ELIXR: Transforming Course Design - Business Math
Select this link to open drop down to add material ELIXR: Transforming Course Design - Business Math to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
ALGEBRA 1 I. PHILOSOPHY Algebra 1 CP covers all the dimensions of the understanding of elementary algebra: its skills, its properties, its uses, and its representations. 1 copy.pdf | 677.169 | 1 |
GeoGebra, a free, open-source, dynamic mathematics software, is rapidly gaining popularity in the teaching and learning of mathematics around the world. Currently, GeoGebra is translated to 55 languages, used in 190 countries, and downloaded by approximately 500,000 users in each month, and clearly making an impact on mathematics education in most countries. This increased use compelled the establishment of the International GeoGebra Institute (IGI) that serves as a virtual organization to support local GeoGebra initiatives and institutes. There are already 132 institutes in 67 countries, which pursue training and support of teachers, develop teaching materials, and carry out research projects. In this talk, I will outline the directions of GeoGebra software development, its extension to STEM subjects, activities of its community, and the work of GeoGebra Institutes. | 677.169 | 1 |
Note: These results are described in terms of a traditional math course sequence, but they apply equally well to an integrated math sequence.
The table shows the 20 topics rated most important as prerequisites
by instructors of credit-bearing first-year college mathematics
courses in the 2012 ACT
National Curriculum Survey.1 Nine—or
45%—of the topics are typically covered in grade 7 or earlier, while
10 are topics from Algebra I. The one remaining topic is typically
taught in Algebra II.
This finding suggests that an important contributor to students'
college and career readiness is the ability of teachers throughout
K–12 to keep strengthening many of the topics students learn
in earlier grades as well as to develop connections, deepen
understanding, or increase fluency. It's said that students don't learn
the skills from one mathematics course until they take the next. It's
not enough for students to stay at the same level.
Every three to five years, the ACT
National Curriculum Survey asks
educators about what they teach (or
don't teach) in their courses and how
important they feel various topics
in their discipline are for students
to know to be successful in future
coursework. The survey also asks
educators for their opinions on
educational topics of current interest,
such as the college readiness of
their students or the implementation
of improved standards like the ACT
College and Career Readiness
Standards or the Common Core
State Standards.
This brief highlights a finding from the
2012 Mathematics survey.
infobrief@act.org for more information or to suggest ideas for future ACT Information Briefs. | 677.169 | 1 |
Browse related Subjects ...
Read More natural world, accompanied by classic and contemporary examples and exercises. The COMAP approach presented in "FAPP" makes contemporary mathematical ideas exciting, relevant, and fun. The text motivates students to think about and appreciate how math affects the world around them. Students learn the basics of management science, statistics, finance, game theory, voting, and other topics in a relatable context, developing the knowledge and skills that will benefit them in future courses, their careers, and their lives. The new edition maintains the strengths that have kept this text a best-seller while also including new examples, new exercises, new pedagogy, and enhanced media tools for students and instructors to support the teaching and learning goals. "Like New. | 677.169 | 1 |
2015-2016 School Calendar
Intro to Secondary I Mathematics
Open to all students entering their first year of high school in September, this course will help students improve their elementary math skills and prepare them for high school math topics in September.
For more information, please download the pdf version of the application form by clicking on the link below: | 677.169 | 1 |
This volume will provide a welcome resource for teachers seeking an undergraduate text on advanced trigonometry, when few are readily available. Ideal for self-study, this text offers a clear, logical presentation of topics and an extensive selection of problems with answers. Contents include the properties of the triangle and the quadrilateral;... more...
Karl Gustafson is the creater of the theory of antieigenvalue analysis. Its applications spread through fields as diverse as numerical analysis, wavelets, statistics, quantum mechanics, and finance. Antieigenvalue analysis, with its operator trigonometry, is a unifying language which enables new and deeper geometrical understanding of essentially every... more...
This volume offers a concise, highly focused review of what high school and beginning college undergraduates need to know to successfully solve the trigonometry problems they will encounter on exams. Rigorously tested examples and coherent, to-the-point explanations are presented in an accessible form and will provide valuable assistance in conquering... more...
CliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. CliffsQuickReview Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry ? whether you need a supplement to your textbook... more...
Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is... more...
Classic graduate-level text discusses the Fourier series in Hilbert space, examines further properties of trigonometrical Fourier series, and concludes with a detailed look at the applications of previously outlined theorems. 1956 edition. more...... more...
Most math and science study guides are a reflection of the college professors who write them-dry, difficult, and pretentious.
The Humongous Book of Trigonometry Problems is the exception. Author Mike Kelley has taken what appears to be a typical trigonometry workbook, chock full of solved problems-more than 750!-and made notes in the margins... more...
This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola. The development of the hyperbolic functions, in addition to those of the trigonometric (circular) functions, appears in parallel columns... more... | 677.169 | 1 |
The GoodQuestions project seeks to improve college calculus instruction by adapting two methods developed in physics instruction — ConcepTests and Just-in-Time-Teaching. GoodQuestions is a pedagogical strategy that aims to raise the visibility of the key concepts and to promote a more active learning environment. Lists available of questions used in and before class (at various levels), class web pages, and other resources. | 677.169 | 1 |
Well, as the headline indicates: Which one should I purchase? It needs to be able to differentiate, integrate etc. (also functions with more than one variable) and if it's able to draw circles, ellipses, 3D-images or similar, then that would be a great plus.
I have a TI-84.
Thanks in advance for your answers.
December 2nd 2008, 06:34 PM
Scary_Joe
Well, I have had my Ti-Nspire CAS for a few months now, and here is what I've concluded.
The display on the Nspire is WAY better than on the 89, and It's a lot more organized and easier to use with Integrals, derivatives, sums, products, limits, etc. because you can type in all of those things directly into the calculator exactly how it looks on the paper, and it's easier to see equations when you're typing it into the calculator because for some things, it puts it into pretty print while you're typing it in. And, the statistics that an Nspire can do is by far the best there is, way better than an 89. Also, you can connect the calculator to the computer and install updates for it directly(I don't know if 89 or voyage 200 can do this).
But, what I don't like is mainly the graphing capabilities. It is completely different than the way the 89 and 83/84 graphs, because you type in your function below the actual graph, and it pastes what the function equation is right next to the graph also. And how it traces the graph is kind of annoying, because you as it goes along the graph, it tells you the max/mins that there are, and intercepts, etc. I liked how the 83 did it with the calc command that you go left bound and right bound, and it gives it to you. Also, I can't stand how it graphs Parametric Equations, and Polar Equations, because I would prefer how you can set up the window in another screen. I particularly liked how you could set up the tstep and θstep on an 83/84
From what I can see, the Nspire cannot graph 3D or Implicitly. I am not sure the 89 can, but I am pretty sure the Voyage 200 can.
so overall, if you dont want to read my ramblings Pros:
Display is neat
Top of the line statistics
can save documents
the ability to update the calculator
higher resolution display
I also LOVE the ability to define functions. (89 and voyage 200 can also) Cons: Graphing (it still can graph functions nicely)
more difficult to calculate max/mins/intercepts/area directly from the graph.
no 3D/Implicit graphing
Im not sure, if TI will make an update so that it can graph 3D and Implicit equations, but it IS possible. Only time will tell. | 677.169 | 1 |
self-instruction book explains angles and triangles, and demonstrates the solutions to right triangle problems. Chapters that follow deal with trigonometric functions of sine, cosine, and tangent, radian measures, Pythagorean and other trigonometric identities, graphs of trigonometric functions, waves, polar coordinates, complex numbers, conic sections, spherical trigonometry, polynomial approximation for sin x and cos x, and more. Exercises follow every chapter with answers given at the back of the bookFeatured Language Guides Spectrum Language Arts for Grade 4 includes focused practice for language arts mastery such as grammar and usage, parts of speech, sentence types, and more. Learn more | See related books
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":13.04,"ASIN":"0764142518","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":14.36,"ASIN":"1438000391","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":14.25,"ASIN":"0764139185","moqNum":1,"isPreorder":0}],"shippingId":"0764142518::%2FU33%2ByqUtTbm16iYbO9xiTcAJXoJXw0dq9hUwSXAmb2Jw38Iz7SZfM2RZGpjZx%2Fo4aG1ddPxm6jSTRSBL6XMIAZz%2FVcjkSvgLD3u3xNC7Dvo541mT97UGA%3D%3D,1438000391::j13%2BWmtmgKrywF2Pt%2BNBOcsnBIWpgcLDkAghyjAzPNqh0jhYq%2BiMlhd1RNH2xRdYeaIf8OZuRlJT30qi7AS6PMDpTC8j%2F9EQ9Vx92KZ%2Bhzlgvcbcv5XXrA%3D%3D,0764139185::pSWTn2ccDmOZ6oqVjHqT2Okw%2FabcfvindIVeMC4KDmt8dTwUscFf%2BIWJGTauFUOq%2FuCCFV2XUw22VI4ejFuYVuyZVCcG2FPpG6YiR9XUf. Explaining trigonometry easy and understandable. Not only give the facts, but explain logically why. It si also fun because it tells a simple story which makes it exiting and applicable. I definitely recomend this book. Its fun to read and makes you an expert without effort.
I purchased the E-Z Geometry book, and LOVED it. This book, however, uses a confusing story with characters to try to teach the concepts of Trigonometry. The story and characters, in my opinion, hinder the learning process. I wish this book would've used the approach used in the Geometry book.
I do not recommend this resource, unless you've used other Barron's books with this approach.
The book is good at giving formulas and an explanation of where the formula came from, but it does not include enough examples of working the problems. It is especially hard to figure out multi-step problems that build initially on the chapter concept. The answers are included in the back of the book, which is a nice feature, but I did find myself helping my teenager by working backwards from the answer to explain to her how the problem works. | 677.169 | 1 |
Jim Kouzes and Barry Posner are authors of the award-winning and best-selling books, The Leadership Challenge , Encouraging the Heart and Credibility as well as the widely-used Leadership Practices Inventory (LPI) . The launch of LPI Online, combined with their other publications, truly make them the most trusted source on becoming a better... more...
From differentiation to integration - solve problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? Have no fear! This hands-on guide focuses on helping you solve the many types of calculus problems you encounter in a focused, step-by-step manner. With... more...
An essential reference guide for Math, Science, and Engineering students. You will use it from high school all the way to graduate school and beyond. FREE Functions, Equations, and Table of Derivatives in the trial version. Algebraic formulas. Trigonometric formulas. Geometric formulas. Linear Algebra formulas. Calculus formulas. Statistical formulas.... more...
Can you resist the allure of Edward's myriad charms—his ocher eyes and tousled hair, the cadence of his speech, his chiseled alabaster skin, and his gratuitous charm? Will you hunt surreptitiously and tolerate the ceaseless deluge in Forks to evade the sun and uphold the facade ? Join Edward and Bella as you learn... more...
Want to become an accountant? Own a small business but need help balancing your books? Worried about managing your finances under the cloud of the recession? This hands-on workbook gets you up to speed with the basics of business accounting, including reading financial reports, establishing budgets, controlling cash flow, and making wise financial... more...
Hands-on practice in solving quantum physics problems Quantum Physics is the study of the behavior of matter and energy at the molecular, atomic, nuclear, and even smaller microscopic levels. Like the other titles in our For Dummies Workbook series, Quantum Physics Workbook For Dummies allows you to hone your skills at solving the difficult and... more...
English Grammar Workbook For Dummies, UK Edition is grammar First Aid for anyone wanting to perfect their English and develop the practical skills needed to write and speak correctly. Each chapter focuses on key grammatical principles, with easy-to-follow theory and examples as well as practice questions and explanations. From verbs, prepositions... more...
The most captivating way to master vocabulary for the SAT, ACT, GED, and SSAT exams Join Bella, Jacob, and Edward as you learn more than 600 vocabulary words for the SAT, ACT, GED, and SSAT! With hundreds of new vocabulary words, this book can be used completely on its own or as a follow-up to Defining Twilight and Defining New Moon . You'll... more... | 677.169 | 1 |
WildLinAlg12: Generalized dilations and eigenvectors This video introduces the important idea of changing coordinates in Linear Algebra. A linear transformation can be described using many different matrices, depending on the underlying coordinate system, or ordered basis, which is used to describe the space.
The simplest case is when the linear transformation is in diagonal form. Finding such a diagonal form requires finding the eigenvalues and eigenvectors of a matrix, which we introduce in this video. We also discuss change of basis matrices. Author(s): No creator set
License information
Related content
No related items provided in this feed
WildLinAlg10: Equations of lines and planes in 3D This video shows how we work with lines in the plane and planes in 3D space in Linear Algebra. A line in 2D is represented by a linear equation in x and y, a plane in 3D by a linear equation in x,y and z. Both can also be described in parametric form. It is important to be able to change from a Cartesian to a parametric form.
The space of all lines in the plane has a curious connection with the Mobius band. Lines in 3D are somewhat trickier to describe since they require two linear equations.
Author(s): No creator set
License information
Related content
No related items provided in this feed
Why did the General Strike not Succeed? Why did the General Strike not Succeed? Dr. Hester Barron, University of Sussex: Introduction; How far were the leaders of the TUC General Council responsible?; Government position (maintenance of...
History as written and presented by current historians. Visit thehistoryfaculty.com for free downloads and more information.
12 Glossary Additional resources Test your knowledge How things change4 When surgery is required3 Issues with medications Secondary prevention using drugs1 Interventions and conditions), this content is made available under a
No related items provided in this feed
7 Immediate treatment of cardiovascular diseases Cardiopulmonary resuscitation (CPR) Early warning signs7 Special circumstances6 A balanced diet What can individuals do.1 Measures of adiposity | 677.169 | 1 |
11Functions Modeling Change: A Preparation for Calculus
Summary
This text provides a strong foundation to precalculus that focuses on a small number of key topics thereby emphasizing depth of understanding rather than breath of coverage. It provides a solid way to motivate concepts and develop critical thinking skills. The new fourth edition emphasizes functions as models of change. It contains superior exercises and applications that motivate the concepts readers can use to fully grasp precalculus. | 677.169 | 1 |
This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks?a sort of potato-stamp method?Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling... more study of various types of abstract (multi-term) fractional... more...
Difference Equations: Theory, Applications and Advanced Topics, Third Edition provides a broad introduction to the mathematics of difference equations and some of their applications. Many worked examples illustrate how to calculate both exact and approximate solutions to special classes of difference equations. Along with adding several advanced... more...
This book provides a systematic, rigorous and self-contained treatment of positive dynamical systems. A dynamical system is positive when all relevant variables of a systemare nonnegative in a natural way. This is in biology, demography or economics, where the levels of populations or prices of goods are positive. The principle also finds... more...
This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves... more... | 677.169 | 1 |
Dawn Griffiths
Dawn Griffiths started life as a mathematician at a top UK university where she was awarded a First-Class Honours degree in Mathematics. She went on to pursue a career in software development, and has over 15 years experience working in the IT industry. Dawn has written several books, including Head First C, Head First Statistics and Head First 2D Geometry.
"If you are looking for a book that shows you fun and real-life applications of 2D geometry while explaining its concepts lucidly, then Head First 2D Geometry by Lisa Fallow and Dawn Griffiths is definitely for you. There is not one thing about this book that is dull or boring - the words most frequently associated with a traditional geometry textbook."
--Heartcrossings.blogspot.com
"I challenge anyone to find a better book for learning basic probability and statistics! ...This book is clearly a labour of love, as it is a low effort and fun way to learn probability and statistics! "
--Mitch Wheat, Amazon.com
"The study of statistics can be dry and difficult, but not with this non-intimidating, heavily illustrated, and entertaining guide. The author uses real-world examples, stories, and puzzles to engage the reader with the content equivalent to a first-year high school or college statistics course. The book is different, and, for the beginner, better than any statistics textbook."
--Michael L. Kleper, The Kleper Report on Digital Publishing
"...if you are after a tutorial it will cover not only the
topics you need but in a format that makes learning a pleasure..."
--Neil Davis, Amazon.co.uk
"Don't know the difference between the mean, the median and the average? Don't worry, hardly anybody does. But you'll learn about it here all right. "
--Bob and Joy Schwabach, On Computers
"This book has considerable more meat than a Dummies or Idiots offering while maintaining the fun and readability that have drawn so many people to that genre. "
--John Bollinger, The Capital Growth Letter
"This book covers the full range of topics dealt with in first-year statistics. If only it really *were* the textbook used, statistics would be more firmly set as a common skill -- and Wall Street would be much more circumspect in how it pleads poverty."
--Brett Merkey, Amazon.com
"...the ideal background text for high school and
college students taking statistics seriously."
--Michael Shaw, Macintosh Users East (MaUsE)
"This book will be an important tool for all students of the experimental sciences."
--Ira Laefsky, Amazon.com
"This is an educational and entertaining guide, which present the material in an engaging manner. "
--Nadia Russ, Wonderpedia | 677.169 | 1 |
expert EQUATIONS
By Daniel McElholum
Description
This app extends a student's understanding of algebra by covering the concept of equations. It contains a number of related activities and is useful for math students in levels 8 to 12. It contains a unique 'equation solving calculator'. | 677.169 | 1 |
76
Comment: ELIGIBLE FOR *FREE* SUPER SAVER SHIPPING. AMAZON CUSTOMER SERVICE WITH DELIVERY TRACKING. Book may have moderate wear to corners-- More than 44,000 copies sold of second edition-- More than 230,000 students are enrolled in trigonometry courses-- Required study for all mathematics majors-- Hundreds of practical problems solved step by step-- Complements most popular textbooks
Editorial Reviews
From the Back Cover
Get a firm grasp of trigonometry with this simple-to-use guide! It can help you pump up your problem-solving skills, ace your exams, and reduce the time you need to spend studying. Students love Schaum's Outlines! Each and every year, students purchase hundreds of thousands of the best study guides available anywhere. Students know that Schaum's delivers the goodsin faster learning curves, better test scores, and higher grades!
If you don't have a lot of time but want to excel in class, this book helps you:
Brush up before tests
Find answers fast
Study quickly and more effectively
Get the big picture without spending hours poring over lengthy texts
Schaum's Outlines give you the information teachers expect you to know in a handy and succinct formatwithout overwhelming you with unnecessary details. You get a complete overview of the subjectand no distracting minutiae. Plus, you get plenty of practice exercises to test your skill. Compatible with any classroom text, Schaum's lets you study at your own pace and reminds you of all the important facts you need to rememberfast! And Schaum's is so complete it's the perfect tool for preparing for graduate or professional exams!
Inside, you will find:
Hundreds of detailed problems, including step-by-step solutions
Hundreds of additional practice problems, with answers supplied
Clear explanations of trigonometry and the underlying algebra
Understandable coverage of all relevant topics
If you want top grades and excellent understanding of trigonometry, this powerful study tool is the best tutor you can have!
--This text refers to an out of print or unavailable edition of this title.
About the Author
McGraw-Hill authors represent the leading experts in their fields and are dedicated to improving the lives, careers, and interests of readers worldwide
--This text refers to an out of print or unavailable edition of this title.
Most Helpful Customer Reviews
I found the book very useful for brushing up one's knowledge of trigonometry.It gets down to the point right away without wasting any time on unnecessary theory.After all, its your problem solving ability that counts and not how well you know the theory.:) What I like most about this book is that it treats the subject matter from a student's point of view (like it teaches you how to get the answer using a calculator STEP by STEP.....it actually tells you which buttons to push).Also, throughout the book ,the application of trigonometry in surveying, construction,astronomy and air-navigation is emphasized. I recommend this book for anyone who's taking computer science/engineering/technology courses in college.
Trigonometry was never a good subject for me--I never "got" it. But when I was taking advanced math and science courses, I needed trig. This book helped me to "get" it, finally, and be able to solve trigonometric problems. It's very clear, up-to-date, and well-written.
I am observing that my test scores on tests on tests involving trigonometery are increasing, thanks to this thin aid. It is thin, yet it is good. That is almost impossible. This book is one of the best created. Trigonometry is hard, and is mentioned and applied almost everywhere. This book is comphrehensive and is both easy and advanced. Everyone should have this. You can have knowledge of math, science, and computers with it.
I used Schaum's for a Summer School Trigonometry course. It doesn't replace a textbook but it covered all the necessary topics effectively and provided a good alternative to derivations and problems found in the text. I relied on it as a backup and a good check on the work being covered in class.
5/13: Schaum's Outlines: Trigonometry is so good that it's actually my basis of what I know about trigonometry. Thanks to the book: my foundation of trigonometry translated to success in Calculus. Everything I know about trigonometry starts with Schaum's Outlines: Trigonometry. I solved every problem in the book, and the clouds got smaller and smaller every time I do a problem. The explanations by the authors are very clear and concise. Probably the only missing segment in the book is what happens when you see a segment of a sinusoid and are asked to derive an equation from that. Not to worry, Schaum's Outlines: Trigonometry covers about 95% of the essential materials about trigonometry. All in all, do every problem in Schaum's Outlines: Trigonometry from cover to cover, and you will be surprised how well prepared you will be for Calculus and beyond.
I purchased this book and Schaum's Precalculus for a self-study of precal over the summer, and it is NECESSARY to use the PRECALCULUS book even for a regular trig course since it covers vectors, polar and parametric EQs, and complex roots in radians. As for this book, it covers area of triangles, more identities, and more applications than the precal book, but is still not sufficient for a complete self-study. Taking the actual course, whether it be trigonometry or precalculus is necessary to be prepared for the rigor of calculus, yet for a self-study, it would be necessary to DO ALL THE PROBLEMS in this book, and purchase a separate text with more practice problems. In conclusion, this is an excellent book for review or as a supplement to a full course.
The preface claims that the book can be used by students who are studying trig for the first time. What a load of bunk. As a "first time" trig student, I couldn't get through the first chapter without getting confused. I moved on to the second chapter, and the information was just as muddled. This book may be more useful as a study guide to be used together with a real trig textbook, but I don't believe it is useful for the beginner. Kahn Academy on Youtube gave more clearer explanations and examples on the subject. Beginners should stay away from this book! | 677.169 | 1 |
We begin with function notation, a basic toolkit of functions, and the basic operation with functions: composition and transformation. Building from these basic functions, as each new family of functions is introduced we explore the important features of the function: its graph, domain and range...
Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and introductory algebra.
According to the introduction of Elementary Linear Algebra, "this is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra." | 677.169 | 1 |
You are here
MATH FOR ELEMENTARY SCHOOL TEACHERS I
First of two courses designed for prospective elementary teachers. Emphasizes the real number system, arithmetic operations, and algebra. Explorations focus on representing, analyzing, generalizing, formalizing, and communicating patterns and structures. Content is presented using problem solving and exploration. Prerequisite: ACT math score of 23 or above, SAT math score of 530 or above; MATH 140 with a minimum grade of 'C-'; or Accuplacer College-Level Mathematics test with a score of 65 or above. | 677.169 | 1 |
13.1 Additional Graphs of Functions 13.2 The Circle and the Ellipse 13.3 The Hyperbola and Functions Defined by Radicals 13.4 Nonlinear Systems of Equations 13.5 Second-Degree Inequalities and Systems of Inequalities | 677.169 | 1 |
Topology is generally considered to be one of the three linchpins of modern abstract mathematics (along with analysis and...
see more
Topology is generally considered to be one of the three linchpins of modern abstract mathematics (along with analysis and algebra). Recently, topology has an important components of applied mathematics, with many mathematicians and scientists employing concepts of topology to model and understand real-world structures and phenomena. Topology is the study of shapes:Including their properties, Deformations applied to them, Mappings between them and Configurations composed of them. Here we can learn this course with the help of provided study materialology to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Topology
Select this link to open drop down to add material Topology to your Bookmark Collection or Course ePortfolio
This module (eCouse) is designed to standalone as an instructional tool to cover the topic of Linear Equations from point and...
see more
This module (eCouse) is designed to standalone as an instructional tool to cover the topic of Linear Equations 2: Linear Equations I to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Beginning Algebra Module 2: Linear Equations I
Select this link to open drop down to add material Beginning Algebra Module 2: Linear Equations I to your Bookmark Collection or Course ePortfolio
This module (eCouse) is designed to stand alone as an instructional tool to cover the topic of Linear Equations II from point...
see more
This module (eCouse) is designed to stand alone as an instructional tool to cover the topic of Linear Equations II Variable Expresions using...
see more
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of Variable Expresions using properties of real numbers and rules of exponentiation. Learners would benefit by reviewing modules in sequence for the related course in 5: Variable Expression I to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Beginning Algebra Module 5: Variable Expression I
Select this link to open drop down to add material Beginning Algebra Module 5: Variable Expression I to your Bookmark Collection or Course ePortfolio
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of Variable Expressions....
see more
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of Variable Expressions factoring polynomials....
see more
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of factoring polynomials 7: Factoring to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Beginning Algebra Module 7: Factoring
Select this link to open drop down to add material Beginning Algebra Module 7: Factoring to your Bookmark Collection or Course ePortfolio
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of Graphing Polynomial...
see more
This module (eCourse) is designed to stand alone as an instructional tool to cover the topic of Graphing Polynomial Equations eCourse (module) is designed to stand alone as an instructional tool to cover the topic of First-Degree Equations and...
see more
This eCourse (module) is designed to stand alone as an instructional tool to cover the topic of First-Degree Equations and Inequalities in Beginning Algebra. Learners would benefit by reviewing modules in 1: First Degree Equations and Inequalities with One Variable to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Beginning Algebra Module 1: First Degree Equations and Inequalities with One Variable
Select this link to open drop down to add material Beginning Algebra Module 1: First Degree Equations and Inequalities with One Variable Welding Safety Standards and Procedures to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material WEL100 Safety for Welders: Welding Safety Standards and Procedures
Select this link to open drop down to add material WEL100 Safety for Welders: Welding Safety Standards and Procedures Intro to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material WEL100 Safety for Welders: Intro
Select this link to open drop down to add material WEL100 Safety for Welders: Intro to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
Book Description
Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
Buy A+ National Pre-apprenticeship Maths and Literacy for Hospitality book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books.
You might also like...
Finally there is a key concepts book in hospitality management available on the market! Tailored to your course structure and written with your needs in mind, as well as being international in its core (contributors from around the globe), this makes out for an excellent companion throughout your hospitality degree.
This text now includes updates to all statistics, information on job design and empowerment, updated coverage of trade unionism and a new chapter on business ethics. It matches new NVQ requirements and incorporates new material relevant to courses and learning needs.
This book celebrates what has been already achieved in moving the meetings and events industry towards a more sustainable future, and pushes the boundaries of imagination to visualize what might be possible in the years to come.
Books By Author Andrew Spencer
Current approaches to morphology, Andrew Spencer argues, are flawed. He uses intermediate types of lexical relatedness in different languages to develop a morphologically-informed model of the lexical entry. He uses this to build a model of lexical relatedness consistent with paradigm-based models. A book for all morphologists and lexicographers.
Helps learners' improve their Maths and English skills and help prepare for Level 1 and Level 2 Functional Skills exams. This title enables learners to improve their maths and English skills and real-life questions and scenarios are written with an automotive context to help learners find essential Maths and English theory understandable.
This Handbook provides a comprehensive account of current research on case and the morphological and syntactic phenomena associated with it. Scholars from all over the world provide overviews of current theoretical, typological, diachronic, and psycholinguistic research and assess cross-linguistic work on case and case-systems | 677.169 | 1 |
Developmental Math 2
The course focuses on the algebraic concepts and skills for solving equations and inequalities, applying the laws of exponents to simplify polynomials, factoring polynomial expressions and using factoring to solve equations, graphing linear equations in two variables, and performing basic operations with radical expressions.
Textbook Information
TCC uses a customized textbook. The contents are available to TCC students through their Connect links. Please access Connect through Blackboard. | 677.169 | 1 |
How to Solve Word Problems in Geometry - 00 edition
to solve--this guide offers detailed, easy-to-follow solution procedures. Emphasizes the mechanics of problem-solving. Includes worked-out problems and a 50-question self-test with answers. to solve--this guide offers detailed, easy-to-follow solution procedures. Emphasizes the mechanics of problem-solving. Includes worked-out problems and a 50-question self-test with answers. ...show less
Edition/Copyright:00 Cover: Publisher:McGraw-Hill Publishing Company | 677.169 | 1 |
8th grade pre algebra worksheets English | Español Try our Free Online Math Solver! Algebrator program helps a lot of homeschoolers get rid off their fear of math. Here are a few choice keyword phrases typed in today to access our this page: mathematics torrent type in a math problem get answer for parents rule method in college algebra adding and subtracting integers for dummies how do you find the value of x when to angles are parallel to eachother ti 83 rational expressions program ... | 677.169 | 1 |
82,"ASIN":"0486675491","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":9.9,"ASIN":"0872209547","moqNum":1,"isPreorder":0}],"shippingId":"0486675491::gMdL7A6sJ1fUfDtzo08iIQY7EAVO6D3519uEtEyKqtRP8w0z10%2FJRV4ImzAAfGQ9QTYxSWpQdgNGg6t8n%2BSLe3S7rjtpTpHaczmVyd33kxk%3D,0872209547::4qimhW%2BmfNabd9iSHEzaxGUGz7a7pdl5lK68PeKkdmFMSs6ldhWgdm0GaIZTKjYnJW4SOC9RfvgFvUYenzVvq%2FSdluTCm%2B4Atd8pQxW5cFreund's text, which is based on a course that the author taught to university students fulfilling their general education requirement, is a clearly written and carefully constructed introduction to basic discrete probability. Each topic is placed in context and is illustrated by copious examples that demonstrate both the relevance and utility of probability. The exercises at the end of each section, which are generally straightforward applications of the material covered in that section, reinforce the reader's understanding of the material. Answers are provided to the odd-numbered exercises, making the text suitable for self-study. This text is a good entry point to the study of probability. However, the scope of the text is limited. The emphasis is on how to solve problems rather than the underlying theory. Freund succeeds in making the text as widely accessible as possible, albeit at the expense of a deeper understanding of the material.
The text begins with a chapter on enumerative combinatorics that covers tree diagrams, the Multiplication Principle, factorials, permutations, combinations, and indistinguishable objects. Freund then introduces the classical, frequentist, and subjective (Bayesian) approaches to probability. He contrasts the different approaches, demonstrates how each is applied, discusses their limitations, and shows that they lead to equivalent results. In the following chapter on the mathematical expectation of an event, Freund illustrates how probability is used in making business decisions. Next, Freund puts probability on a formal footing, discussing events, sample spaces, compound events, mutually exclusive events, and probability measures.Read more ›
I've never seen a probability book with such good examples. Most books on probability give you all of the equations, but they don't really tell you how to apply them to real situations. This book has nothing but real examples. It is the book on probability that I have been looking for
This book is very good for those who have little knowledge in Probability but do manage some basic math concepts: polinomials, factorials, limits, etc. I bought it because I was looking for a mathematical course in Probability but this book is not for that, it is very simple. It is not a "definition-theorem-proof" book.
As an introduction to probability, this is an excellent book: easy to read, easy to follow, good contents, etc. The title does say introduction, so don't expect to emerge as the master of probability/statistics after reading this book. | 677.169 | 1 |
Beginning and Intermediate Algebra Activities - 2nd edition
ATYC in "Crossroads in Mathematics Standards for Introductory College Mathematics Before Calculus." Providing maximum flexibility, the activities can be used during class or in a laboratory setting to introduce, teach, or reinforce a topic."ATYC in "Crossroads in Mathematics Standards for Introductory College Mathematics Before Calculus." Providing maximum flexibility, the activities can be used during class or in a laboratory setting to introduce, teach, or reinforce a topic | 677.169 | 1 |
Search Results
This algebra lesson helps students make the connection between functions and their graphs. The model of the level of water in a bathtub is used. Students will watch the graph and a chart of the depth of the water at...
This lesson uses data from the United States Federal Statistics database to help students in understanding how to determine a mathematical model that best fits a data set. Students will plot building permit data over a...
Students in need of experience constructing and interpreting statistical graphs will find this exercise useful. The lesson uses data from past presidential elections; students will construct a variety of graphs (bar...
This algebra lesson demonstrates exponential growth and decay. The document includes three different ways in which students will retrieve data from the internet, formulate a function, perform calculations and then...
With this algebra lesson, students will gather data about different airlines, including flight delays, mishandled baggage and other consumer complaints. The material helps students learn how to effectively analyze data,... | 677.169 | 1 |
While differential equations happen all the time in calculus and chemical engineering, you MUST do all of your algebra correctly to get the right answer. I spent years becoming an algebra ninja! I have been playing golf for 26 years and took professional lessons for roughly 10 through various PGA teaching professionals, some for as long as three years at a time. | 677.169 | 1 |
Mathematical Ideas
9780321361462
ISBN:
0321361466
Pub Date: 2007 Publisher: Addison-Wesley
Summary: One of the biggest issues college math instructors face is capturing and keeping student interest. Over the years, John Hornsby has refined a creative solution--bringing the best of Hollywood into his mathematics classroom. Mathematical Ideas applies this same strategy of engaging students through descriptions of video clips from popular cinema and television to the textbook. Alongside fresh data and tools, this Elev...enth Edition uses up-to-the-minute images as well as old favorites of math being done in Hollywood. In addition, examples are clarified with additional annotations, the margin notes have been freshened, and Chapter 14: Personal Financial Management has been updated to meet the needs of today's students. With great care and effort, Vern Heeren and John Hornsby have crafted this new edition to serve the needs of today's students and instructors.
Miller, Charles David is the author of Mathematical Ideas, published 2007 under ISBN 9780321361462 and 0321361466. Seventy three Mathematical Ideas textbooks are available for sale on ValoreBooks.com, sixty nine used from the cheapest price of $3.02, or buy new starting at $26Jackson, NJShipping:StandardComments:ALTERNATE EDITION: New 11th Edition and Expanded 11th Edition. Never Used. May have slight Shelfwear but MINT Cond... [more]) [ New 11th Edition and Expanded 11th Edition. Never Used. May have slight Shelfwear but MINT Condition Inside. This is an Instructors edition and May Conatin Publisher Notes [more]
)[less] | 677.169 | 1 |
The Exponentiations Applet allows users to input complex numbers in either cartesian or exponential form and display them in vector form. The applet also shows the results of applying powers and roots to complex numbers. The applet, constructed using GeoGebra, requires Java 1.4.2 or later. | 677.169 | 1 |
1934968390","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.95,"ASIN":"1933241586","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.95,"ASIN":"1933241578","moqNum":1,"isPreorder":0}],"shippingId":"1934968390::i%2BcrsAB2oBjuy1xjB9l6Gf4GPauGxqeJHYZ9cdsmHG3P7z7t95GvgYqd0T7g9d1c%2F089H4rHIZ%2B%2FCBJLZKne6ShRS2G0%2BS4Evtyz29hLqRg%3D,1933241586::VUPi%2B9rIZ7g9zdp0CMjUvNGs9rmGzMYvcbWMNtJN3q%2FQ2kjuCR667wLy2QztLJlUqWcLEG5l4vYHPUwbZNpIlYvgIbX9kjfdD9CxZ7zbuqw%3D,1933241578::VUPi%2B9rIZ7g9zdp0CMjUvDu4Cx5E47i65zyBLHYD9ntLbLI8kJupO2iUwShP78yMIgez9%2FufdMwOGYILO1Jj3VlD87BfEPne3RP%2FUUIvd a word problems workbook, I would have expected problems that require more concept-level thinking (what operation is needed here?). The problems provide plenty of repeat practice for basic math functions, but there's not much mix of types of problems in each lesson, so the student isn't really required to do much critical thinking in figuring out what approach to use in solving the problem. Once the student solves the first problem or two, the rest of the problems are pretty predictable. I would prefer to have quicker math functions involved if it meant my daughter had to spend more time critically evaluating the path to the solution.
Will write a second review later; but, just wanted to let parents/readers know that I browsed through the book and it looks great. My daughter did 2 pages of it and the questions in books are already picking her brain!!! It is less boring (better than plain old additions/substractions/multiplication/division exercise rut)! She loves to check on the net to see how many meters are in a kilometer to solve the problems in the book. So far so good; keeping my fingers crossed.
I got this to help my daughter keep up on her math during the summer. I have used the Kumon workbooks since my daughter was in Kindergarten. She is a fifth grader now and I think these books are a great helper in addition to her school studies.
i bought this series of books last year for my fourth grader.Though the way of explaining was good. The problems were too easy for fourth grader and they don't have to use any critical thinking to solve the problems. So my main mission was to teach her some tough problems or develop her brain in competitive way but she was like, its too easy and repetitive.
I love the kumon books great work and I recommend it to kids who need practice my son loves the little pictures but its mostly little examples and worksheets I bought almost all they offer in math and love all of them
If your kid has been to Kumon, you know what is involved - yes, the daily grind for the kid and the parents. If you like that grind (each of the Kumon franchise will tell you how essential it is) then of course you do not have to buy this book.
However, for rest of us, who questions that grind, this is an easier way to get the juice of the course but without the endless repetitions - at a fraction of the cost. The problems are not same as what a Kumon franchise offers. But that really did not matter. All problems are very nicely created. The difficulty level progresses through the course. And you can always go back to a previous level if you need to. I suggest you do not let your kid to mark the book to make this last part easier.
The price of the book is right. However, I wish the book should have been thicker with more problems - of course a 80 page book cannot be sufficient for the entire year even if the price goes up a bit. | 677.169 | 1 |
24.29
Slay the calculus monster with this user-friendly guide Calculus For Dummies, 2nd Edition makes calculus manageable-even if you're one of the many students who sweat at the thought of it. By breaking down differentiation...
$ 12.99
Master pre-calculus from the comfort of home! Want to "know it ALL" when it comes to pre-calculus? This book gives you the expert, one-on-one instruction you need, whether you're new to pre-calculus or you're...
$ 19.49
For students who need to polish their calculus skills for class or for a critical exam, this no-nonsense practical guide provides concise summaries, clear model examples, and plenty of practice, practice, practice....
$ 15.29
Take the FEAR OUT of Business Calculus Business Calculus Demystified clarifies the concepts and processes of calculus and demonstrates their applications to the workplace. Best-selling math author Rhonda Huettenmueller...
$ 15.29
In its largest aspect, the calculus functions as a celestial measuring tape, able to order the infinite expanse of the universe. Time and space are given names, points, and limits; seemingly intractable problems...
$ 24.29
This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with...
$ 9.29
The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these...
$ 14.99
Self-contained and suitable for undergraduate students, this text offers a working knowledge of calculus and statistics. It assumes only a familiarity with basic analytic geometry, presenting a coordinated study...
This early work on calculus is both expensive and hard to find in its first edition. It details the mathematical techniques of successive differences, relative growing, curvature of curves, and includes numerous...
$ 8.79
Highly readable, self-contained text provides clear explanations for students at all levels of mathematical proficiency. Over 1,600 problems, many with detailed answers. Corrected 1969 edition. Includes 394...
$ 25.99
This classic treatise on the calculus of finite differences offers a thorough discussion of the basic principles of the subject, covering nearly all the major theorems and methods. Over 200 problems. 1872 edition....
$ 12.29
Comprehensive but concise, this workbook is less rigorous than most calculus texts. Topics include functions, derivatives, differentiation of algebraic functions, partial differentiation, indeterminate forms,...
$ 12.29
Ideal for self-instruction as well as for classroom use, this text improves understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. Over 1,200 problems, with hints and complete... | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
SAT II Mathematics Level IC Test score of 480 or a grade of at least a C- in MATH 1314 or an
equivalent course.
Course Description
Functions and their graphs and domains, polynomial and rational functions, exponential and
logarithmic functions, trigonometric functions, analytic geometry, and linear systems of equations
will be covered.
Student Learning Objectives/Outcomes
1. Students will evaluate functions, determine their domains, and be able to find the inverse
function if one exists.
2. Students will perform algebraic operations with polynomial and rational functions, and
determine the domains and asymptotes of rational functions.
3. Students will evaluate and recognize exponential and logarithmic functions, and use their
properties to solve exponential and logarithmic equations.
4. Students will evaluate trigonometric functions, use fundamental trigonometric identities, and
evaluate inverse trigonometric functions.
5.Students will solve systems of linear equations.
Required Textbooks and Materials
Precalculus, 7th Ed., Larson & Hostetler
Students Solutions Manual is recommended.
Course Syllabus
Page 1
Assignments & Academic Calendar
There will be three midterm examinations and a comprehensive final examination.
Midterm I
Feb. 6
Midterm II
Mar. 4
Midterm III
Apr. 6
Final Exam
2:00 Monday, May 11
To succeed in this course one has to do a large number of problems.
Here is a list of selected problems which all the students should do. The answers
to these problems are in the book and solutions are in the Solution Manual except for a very few
even numbered problems for which solutions will be provided in class. In addition to the
problems from the text listed below, some additional homework problems will be given in class.
The lowest midterm grade will be dropped. Note that the final examination is comprehensive.
Course & Instructor Policies
There will be no make-ups except in extraordinary circumstances.
Field Trip Policies
Off-campus Instruction and Course Activities
Off-campus, out-of-state, and foreign instruction and activities are subject to state law andUniversity policies and procedures regarding travel and risk-related activities.Informationregarding these rules and regulations may be found at the website address
information is
available from the office of the school dean.Below is a description of any travel and/or risk-
related activity associated with this course.
Student Conduct & Discipline
The University of Texas System and The University of Texas at Dallas have rules and regulationsfor the orderly and efficient conduct of their business.It is the responsibility of each student andeach student organization to be knowledgeable about the rules and regulations which govern | 677.169 | 1 |
Everything that we do with computers, from Word, to the internet, every page of the internet, games, etc. Everything on a computer or even a handheld device (cell phone, etc.) has a program on it. That program was created by a computer programmer, writing in a programming language like C++, Java, Pascal, Javascript, Perl, etc.
...I try to take the approach of first gauging where the student is at in terms of knowledge and understanding and then building on that foundation to get to the point where he or she needs to be. I have always held myself to a higher standard than most, and the word complacency is not in my vocabulary. That being said, I would very much appreciate any feedback that I can get.
...As noted above, trigonometry is usually encountered as a part of a pre-calculus course. In my view, much of the traditional material associated with trigonometry should be replaced by an introduction to the linear algebra of vectors, which provides alternative methods of solving many of the prob... | 677.169 | 1 |
This book provides explanatory text, illustrative mathematics and algorithms, demonstrations of the iterative process, pseudocode, and well-developed examples for (familiar as well as novel) applications of the branch-and-bound paradigm to relevant problems in combinatorial data analysis. more...
A collection of the various old and new results, centered around the following simple observation of J L Walsh. This book is particularly useful for researchers in approximation and interpolation theory. more...
Mathematics for Dyslexics: Including Dyscalculia, 3rd Edition discusses the factors that contribute to the potential difficulties many dyslexic learners may have with mathematics, and suggests ways of addressing these difficulties. The first chapters consider the theoretical background. The later chapters look at practical methods, which may help dyslexic... more...
Veteran educators share proven solutions to guide a new secondary math teacher through the challenging first few months and provide the more experienced teacher with interesting alternatives to familiar methods. more...
Shows how well-meant teaching strategies and approaches can in practice exacerbate underachievement in maths by making inappropriate demands on learners. As well as criticizing some of the teaching and grouping practices that are considered normal in many schools, this book also offers an alternative view of attainment and capability. more...
Tough Test Questions? Missed Lectures? Not Rnough | 677.169 | 1 |
Numerical Methods for EngineersInstructors loveNumerical Methods for Engineersbecause it makes teaching easy! Students love it because it is written for them--with clear explanations and examples throughout. The text features a broad array of applications that span all engineering disciplines. ..The sixth edition retains the successful instructional techniques of earlier editions. Chapra and Canale's unique approach opens each part of the text with sections called Motivation, Mathematical Background, and Orientation. This prepares the student for upcoming problems in a motivating and engaging manner. Each part closes with an Epilogue containing Trade-Offs, Important Relationships and Formulas, and Advanced Methods and Additional References. Much more than a summary, the Epilogue deepens understanding of what has been learned and provides a peek into more advanced methods. Helpful separate Appendices. "Getting Started with MATLAB" abd "Getting Started with Mathcad" which make excellent references...Numerous new or revised problems drawn from actual engineering practice, many of which are based on exciting new areas such as bioengineering. The expanded breadth of engineering disciplines covered is especially evident in the problems, which now cover such areas as biotechnology and biomedical engineering. Excellent new examples and case studies span asll areas of engineering disciplines; the students using this text will be able to apply their new skills to their chosen field...Users will find use of software packages, specifically MATLAB�, Excel� with VBA and Mathcad�. This includes material on developing MATLAB� m-files and VBA macros. . . . | 677.169 | 1 |
This site can be used equally well for demonstration materials in lectures and for tutorial purposes. Topics range from basic...
see more
This site can be used equally well for demonstration materials in lectures and for tutorial purposes. Topics range from basic algebra through first-year calculus. Solved exercises throughout make the site especially appropriate for individual study. Interactive elements are added using LiveMath and Macromedia Flash Mathematics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Interactive Mathematics
Select this link to open drop down to add material Interactive Mathematics to your Bookmark Collection or Course ePortfolio
Creative Geometry is a set of web pages designed by a geometry teacher and written for both geometry teachers and geometry...
see more
Creative Geometry is a set of web pages designed by a geometry teacher and written for both geometry teachers and geometry students. In these web pages, teachers and students will find creative and interesting "hands-on" projects for most topics in the geometry curriculum. Each project is designed to help students understand, remember, and find value in the concepts of geometry Geometry to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Creative Geometry
Select this link to open drop down to add material Creative Geometry to your Bookmark Collection or Course ePortfolio
A collection of puzzles whose answers involve the Fibonacci numbers. The puzzles are of two types: easier and harder; the...
see more
A collection of puzzles whose answers involve the Fibonacci numbers. The puzzles are of two types: easier and harder; the goal is to explain WHY the puzzles have their respective answers. A link to the page containing the harder puzzles is provided. This site is a sub-page of the larger site, Fibonacci Numbers and the Golden Section Easier Fibonacci Puzzles to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Easier Fibonacci Puzzles
Select this link to open drop down to add material Easier Fibonacci Puzzlesework and Study Help to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Homework and Study Help
Select this link to open drop down to add material Homework and Study Help to your Bookmark Collection or Course ePortfolio
This site provides a comprehensive record of M.C. Escher's life, including links to approximately twenty images of his works...
see more
This site provides a comprehensive record of M.C. Escher's life, including links to approximately twenty images of his works of art as well as to a number of additional resources. ExcellentC. Escher to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material M.C. Escher
Select this link to open drop down to add material M.C. Escher History of Pi to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material A History of Pi
Select this link to open drop down to add material A History of Pi to your Bookmark Collection or Course ePortfolio
This site contains reference material in Matrix Algebra. Topics covered include matrix operations, linear equations,...
see more
This site contains reference material in Matrix Algebra. Topics covered include matrix operations, linear equations, determinants, eigenvectors and eigenvalues. S.O.S. Mathematics--Matrix Algebra is a part of an independent, commercial site that offers straightforward technical assistance primarily to high school andMatrix Algebra to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material S.O.S. Mathematics--Matrix Algebra
Select this link to open drop down to add material S.O.S. Mathematics--Matrix Algebra to your Bookmark Collection or Course ePortfolio
Decision-Making is central to human activity. Thus, we are all decision-makers. However, a "good" decision making starts with...
see more
Decision-Making is central to human activity. Thus, we are all decision-makers. However, a "good" decision making starts with a consecutive-focused-thinking process that encompasses many disciplines of study. This site offers information on applied management science and an introduction to general operations research. It describes deterministic; and probabilistic models; lists books on operations research and management science; and links to related Management Science to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Applied Management Science
Select this link to open drop down to add material Applied Management Science to your Bookmark Collection or Course ePortfolio
This is a complete set of videos for my on-line differential equations class. All standard topics are addressed: modeling,...
see more
This is a complete set of videos for my on-line differential equations class. All standard topics are addressed: modeling, first order equations, constant coefficient linear equations, Euler and Runge Kutta methods, Laplace transforms, partial differential equations, series solutions and first order Videos to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Differential Equations Videos
Select this link to open drop down to add material Differential Equations Videos to your Bookmark Collection or Course ePortfolio
This webpage uses the criminal trials in the US Justice system to illustrate hypothesis testing. An applet allows the user...
see more
This webpage uses the criminal trials in the US Justice system to illustrate hypothesis testing. An applet allows the user to examine the probability of a Type I and Type II errors under various conditions. An applet allows users to visualize p-values and the power of a test I and Type II Errors - Making Mistakes in the Justice System to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Type I and Type II Errors - Making Mistakes in the Justice System
Select this link to open drop down to add material Type I and Type II Errors - Making Mistakes in the Justice System to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
A Survey of Mathematics with Applications (9th Edition)
9780321759665
ISBN:
0321759664
Edition: 9 Pub Date: 2012 Publisher: Pearson
Summary: This textbook serves as a broad introduction to students who are looking for an overview of mathematics. It is designed in such a way that students will actually find the text accessible and be able to easily understand and most importantly enjoy the subject matter. Students will learn what purpose math has in our lives and how it affects how we live and how we relate to it. It is not heavy on pure math; its purpose ...is as an overview of mathematics that will enlighten students without an intense background in math. If you want to obtain this and other cheap math textbooks we have many available to buy or rent in great condition online.
Allen R. Angel is the author of A Survey of Mathematics with Applications (9th Edition), published 2012 under ISBN 9780321759665 and 0321759664. Five hundred fourteen A Survey of Mathematics with Applications (9th Edition) textbooks are available for sale on ValoreBooks.com, two hundred thirty used from the cheapest price of $93.47, or buy new starting at $140 shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
To me, the least helpful part was the first few pages of each section. It was almost as if trying too hard to explain something which would make easy things seem complicated. I deffinatly the practice questions were great practice for preparing for tests/quizzes and having some answers in the back of the book were helpful in checking myself.
The primary subject of this book was general college math and it was deffinatly effective. There is a lot of information in here that was very simple for me to understand and then there were other lessons that were much more complicated and requiread a lot more practice and hair pulling for me.
Honestly there is nothing I would change about this book. It worked great for the class I took. It explained the lesson in the beginning of the chapter then gave problems pertaining to the lesson. It was very helpful. Having the odd number answers in the back of the book helped a lot especially when you couldn't figure out how to do a problem. I would take the answer and answer the problem until I got that answer.
I learned how to do numerous things in this book, such as writing expressions, solving for X, rise over run and other problems. This book helped me a lot in my math skills. I haven't had math in a couple years, and this book helped refresh my math skills. Thanks to this book and the teacher I passed my HESI test for nursing with flying colors.
Adding because it help me learn the value of money and putting he decimals in the right place.It help me focus on adding nonstandard getting the right change back if you buy something in the near future.
I learn how to add, multiply, subtracting, and division in many ways the purpose of that is to help me build focus in this skill.Math is my favorite subject in this world I love it because without it we would not have paychecks ,banks or even stores who accept money. | 677.169 | 1 |
DescriptionApplication designed for beginners to learn the basics of trigonometry. This explains completely its mathematical and geometrical interpretation and physical significances. Deals all about trigonometric formulas and identities. Very helpful tutorials for students to solve mathematical problems under the category of heights and distances…try out now…
Solve equations and system of equations Find derivatives, definite and indefinite integral Support for trigonometric inverse-trigonometric and hyperbolic functions
No network access required! Works offline It has a offline Catalog with details of each function with example. Note: Free version contains ads! In-app upgrade to PRO version to make it ad-free.
* Enter multiple comma separated functions to plot them. * Press back button to make graph full screen by hiding keyboard * Press on any example in Catalog to copy it to Solver tab * Trigonometric functions default input is in radian i.e to calculate Sin(90Degree) to calculate Sine of 90 degree.
Note- 1:- It doesn't solve word problems. Note- 2:- It doesn't show steps to solve a problem. It solve shows final answer. | 677.169 | 1 |
This book broadens AP Calculus with new applications of volume methods. The book develops a thorough introduction to Multivariable Calculus, helps students to visualize three-dimensional surfaces, and provides meaningful curriculum for those weeks following the AP Exam | 677.169 | 1 |
Riemann Surfaces : A Primer textbook, aimed at advanced undergraduate or beginning graduate students in mathematics, introduces both the theory of Riemann surfaces, and of analytic functions between Riemann surfaces. The first half of the book describes the basic theory, the second half develops the theory of harmonic and subharmonic functions on a Riemann surface, and culminates with a detailed proof of the famous Uniformisation Theorem and some of its applications to Riemann surface theory. The book is a major revision of the author's earlier 'Primer', with new chapters and more exercises and examples. | 677.169 | 1 |
Elementary Geometry
9780471510024
ISBN:
0471510025
Edition: 3 Publisher: John Wiley & Sons Inc
Summary: Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a; expanded coverage of analytic ...geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms;Gustafson, R. David is the author of Elementary Geometry, published under ISBN 9780471510024 and 0471510025. Three hundred twenty three Elementary Geometry textbooks are available for sale on ValoreBooks.com, sixty eight used from the cheapest price of $101.50, or buy new starting at $192 | 677.169 | 1 |
A 4-6 week unit for use with college-bound high school students, combining the introduction of chemistry with a methodical method of problem solving and a review of the mathematics needed for high school chemistry. It...
The Science and Mathematics Initiative for Learning Enhancement, or SMILE program, is a project of the Illinois Institute of Technology (IIT) Center and is funded by a grant from the Lucent Technologies Foundation. The...
This online tutorial is intended for college students taking an early course in mathematical optimization or linear differential equations. Although it is written by a professor of economics, little economic theory is...
A unit that addresses the sheer volume of incomprehensible numbers (speed, distance, age) in the natural world, helping students to understand the scale of the world using the concepts of rates, proportions and...
Looking back to the late nineteenth century, one can find traces of the earliest distance education learning programs at the university level at places like the University of Chicago and Columbia University. It would... | 677.169 | 1 |
The sitcom, "The Simpsons" "contains over a hundred instances of mathematics ranging from arithmetic to geometry to calculus,...
see more
The sitcom, "The Simpsons" "contains over a hundred instances of mathematics ranging from arithmetic to geometry to calculus, many designed to expose and poke fun at innumeracy." This site offers several "ways to introduce important concepts to students, and to reduce math anxiety and motivate students in courses for non-majors Simpsons Math to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material The Simpsons Math
Select this link to open drop down to add material The Simpsons MathVisually searchable database of algebra 1 videos. Click on a problem to see the solution worked out on YouTube. The...
see more
Visually searchable database of algebra 1 videos. Click on a problem to see the solution worked out on YouTube. The solutions are meant to accompany the free and open textbook Elementary Videos to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra 1 Videos
Select this link to open drop down to add material Algebra 1 Videos to your Bookmark Collection or Course ePortfolio
Visually searchable collection of algebra 2 videos. Click on a problem to see the solution worked out on YouTube. These...
see more
Visually searchable collection of algebra 2 videos. Click on a problem to see the solution worked out on YouTube. These videos are meant to accompany the free and open textbook Intermediate 2 Videos to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra 2 Videos
Select this link to open drop down to add material Algebra 2 Videos theV3) Configuration to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material (V3) Configuration
Select this link to open drop down to add material (V3) Configuration to your Bookmark Collection or Course ePortfolio
This video was recorded at Is all about Math. Explain the Method of Mathematical Induction. Francesco Maurolico, Pascal and...
see more
This video was recorded at Is all about Math. Explain the Method of Mathematical Induction. Francesco Maurolico, Pascal and John Wallis. Applying the method of Induction to prove the sum of odd numbers is a square,2,3, .. infinity. Mathematical Induction to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material 1,2,3, .. infinity. Mathematical Induction
Select this link to open drop down to add material 1,2,3, .. infinity. Mathematical Induction to your Bookmark Collection or Course ePortfolio
This video was recorded at AAAI 2009: AI Video Competition. This research deals with motion planning and control for...
see more
This video was recorded at AAAI 2009: AI Video Competition. This research deals with motion planning and control for Deformable Linear Objects (DLOs). It is still a complex task to get a robot manipulate a rope or cloth. To realise this vision of getting a robot to handle dexterous objects, we have taken the simplest object i.e. a DLO for our study purpose. The operations performed on the DLO are knot-(un)tying. The DLO is thus parameterised as a Knot and we make the DLO (un)tied into various knot types. The mathematical branch of Knot Theory is extensively used here to realise the Knots. The configuration space that the Knot can move is computed by the Knot Energy. We use the Minimum Distance Knot energy here. With this, we create a hierarchical graph structure with nodes corresponding to optimal knot configurations obtained by optimising this Knot energy functional. Thus by navigating this graph, we are able to (un)tie various knots. The study looks into 3 simple and 2 complex knot types. Motion control while (un)tying is also brought about using the SARSA(λ) [Reinforcement Learning] algorithm. The motion planner is resilient to perturbations as well. Thus by devising Knot Energy together with SARSA(λ), we have built a multi-scale, reactive knot (un)tying motion planner. Results show that our method is incredibly faster than normal Probabilistic and feedback control methods. For more, please refer to my MSc thesis titled 'Multi-scale, Reactive Motion Planning with Deformable Linear Objects' at or for the complete version including Motion control at thesis - SARSA and Knot energy version.pdf 7. Motion Synthesis and Control Learning for (Un)Knotting Deformable Linear Objects to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material 7. Motion Synthesis and Control Learning for (Un)Knotting Deformable Linear Objects
Select this link to open drop down to add material 7. Motion Synthesis and Control Learning for (Un)Knotting Deformable Linear ObjectsThis video was recorded at Video Journal of Machine Learning Abstracts - Volume 2. We consider the problem of estimating...
see more
This video was recorded at Video Journal of Machine Learning Abstracts - Volume 2. We consider the problem of estimating neural spikes from extracellular voltage recordings. Most current methods are based on clustering, which requires substantial human supervision and produces systematic errors by failing to properly handle temporally overlapping spikes. We formulate the problem as one of statistical inference, in which the recorded voltage is a noisy sum of the spike trains of each neuron convolved with its associated spike waveform. Joint maximum-a-posteriori (MAP) estimation of the waveforms and spikes is then a blind deconvolution problem in which the coefficients are sparse. We develop a block-coordinate descent method for approximating the MAP solution. We validate our method on data simulated according to the generative model, as well as on real data for which ground truth is available via simultaneous intracellular recordings. In both cases, our method substantially reduces the number of missed spikes and false positives when compared to a standard clustering algorithm, primarily by recovering temporally overlapping spikes. The method offers a fully automated alternative to clustering methods that is less susceptible to systematic errors blind deconvolution method for neural spike identification to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material A blind deconvolution method for neural spike identification
Select this link to open drop down to add material A blind deconvolution method for neural spike identification to your Bookmark Collection or Course ePortfolio | 677.169 | 1 |
Beginning Algebra 5th edition
0072316934
9780072316933
Details about Beginning Algebra:
This interactive CD-ROM is a self-paced tutorial specifically linked to the text and reinforces topics through unlimited opportunities to review concepts and practice problem solving. The CD-ROM contains chapter-and section-specific tutorials, multiiple choice questions with feedback, as well as algorithmically generated questions. It requires virtually no computer training on the part of students and supports IBM and Macintosh computers. In addition, a number of other technology and Web-based ancillaries are under development; they will suppot the ever-changing technology needs in developmental mathematics.
Back to top
Rent Beginning Algebra 5th edition today, or search our site for James textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by McGraw-Hill Companies, The. | 677.169 | 1 |
Through previous editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. Now, ADVANCED ENGINEERING MATHEMATICS features revised examples and problems as well as newly added content that has been fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts and problem sets. In this new edition, computational assistance in the form of a self contained Maple Primer has been included to encourage students to make use of such computational tools. The content has been reorganized into six parts and covers a wide spectrum of topics including Ordinary Differential...
Less
Free Delivery Worldwide : Advanced Modern Engineering Mathematics : Paperback : Pearson Education Limited : 9780273719236 : 0273719238 : 13 Jan 2011 : Building...
Free Delivery Worldwide : Advanced Engineering Mathematics : Paperback : Palgrave MacMillan : 9780230275485 : 0230275486 : 17 May 2011 : This bestselling textbook is a comprehensive course for undergraduates in engineering and science from second year level onwards. Its highly successful technique- oriented approach guides the student through the development of each topic. There are hundreds of worked examples and exercises. New material has been added throughout this edition.
Free Delivery Worldwide : Advanced Engineering Mathematics : Paperback : John Wiley and Sons Ltd : 9780470646137 : 0470646136 : 03 Nov 2011 : The tenth edition of this bestselling text includes examples in more detail and more applied exercises; both changes are aimed at making the material more relevant and accessible to readers. Kreyszig introduces engineers and computer scientists to advanced math topics as they relate to practical problems.
The eight-board Mathematics Domain-Specific Vocabulary Words Mini Bulletin Board Set features 48 mathematics Includes domain-specific vocabulary words address the Mathematics Common Core State Standards. Includes resource guide with activities, display tips, and additional vocabulary words
Free Delivery Worldwide : Basic Engineering Mathematics : Paperback : Taylor & Francis Ltd : 9780415662789 : 0415662788 : 30 Apr 2014 : John Bird's approach to mathematics, based on numerous worked examples and interactive problems, is ideal for vocational students who require an entry-level textbook.Theory is kept to a minimum, with the emphasis firmly placed on problem-solving skills, making this a thoroughly practical introduction to the basic mathematics engineering that students need to master. The extensive and thorough topic coverage makes this an ideal introductory textbook for vocational engineering courses, including the BTEC National Specifications. Now in its sixth edition, Basic Engineering Mathematics has helped thousands of students to succeed in their exams. The new...
Less
Free Delivery Worldwide : Engineering Mathematics : Paperback : Palgrave MacMillan : 9781137031204 : 1137031204 : 01 Apr 2013 : Engineering Mathematics is the bestselling book of its kind with over half a million copies worldwide. Its unique programmed approach takes you through the mathematics with a wealth of worked examples and exercises. The online Personal Tutor guides you through hundreds of practice questions with instant feedback.
Free Delivery Worldwide : Modern Engineering Mathematics with Global Student Access Card : Mixed media product : Pearson Education Limited : 9780273734130 : 027373413X : 10 Dec 2010 : Suitable for a first year course in the subject, this book is an introduction to the field of Engineering Mathematics. The book is accompanied by online bridging chapters - refresher units in core subjects to bring students up to speed with what they'll need to know before taking the engineering mathematics | 677.169 | 1 |
conc
rete terms and not abstractly. This text helps students learn mathematics better by moving from the concrete to the abstract. It makes use of multiple representations (verbal, graphical, numerical, and symbolic), applications, visualization, and technology.
concrete terms and not abstractly. This text helps students learn mathematics better by moving from the concrete to the abstract. It makes use of multiple representations (verbal, graphical, numerical, and symbolic), applications, visualization, and technology. ...show less
Numbers, Variables, and Algebraic Expressions. Fractions. Exponents and Order of Operations. Real Numbers and the Number Line. Addition and Subtraction of Real Numbers. Multiplication and Division of Real Numbers. Properties of Real Numbers. Simplifying and Writing Algebraic Expressions | 677.169 | 1 |
S. S. M. Precalculus
9780495382874
ISBN:
0495382876
Edition: 11 Pub Date: 2007 Publisher: Cengage Learning
Summary: Check your work-and your understanding-with this manual, which provides solutions for all of the odd-numbered exercises in the text. You will also find strategies for solving additional exercises and many helpful hints and warnings.
Cole, Matt is the author of S. S. M. Precalculus, published 2007 under ISBN 9780495382874 and 0495382876. One hundred fifteen S. S. M. Precalculus textbooks are available for sal...e on ValoreBooks.com, fifteen used from the cheapest price of $2.11, or buy new starting at $94.46 | 677.169 | 1 |
Interactive teaching methods help students master tricky calculus
Date:
June 5, 2014
Source:
University of British Columbia
Summary:
The key to helping students learn complicated math is to understand how to apply it to new ideas and make learning more interactive, according to a new study. Pre-class assignments, small group discussions and clicker quizzes improve students' ability to grasp tricky first-year calculus concepts.
Share:
Total shares:
FULL STORY
Pre-class assignments, small group discussions and clicker quizzes improve students' ability to grasp tricky first-year calculus concepts, according to a new study by UBC researchers.
Students taught in such active-engagement classes were 10 per cent more likely to understand key concepts on subsequent quizzes, according to the study published in The International Journal on Mathematics Education. This was true even when compared to students in classes already incorporating modest levels of clicker use and interactive discussion. They were also better able to apply their knowledge to new ideas.
"With the right support, you don't need a great deal of instructional experience to introduce the techniques," said UBC mathematician and educational strategist Warren Code, lead author of the paper.
As part of UBC's ongoing efforts to improve undergraduate teaching and learning, Code and colleagues selected two especially difficult topics covered in large first-year calculus classes, and designed week-long 'teaching interventions' to more actively engage students. They then measured the impact on student comprehension of the tricky topics using quizzes and mid-term exams.
The study compared the performance of two sections, a total of 350 students. The demographics, attitudes and math background of both sections were similar. Each student was only exposed to enhanced active teaching methods for one of the two topics.
"You can't replicate perfect lab conditions in the classroom," says Code. "But we designed the observations so students acted as their own control, and each section outperformed the other on the topic for which it received the intervention. So to the degree possible, we're comparing apples to apples."
University of British Columbia. "Interactive teaching methods help students master tricky calculus." ScienceDaily. ScienceDaily, 5 June 2014. <
University of British Columbia. (2014, June 5). Interactive teaching methods help students master tricky calculus. ScienceDaily. Retrieved July 5, 2015 from
University of British Columbia. "Interactive teaching methods help students master tricky calculus | 677.169 | 1 |
MATH 11012: INTUITIVE CALCULUS
SOME SUGGESTIONS BY NANCY SMITH (FORMER COORDINATOR)
It is important that you cover the entire departmental syllabus. You'll need to make a
day-by-day schedule for yourself in order to keep up with this departmental syllabus.
If you're teaching a 2-day/week class you may want to remind students (and
yourself?) that each class meeting is 11/2 regular classes (meaning 3 hours homework
for the next class is not unreasonable).
COMMENTS: Your students will have poor algebra skills. Do not take class time to
review algebra skills, but you might:
give review sheets, or
assign problems from Chapter 1.
Do explain algebra as you work calculus. Stress 'why' not 'proof'. This is "Intuitive" or
experience-based. Remember to teach by example.
Notation: It is important throughout the course to stress notation (standard notation not
student inventions!). Help them to understand the meaning and value of notation and
how to use it correctly. I try to stress writing the solution not just finding the answer.
Chapter 2. You might want to briefly review (x + 1)3 = . . . , fractions, point-slope line
form, supply, demand, and split functions as they come up.
Try to help students separate notions of limit, continuity, and differentiability. Try
to help students make the connection between geometric, algebraic and numeric
explanations. Draw lots of pictures and graphs. Work lots of problems. The students
have a hard time with the notion of function. I use: "X is 'where' and Y is 'how tall.'" This
text does not do limits as x approaches until chapter 6. You might consider introducing
this idea here, enabling you to relate horizontal asymptotes and limits in chapters 3 and
4.
Chapter 2. Section 1. There is a nice (for both traditional and calculator sections)
calculator example showing a hole in a function on page 85, Graphing Calculator
Exploration. Because the hole in the example occurs at an integer one can use
ZINTEGER or ZDECIMAL on the TI-82 or 83 (TI-81: Zoom: INTEGER).
Chapter 3. Sections 1-2 (graphing). Teach by example. I attempt to quickly cover the
complete graph. Make sure students can graph polynomials, then take on corners or
rational functions. To help students get organized, make a list of steps and always follow
this list.
Note: Using the graphing calculator does not seem to make this topic significantly easier
for students. The object is not to create the perfect graph, but to relate the derivative
and the graph. I tend to take off extra points if a student has the correct graph, but
incorrect intervals of increasing/decreasing for example. Probably the most useful
feature of the graphing calculator is the table (TI-82&83) for functional values.
Chapter 3, Sections 3-5 (optimization). Our students have a very hard time with story
problems. Many tell me they can't do them or they never have done them. I tell them
now they will learn! Try to help them to break down the story phrase by phrase.
Sometimes I have them price each piece: cost of front fence, cost of side fence, etc. To
help them get organized give the steps and do every problem following those steps. I
have students write their plan: To Max Area: A' = 0
Be sure to do economic problems, remembering that most of our students are business
majors. If you've extra time Chapter 3 can use it.
Chapter 4 (logs & exponents). These students have studied logarithms and exponents,
they just forgot! Don't get bogged down here. Review quickly. They need to know the
derivative of the natural logarithm and exponential functions. I encourage my students
to use their calculator and to give decimal solutions so they have some feel for size of
their answer. (I require a calculator in my traditional classes.) I go over the use of the log
and exponential features of the various calculators. The notion of present value is
important.
Notation: The book uses some nonstandard notation here. In Section 1, the usual A=Pert
is given as (Value after n years)=Pern. Compound interest is given as (Value after n
periods)=P(1+r)n. Also in Section 4.3 E(p) denotes consumer expenditure while in Section
4.4 E(p) denotes elasticity of demand. I suggest CE(p) for the former.
Chapter 4, Section 4 (elasticity): (See preceding comment on notation.) Be sure to
cover relative rates first. The book does a nice job of explaining elasticity using relative
rates. If your class has students who have had economics, you might ask for their input.
Students have offered insulin as very elastic and heroin as 'perfectly' elastic starting a
nice conversation. The Application Preview, blue section, looks interesting.
Chapter 5 (integration).. Consider introducing section 6 (substitution) earlier in the
chapter.
Chapter 5, Section 4 (areas): Asking for "set-up" only is a good ploy for tests. (Some
instructors of calculator sections allow students to use the numeric integration feature for
2 3
some of the area problems.
0
x dx is approximated by math 9: fnInt (x3,x,0,2) on the TI-
82 and 83. Be aware some functions take a very long time or give inaccurate results.)
You may wish to supplement this exercise set. Goldstein has good exercises.
Chapter 5. Section 6 (substitution): For students of weak algebra skills this section is very
hard and may take several days or returning to the subject several times. You might wish
to supplement the exercise set with more difficult substitution problems. (Goldstein has
some.) Students must be given problems of sufficient difficulty to force them to write out
all the steps, otherwise they think they understand when they do not. Be aware that the
exercise set in our text includes a number of examples which cannot be done with
substitution.
Chapter 5. Section 5. Be sure to do Consumers' and Producers' Surplus. (I have other
texts with explanations, if you'd like to read up on the subject.) It is important to our
students that they understand that this is a sum and an area. You might ask members of
your class who have had economics to explain equilibrium point. Students tell me that
they calculate Consumers' and Producers' Surplus in their economics class. They use
linear supply and demand functions and find areas of triangles.
The Gini Index of Income Distribution is both accessible and interesting to our
students. It does not take long to explain.
Note: Recall that the Business School has specifically asked us to cover many
applications in this course.
By Nancy Smith, December, 2002
Revised August 2005 by Darci Kracht
2 | 677.169 | 1 |
1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:02
The following content is
provided under a Creative
3
00:00:02 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:05
Your support will help MIT
OpenCourseWare continue to
5
00:00:05 --> 00:00:10
offer high-quality educational
resources for free.
6
00:00:10 --> 00:00:12
To make a donation, or to view
additional materials from
7
00:00:12 --> 00:00:15
hundreds of MIT courses, visit
MIT OpenCourseWare
8
00:00:15 --> 00:00:20
at ocw.mit.edu.
9
00:00:20 --> 00:00:21
AUDIENCE: OK.
10
00:00:21 --> 00:00:26
I hoped I might have Exam 2
for you today, but it's not
11
00:00:26 --> 00:00:28
quite back from the grader.
12
00:00:28 --> 00:00:33
It's already gone to the second
grader, so it will not be long.
13
00:00:33 --> 00:00:40
And I hope you've had a look
at the MATLAB homework for
14
00:00:40 --> 00:00:43
a variety of possible.
15
00:00:43 --> 00:00:49
I think we've got, there were
some errors in the original
16
00:00:49 --> 00:00:52
statement, location of the
coordinates, but I think
17
00:00:52 --> 00:00:53
they're fixed now.
18
00:00:53 --> 00:00:55
So ready to go on that MATLAB.
19
00:00:55 --> 00:00:58
Don't forget that it's four on
the right-hand side and not
20
00:00:58 --> 00:01:04
one, so if you get an answer
near 1/4 at the center of the
21
00:01:04 --> 00:01:07
circle, that's the reason.
22
00:01:07 --> 00:01:12
Just that factor four
is to remember.
23
00:01:12 --> 00:01:15
I'll talk more about the
MATLAB this afternoon in the
24
00:01:15 --> 00:01:18
review session right here.
25
00:01:18 --> 00:01:22
Just to say, I'm highly
interested in that problem.
26
00:01:22 --> 00:01:28
Not just increasing N, the
number of mesh points in the
27
00:01:28 --> 00:01:34
octagon, but also increasing
the number of sides.
28
00:01:34 --> 00:01:42
So there are two numbers there,
we had N points on a ray,
29
00:01:42 --> 00:01:44
out from the center.
30
00:01:44 --> 00:01:49
But we have M sides
of the polygon.
31
00:01:49 --> 00:01:55
And I'm interested in both
of those, getting big.
32
00:01:55 --> 00:01:57
Growing.
33
00:01:57 --> 00:01:58
I don't know how.
34
00:01:58 --> 00:02:08
And maybe a reasonable balance
is to take, I think N
35
00:02:08 --> 00:02:11
proportional to M is a
pretty good balance.
36
00:02:11 --> 00:02:14
So I'd be very happy;
I mean I'm very happy
37
00:02:14 --> 00:02:15
with whatever you do.
38
00:02:15 --> 00:02:19
But I'm really interested
to know what happens as
39
00:02:19 --> 00:02:22
both of these increase.
40
00:02:22 --> 00:02:25
How close, how quickly do you
approach the eigenvalues
41
00:02:25 --> 00:02:26
of a circle.
42
00:02:26 --> 00:02:29
And you might keep the
two proportional as
43
00:02:29 --> 00:02:31
you increase them.
44
00:02:31 --> 00:02:34
So let me say more about that
this afternoon, because it's a
45
00:02:34 --> 00:02:37
big day today, to
start Fourier.
46
00:02:37 --> 00:02:41
Fourier series, the new
chapter, the new topic.
47
00:02:41 --> 00:02:44
In fact, the final major
topic of the course.
48
00:02:44 --> 00:02:52
So I tried to list here, so
here I'm in Section 4.1, so I'm
49
00:02:52 --> 00:02:54
talking about Fourier series.
50
00:02:54 --> 00:02:58
So Fourier series is for
functions that have period 2pi.
51
00:02:59 --> 00:03:05
It involves things like sin(x),
like cos(x) like e^(ikx), all
52
00:03:05 --> 00:03:11
of those if I increase x by
2pi, I'm back where I started.
53
00:03:11 --> 00:03:15
So that's the sort of functions
that have Fourier series.
54
00:03:15 --> 00:03:21
Then we'll go on to the other
two big forms, crucial
55
00:03:21 --> 00:03:23
forms of the Fourier world.
56
00:03:23 --> 00:03:28
But 4.1 starts with the
classical Fourier series.
57
00:03:28 --> 00:03:34
So I realize, you will have
seen, many of you will have
58
00:03:34 --> 00:03:36
seen Fourier series before.
59
00:03:36 --> 00:03:40
I hope you'll see some
new aspects here.
60
00:03:40 --> 00:03:48
So, let me just get organized.
61
00:03:48 --> 00:03:53
It's nice to have some examples
that just involve sine.
62
00:03:53 --> 00:03:57
And since the sine is an odd
function, that means it's sort
63
00:03:57 --> 00:04:01
of anti-symmetric across zero,
those are the functions
64
00:04:01 --> 00:04:03
that will have only sine.
65
00:04:03 --> 00:04:05
That will have a
sine expansion.
66
00:04:05 --> 00:04:07
Cosines are the opposite.
67
00:04:07 --> 00:04:10
Cosines are symmetric
across zero.
68
00:04:10 --> 00:04:12
Like a constant,
or like cos(x).
69
00:04:12 --> 00:04:15
Zero comes right at
the symmetric point.
70
00:04:15 --> 00:04:18
So those will have
only cosines.
71
00:04:18 --> 00:04:22
And a lot of examples fit in
one or the other of those,
72
00:04:22 --> 00:04:24
and it's easy to see them.
73
00:04:24 --> 00:04:28
The general function, of
course, is a combination
74
00:04:28 --> 00:04:30
odd and even.
75
00:04:30 --> 00:04:33
It has cosines and it has
sines, it's just the
76
00:04:33 --> 00:04:35
some of the two pieces.
77
00:04:35 --> 00:04:41
So, this is the standard
Fourier series, which I
78
00:04:41 --> 00:04:45
couldn't get onto one line,
but it has all the cosines
79
00:04:45 --> 00:04:49
including this slightly
different cos(0),
80
00:04:49 --> 00:04:51
and all the sines.
81
00:04:51 --> 00:04:57
But because this one has these
three different pieces, the
82
00:04:57 --> 00:05:02
constant term, the other
cosines, all the sines, three
83
00:05:02 --> 00:05:07
slightly different formulas,
it's actually nicest of all,
84
00:05:07 --> 00:05:10
to use this final form.
85
00:05:10 --> 00:05:12
Because there's
just one formula.
86
00:05:12 --> 00:05:13
There's just one kind.
87
00:05:13 --> 00:05:18
And I'll call its coefficient
c_k, and now they multiply
88
00:05:18 --> 00:05:22
e^(ikx), so we have to
get used to e^(ikx).
89
00:05:24 --> 00:05:29
We may be more familiar with
cos, sin(kx) and cos(kx),
90
00:05:29 --> 00:05:34
but everybody knows e^(ikx)
is a combination of them.
91
00:05:34 --> 00:05:38
And if we let k go from minus
infinity to infinity, so
92
00:05:38 --> 00:05:40
we've got all the terms.
93
00:05:40 --> 00:05:48
Including e^(-i3x), and
e^(+i3x), those would
94
00:05:48 --> 00:05:51
combine to give cosines
and sines of 3x.
95
00:05:52 --> 00:05:54
We get one nice formula.
96
00:05:54 --> 00:05:57
There's just one
formula for the C's.
97
00:05:57 --> 00:06:01
So that's one good reason to
look at the complex form.
98
00:06:01 --> 00:06:05
Even if our function
is actually real.
99
00:06:05 --> 00:06:09
That form is kind of neat, and
the second good reason, the
100
00:06:09 --> 00:06:13
really important reason, is
then when we go to the discrete
101
00:06:13 --> 00:06:18
Fourier transform, the DFT,
everybody writes that
102
00:06:18 --> 00:06:20
with complex numbers.
103
00:06:20 --> 00:06:25
So it's good to see complex
numbers first and then we
104
00:06:25 --> 00:06:30
can just translate
the formulas from.
105
00:06:30 --> 00:06:33
And these are also almost
always written with
106
00:06:33 --> 00:06:34
complex numbers.
107
00:06:34 --> 00:06:39
So this is the way to see it.
108
00:06:39 --> 00:06:44
OK, so what do we do
about Fourier series?
109
00:06:44 --> 00:06:46
What do we have to know
how to do and what
110
00:06:46 --> 00:06:47
should we understand?
111
00:06:47 --> 00:06:53
Well, if you've met Fourier
series you may have met the
112
00:06:53 --> 00:06:56
formula for these coefficients.
113
00:06:56 --> 00:06:58
That's sort of like step one.
114
00:06:58 --> 00:07:01
If I'm given the function,
whatever the function might be,
115
00:07:01 --> 00:07:02
might be a delta function.
116
00:07:02 --> 00:07:04
Interesting case, always.
117
00:07:04 --> 00:07:07
Always interesting.
118
00:07:07 --> 00:07:08
Always crazy right?
119
00:07:08 --> 00:07:13
But it's always interesting,
the delta function.
120
00:07:13 --> 00:07:16
The coefficients
can be computed.
121
00:07:16 --> 00:07:21
The coefficients, you'll see,
I'll repeat those formulas.
122
00:07:21 --> 00:07:26
They involve integrals.
123
00:07:26 --> 00:07:29
What I want to say right
now is that this isn't a
124
00:07:29 --> 00:07:31
course in integration.
125
00:07:31 --> 00:07:35
So I'm not interested in doing
more and more complicated
126
00:07:35 --> 00:07:39
integrals and finding
Fourier coefficients
127
00:07:39 --> 00:07:40
of weird functions.
128
00:07:40 --> 00:07:41
No way.
129
00:07:41 --> 00:07:45
I want to understand the
simple, straight, the
130
00:07:45 --> 00:07:47
important examples.
131
00:07:47 --> 00:07:52
And here's a point that's
highly interesting.
132
00:07:52 --> 00:07:56
In practice, in computing
practice, we're close to
133
00:07:56 --> 00:07:57
computing practice here.
134
00:07:57 --> 00:07:59
In everything we do.
135
00:07:59 --> 00:08:03
I mean, this is really
constantly used.
136
00:08:03 --> 00:08:07
And one important question
is, is the Fourier series
137
00:08:07 --> 00:08:09
quickly convergent?
138
00:08:09 --> 00:08:11
Because if we're going to
compute, we don't want to
139
00:08:11 --> 00:08:14
compute a thousand terms.
140
00:08:14 --> 00:08:19
Hopefully ten terms, 20 terms
would give us good accuracy.
141
00:08:19 --> 00:08:24
So that question comes down to
how quickly does those a's
142
00:08:24 --> 00:08:27
and b's and c's go to zero?
143
00:08:27 --> 00:08:28
That's highly important.
144
00:08:28 --> 00:08:32
And you'll connect this decay
rate, we'll connect this with
145
00:08:32 --> 00:08:34
the smoothness of the function.
146
00:08:34 --> 00:08:38
Oh, I can tell you
even at a start.
147
00:08:38 --> 00:08:42
OK, so I just want to
emphasize this point.
148
00:08:42 --> 00:08:50
We'll see it over and over that
like for a delta function,
149
00:08:50 --> 00:08:56
which is not smooth at all,
we'll see no decay at all.
150
00:08:56 --> 00:08:58
In the coefficients.
151
00:08:58 --> 00:09:03
They're constant.
152
00:09:03 --> 00:09:06
They don't decrease as we go to
higher and higher frequencies.
153
00:09:06 --> 00:09:13
I think of k here, I'll use
the word frequency for k.
154
00:09:13 --> 00:09:18
So high frequency means high k,
far off the Fourier series, and
155
00:09:18 --> 00:09:22
the question is, are the
coefficients staying up there
156
00:09:22 --> 00:09:24
big, and we have to
worry about them.
157
00:09:24 --> 00:09:26
Or do they get very small?
158
00:09:26 --> 00:09:29
So a delta function
is a key example and
159
00:09:29 --> 00:09:32
then a step function.
160
00:09:32 --> 00:09:34
So what will be the
deal with those?
161
00:09:34 --> 00:09:38
If I have a function that's
a step function, I'll have
162
00:09:38 --> 00:09:41
decay at rate is 1/k.
163
00:09:41 --> 00:09:44
164
00:09:44 --> 00:09:46
So they do go to zero.
165
00:09:46 --> 00:09:53
The thousandth coefficient will
be roughly of size 1/1000.
166
00:09:53 --> 00:09:54
That's not fast.
167
00:09:54 --> 00:10:02
That's not really fast
enough to compute with.
168
00:10:02 --> 00:10:08
Well, we meet step functions,
I mean, functions with jumps.
169
00:10:08 --> 00:10:11
And we'll see that their
Fourier series, the
170
00:10:11 --> 00:10:16
coefficients do go to
zero but not very fast.
171
00:10:16 --> 00:10:19
And we get something
highly interesting.
172
00:10:19 --> 00:10:23
So when we do these examples,
so I've sort of moved on to
173
00:10:23 --> 00:10:27
examples, so these are
two basic examples.
174
00:10:27 --> 00:10:30
What would be the next example?
175
00:10:30 --> 00:10:32
Step function.
176
00:10:32 --> 00:10:35
Well, yeah, or
maybe a hat next.
177
00:10:35 --> 00:10:37
A hat function would
be, you see what I'm
178
00:10:37 --> 00:10:38
doing at each step?
179
00:10:38 --> 00:10:40
I'm integrating.
180
00:10:40 --> 00:10:44
A hat function might be the
next, yeah, a ramp, exactly.
181
00:10:44 --> 00:10:48
Hat function, which is
a ramp with a corner.
182
00:10:48 --> 00:10:50
Now, so that's one
integral better.
183
00:10:50 --> 00:10:55
You want to guess the decay
rate on that one? k squared.
184
00:10:55 --> 00:10:58
Now we're getting better.
185
00:10:58 --> 00:11:00
That's a faster follow-up.
186
00:11:00 --> 00:11:00
1/k^2.
187
00:11:02 --> 00:11:04
And then we integrate
again, we'd get 1/k^3.
188
00:11:06 --> 00:11:11
Then one more integral, 1/k^4
would be a cubic spline with,
189
00:11:11 --> 00:11:14
you remember the cubic
spline is continuous.
190
00:11:14 --> 00:11:16
Its derivative is continuous,
that gives us a 1/k^3.
191
00:11:17 --> 00:11:20
Its second derivative is
continuous, that gives us a
192
00:11:20 --> 00:11:25
1/k^4, and then you really can
compute with that, if you
193
00:11:25 --> 00:11:27
have such a function.
194
00:11:27 --> 00:11:31
So, point, pay attention
to decay rate.
195
00:11:31 --> 00:11:37
That, and the connection
to smoothness.
196
00:11:37 --> 00:11:41
So examples, we'll start
right off with these guys.
197
00:11:41 --> 00:11:44
And then we'll see the
rules for the derivative.
198
00:11:44 --> 00:11:48
Oh yeah, rules for
the derivative.
199
00:11:48 --> 00:11:53
The beauty of Fourier
series is, well, actually
200
00:11:53 --> 00:11:54
you can see this.
201
00:11:54 --> 00:11:56
You can see the rule.
202
00:11:56 --> 00:11:58
Let me just show you
the rule for this.
203
00:11:58 --> 00:12:03
So the rule for derivatives,
the whole point about
204
00:12:03 --> 00:12:08
Fourier is, it connects
perfectly with calculus.
205
00:12:08 --> 00:12:10
With taking derivatives.
206
00:12:10 --> 00:12:16
So suppose I have F(x) equals,
I'll use this form, the
207
00:12:16 --> 00:12:18
sum of c_k*e^(ikx).
208
00:12:18 --> 00:12:22
209
00:12:22 --> 00:12:24
And now I take its
derivative. dF/dx.
210
00:12:25 --> 00:12:29
What do you think is the
derivative, what's the Fourier
211
00:12:29 --> 00:12:33
series for the derivative?
212
00:12:33 --> 00:12:36
Suppose I have the Fourier
series for some function, and
213
00:12:36 --> 00:12:38
then I take Fourier series
for the derivative.
214
00:12:38 --> 00:12:42
So I'm kind of going
the backwards way.
215
00:12:42 --> 00:12:43
Less smooth.
216
00:12:43 --> 00:12:48
I'm going from, the derivative
of the step function involves
217
00:12:48 --> 00:12:53
delta functions, so I'm
going less smooth as
218
00:12:53 --> 00:12:57
I take derivatives.
219
00:12:57 --> 00:13:00
It's so easy, it jumps at you.
220
00:13:00 --> 00:13:01
What's the rule?
221
00:13:01 --> 00:13:05
Just take the derivative of
every term, so I'll have the
222
00:13:05 --> 00:13:10
sum of, now what happens
when I take the derivative?
223
00:13:10 --> 00:13:14
Everybody see what happens when
I take the derivative of that
224
00:13:14 --> 00:13:17
typical term in the
Fourier series?
225
00:13:17 --> 00:13:19
What happens?
226
00:13:19 --> 00:13:22
The derivative brings
down a factor, ik.
227
00:13:23 --> 00:13:32
With k being the thing that, so
it's ik times what we have.
228
00:13:32 --> 00:13:38
So these are the Fourier
coefficients of the derivative.
229
00:13:38 --> 00:13:42
And that again makes exactly
the same point about the
230
00:13:42 --> 00:13:46
decay rate or the opposite,
the non decay rate.
231
00:13:46 --> 00:13:50
As I take the derivative you
got a rougher function, right?
232
00:13:50 --> 00:13:54
Derivative of a step function
is a delta, derivative of a
233
00:13:54 --> 00:13:57
hat would have some steps.
234
00:13:57 --> 00:14:02
We're going less smooth as
we take more derivatives.
235
00:14:02 --> 00:14:06
And every time we do it,
we see, you understand
236
00:14:06 --> 00:14:07
the decay rate now?
237
00:14:07 --> 00:14:14
Because the derivative just
brings a factor ik, so its high
238
00:14:14 --> 00:14:18
frequencies are more present.
239
00:14:18 --> 00:14:20
Have larger coefficients.
240
00:14:20 --> 00:14:22
So and of course, the
second derivative would
241
00:14:22 --> 00:14:24
bring down (ik)^2.
242
00:14:24 --> 00:14:27
243
00:14:27 --> 00:14:35
So that our equations, for
example, let me just do
244
00:14:35 --> 00:14:38
an application here.
245
00:14:38 --> 00:14:41
Without pushing it.
246
00:14:41 --> 00:14:45
Our application, we started
this course with equations
247
00:14:45 --> 00:14:46
like -u''(x)=delta(x-a).
248
00:14:46 --> 00:14:51
249
00:14:51 --> 00:14:52
Right?
250
00:14:52 --> 00:14:55
If we wanted to apply to
a differential equation,
251
00:14:55 --> 00:14:56
how would I do it?
252
00:14:56 --> 00:15:00
I would take the Fourier
series of both sides.
253
00:15:00 --> 00:15:03
I would look at, I'd jump
into what people would
254
00:15:03 --> 00:15:05
call the frequency domain.
255
00:15:05 --> 00:15:10
So this is a differential
equation written as usual
256
00:15:10 --> 00:15:13
in the physical domain.
257
00:15:13 --> 00:15:17
And with physical
variable x position.
258
00:15:17 --> 00:15:18
Or it could be time.
259
00:15:18 --> 00:15:22
And now let me take
Fourier transforms.
260
00:15:22 --> 00:15:23
So what would happen here?
261
00:15:23 --> 00:15:27
If I take the Fourier transform
of this, well, we'll
262
00:15:27 --> 00:15:30
soon see, right?
263
00:15:30 --> 00:15:33
We get Fourier coefficients
of the deltas.
264
00:15:33 --> 00:15:34
Of the delta function.
265
00:15:34 --> 00:15:38
That's a key example,
and you see why.
266
00:15:38 --> 00:15:40
Over here, what will we get?
267
00:15:40 --> 00:15:43
And now I'm taking two
derivatives, so I
268
00:15:43 --> 00:15:45
bring down ik twice.
269
00:15:45 --> 00:15:46
So I'm looking.
270
00:15:46 --> 00:15:51
Here it would be the sum
of whatever the delta's
271
00:15:51 --> 00:15:52
coefficients are.
272
00:15:52 --> 00:15:54
Shall we call those d?
273
00:15:54 --> 00:15:59
The alphabet's coming
out right. d for delta.
274
00:15:59 --> 00:16:03
So the right side has
coefficients, d_k.
275
00:16:03 --> 00:16:05
And what about the left side?
276
00:16:05 --> 00:16:10
What are the coefficients if
the solution u has coefficients
277
00:16:10 --> 00:16:15
c_k, so let's call this u now.
278
00:16:15 --> 00:16:18
Has coefficients c_k, then
what happens to the second
279
00:16:18 --> 00:16:23
derivative? ik, ik again,
that's i squared k
280
00:16:23 --> 00:16:25
squared, the minus sign.
281
00:16:25 --> 00:16:29
So we would have the sum
of k squared c_k*e^(ikx).
282
00:16:29 --> 00:16:33
283
00:16:33 --> 00:16:37
This is if u itself has
coefficient c_k, then -u''
284
00:16:37 --> 00:16:39
has these coefficients.
285
00:16:39 --> 00:16:41
So what's up?
286
00:16:41 --> 00:16:43
How would we use that?
287
00:16:43 --> 00:16:44
It's going to be easy.
288
00:16:44 --> 00:16:48
We'll just match terms.
289
00:16:48 --> 00:16:49
Right?
290
00:16:49 --> 00:16:52
I can see, what's my
formula, what should c_k
291
00:16:52 --> 00:16:54
be if I know the d_k?
292
00:16:54 --> 00:16:58
I'm given the right-hand side.
293
00:16:58 --> 00:17:01
We're just doing what's
constantly happening,
294
00:17:01 --> 00:17:03
this three step process.
295
00:17:03 --> 00:17:04
You're given the right side.
296
00:17:04 --> 00:17:09
Step one, expand it in
Fourier series now.
297
00:17:09 --> 00:17:13
Step two, match the two sides.
298
00:17:13 --> 00:17:14
So what's the formula for c_k?
299
00:17:16 --> 00:17:19
In this application, which
by the way I had no
300
00:17:19 --> 00:17:20
intention to do this.
301
00:17:20 --> 00:17:24
But it jumped into my head and
I thought why not just do it.
302
00:17:24 --> 00:17:28
What would be the
formula for c_k?
303
00:17:30 --> 00:17:35
It'll be d_k divided
by? k squared.
304
00:17:35 --> 00:17:37
You're just matching terms.
305
00:17:37 --> 00:17:43
Just the way, when we expanded
things in eigenvectors, we'd
306
00:17:43 --> 00:17:46
match the coefficients of the
eigenvectors, and that involved
307
00:17:46 --> 00:17:52
just the simple step, here
it's d_k over k squared.
308
00:17:52 --> 00:17:53
Good.
309
00:17:53 --> 00:17:55
And then what's the final step?
310
00:17:55 --> 00:17:59
The final step is, now you
know the right coefficients,
311
00:17:59 --> 00:18:01
add them back up.
312
00:18:01 --> 00:18:03
Add the thing back
up, like here.
313
00:18:03 --> 00:18:10
Only I'm temporarily calling
it u, to find the solution.
314
00:18:10 --> 00:18:11
Right?
315
00:18:11 --> 00:18:13
Three steps.
316
00:18:13 --> 00:18:16
Go into the frequency domain.
317
00:18:16 --> 00:18:21
Write the right-hand side
as a Fourier series.
318
00:18:21 --> 00:18:28
Second quick step is look at
the equation for each separate
319
00:18:28 --> 00:18:30
Fourier coefficient.
320
00:18:30 --> 00:18:34
Match the coefficients
of these eigenvectors.
321
00:18:34 --> 00:18:35
Eigenfunctions.
322
00:18:35 --> 00:18:38
And that's this
quick middle step.
323
00:18:38 --> 00:18:42
And then you've got the answer,
but you're still in Fourier
324
00:18:42 --> 00:18:44
space, you're still
in frequency space.
325
00:18:44 --> 00:18:48
So you have to use these,
put them back to get the
326
00:18:48 --> 00:18:51
answer in physical space.
327
00:18:51 --> 00:18:51
Right?
328
00:18:51 --> 00:18:53
That's the pattern.
329
00:18:53 --> 00:18:54
Over and over.
330
00:18:54 --> 00:18:59
So that's sort of the general
plan of applying Fourier.
331
00:18:59 --> 00:19:02
And when does it work?
332
00:19:02 --> 00:19:03
When does it work?
333
00:19:03 --> 00:19:07
Because, I mean it's
fantastic when it works.
334
00:19:07 --> 00:19:13
So what is it about this
problem that made it work?
335
00:19:13 --> 00:19:15
What is Fourier happy?
336
00:19:15 --> 00:19:18
You know, when does he raise
his hand, say yes I can
337
00:19:18 --> 00:19:20
solve that problem?
338
00:19:20 --> 00:19:26
OK, what do I need here
for this plan to work?
339
00:19:26 --> 00:19:29
I certainly don't need always
just -u'', Fourier could
340
00:19:29 --> 00:19:31
do better than that.
341
00:19:31 --> 00:19:37
But what's the requirement for
Fourier to work perfectly?
342
00:19:37 --> 00:19:40
Well, linear equation, right?
343
00:19:40 --> 00:19:42
If we didn't have linear
equations we couldn't do
344
00:19:42 --> 00:19:45
all this adding and
matching and stuff.
345
00:19:45 --> 00:19:47
So linear equations.
346
00:19:47 --> 00:19:51
Well, OK.
347
00:19:51 --> 00:19:54
Now, what other
linear equations?
348
00:19:54 --> 00:19:56
Could I have a c(x) in here?
349
00:19:56 --> 00:20:01
My familiar c(x), variable
material property
350
00:20:01 --> 00:20:03
inside this equation?
351
00:20:03 --> 00:20:04
No.
352
00:20:04 --> 00:20:05
Well, not easily, anyway.
353
00:20:05 --> 00:20:09
That would really mess things
up if there's a variable
354
00:20:09 --> 00:20:14
coefficient in here then
it's going to have its
355
00:20:14 --> 00:20:15
own Fourier series.
356
00:20:15 --> 00:20:18
We're going to be
multiplying Fourier series.
357
00:20:18 --> 00:20:22
That comes later and
it's not so clean.
358
00:20:22 --> 00:20:25
So we want, it works
perfectly when it's
359
00:20:25 --> 00:20:28
constant coefficients.
360
00:20:28 --> 00:20:33
Constant coefficients in the
differential equations.
361
00:20:33 --> 00:20:36
And then one more thing.
362
00:20:36 --> 00:20:38
Very important other thing.
363
00:20:38 --> 00:20:39
The boundary conditions.
364
00:20:39 --> 00:20:42
Everybody remembers now, it's a
part of the message of this
365
00:20:42 --> 00:20:47
course is that boundary
conditions are often
366
00:20:47 --> 00:20:48
a source of trouble.
367
00:20:48 --> 00:20:51
They're part of the problem,
you have to deal with them.
368
00:20:51 --> 00:20:56
Now, what boundary conditions
do we think about here?
369
00:20:56 --> 00:21:02
Well, fixed-fixed was
where we started.
370
00:21:02 --> 00:21:04
So if we had fixed-fixed
boundary conditions
371
00:21:04 --> 00:21:06
what would I expect?
372
00:21:06 --> 00:21:12
Then things would give me
a sine series, possibly.
373
00:21:12 --> 00:21:14
Because those are the
eigenfunctions we're used to.
374
00:21:14 --> 00:21:19
Fixed-fixed, it's sines that
go from zero back to zero.
375
00:21:19 --> 00:21:24
Fixed-free will have
some sines or cosines.
376
00:21:24 --> 00:21:27
Periodic would be
the best of all.
377
00:21:27 --> 00:21:32
Yeah, so we need nice
boundary conditions.
378
00:21:32 --> 00:21:38
So the boundary conditions,
let me just say,
379
00:21:38 --> 00:21:40
periodic would be great.
380
00:21:40 --> 00:21:51
Or sometimes a fixed-free,
are familiar ones.
381
00:21:51 --> 00:21:55
At least in simple cases
can be dealt with.
382
00:21:55 --> 00:21:57
OK.
383
00:21:57 --> 00:22:04
So now, boy, that board is
already full of formulas.
384
00:22:04 --> 00:22:09
But, let's go back to the
start and say how do we
385
00:22:09 --> 00:22:13
find the coefficients?
386
00:22:13 --> 00:22:15
So because that was
the first step.
387
00:22:15 --> 00:22:18
Take the right-hand side,
find its coefficient.
388
00:22:18 --> 00:22:22
If we want to, just as applying
eigenvalues, the first step
389
00:22:22 --> 00:22:25
is always find eigenvalues.
390
00:22:25 --> 00:22:29
Here, in applying Fourier,
the first step is always
391
00:22:29 --> 00:22:31
find the coefficients.
392
00:22:31 --> 00:22:33
So, how do we do that?
393
00:22:33 --> 00:22:36
And at the beginning it
doesn't look too easy, right?
394
00:22:36 --> 00:22:39
Because let me take the
first guy, sin(x).
395
00:22:40 --> 00:22:43
Let me take an example.
396
00:22:43 --> 00:22:46
Particular S(x).
397
00:22:46 --> 00:22:49
The most important,
interesting function, S(x).
398
00:22:50 --> 00:22:53
I want it to be an odd
function, so that it
399
00:22:53 --> 00:22:55
will have only sine.
400
00:22:55 --> 00:22:57
And I should have
two period, 2pi.
401
00:22:58 --> 00:23:01
So let me just graph it.
402
00:23:01 --> 00:23:08
So it's going to have
coefficients, and I use b
403
00:23:08 --> 00:23:15
for sine, so it's going
to have b_1*sin(x), and
404
00:23:15 --> 00:23:18
b_2*sin(2x), and so on.
405
00:23:18 --> 00:23:23
And so it's got a whole
infinity of coefficients.
406
00:23:23 --> 00:23:24
Right?
407
00:23:24 --> 00:23:25
We're in function space.
408
00:23:25 --> 00:23:27
We're not dealing
with vectors now.
409
00:23:27 --> 00:23:32
So how is it possible to
find those coefficients?
410
00:23:32 --> 00:23:38
And let me chose a particular
S(x) so I'll put, since it's
411
00:23:38 --> 00:23:45
2pi periodic, if I tell you
what it is over a 2pi interval,
412
00:23:45 --> 00:23:47
just, repeat, repeat, repeat.
413
00:23:47 --> 00:23:52
So I'll pick the 2pi interval
to be minus pi to pi here.
414
00:23:52 --> 00:23:57
Just because it's a nice way,
and so that's a 2pi length.
415
00:23:57 --> 00:24:02
There's zero, I want to
function to be odd across zero.
416
00:24:02 --> 00:24:04
And I want it to be simple,
because it's going to be an
417
00:24:04 --> 00:24:07
important example that I
can actually compute.
418
00:24:07 --> 00:24:09
So I'm going to make it a one.
419
00:24:09 --> 00:24:13
And a minus one there.
420
00:24:13 --> 00:24:15
So, a step function.
421
00:24:15 --> 00:24:18
A step function, a square.
422
00:24:18 --> 00:24:22
And if I repeat it, of
course, it would go down,
423
00:24:22 --> 00:24:25
up, down, up, so on.
424
00:24:25 --> 00:24:30
But we only have to
look over this part.
425
00:24:30 --> 00:24:33
OK.
426
00:24:33 --> 00:24:37
Now, well, you might say wait
a minute how are we going to
427
00:24:37 --> 00:24:41
expand this function in sine.
428
00:24:41 --> 00:24:46
Well, sines are odd functions.
429
00:24:46 --> 00:24:48
Everybody knows what odd means?
430
00:24:48 --> 00:24:53
Odd means that S(-x) is -S(x).
431
00:24:56 --> 00:25:01
So that's the anti-symmetric
that we see in that graph.
432
00:25:01 --> 00:25:04
We also see a few
problems in this graph.
433
00:25:04 --> 00:25:11
At x=0, what is our sine
series going to give us?
434
00:25:11 --> 00:25:14
If I plug in x=0 on
the right-hand side I
435
00:25:14 --> 00:25:16
get zero, certainly.
436
00:25:16 --> 00:25:21
So this sine series
is going to do that.
437
00:25:21 --> 00:25:24
And actually Fourier
series tend to do this.
438
00:25:24 --> 00:25:26
In the middle of a jump
it'll pick the middle
439
00:25:26 --> 00:25:27
point of a jump.
440
00:25:27 --> 00:25:31
Fourier series generally, it's
the best possible, will pick
441
00:25:31 --> 00:25:33
the middle point of the jump.
442
00:25:33 --> 00:25:34
And what about at x=pi?
443
00:25:36 --> 00:25:39
At the end of the interval?
444
00:25:39 --> 00:25:41
What does my series
add up at x=pi?
445
00:25:42 --> 00:25:47
Zero again, because sin(pi),
sin(2pi), all zero.
446
00:25:47 --> 00:25:49
And that'll be in the
middle of that jump.
447
00:25:49 --> 00:25:52
So it's pretty good.
448
00:25:52 --> 00:25:57
But now what I'm hoping is that
my sine series is going to
449
00:25:57 --> 00:26:04
somehow get real fast up to
one, and level out at one.
450
00:26:04 --> 00:26:06
We're asking a lot.
451
00:26:06 --> 00:26:12
In fact, when Fourier proposed
this idea, Fourier series,
452
00:26:12 --> 00:26:17
there was a lot of doubters.
453
00:26:17 --> 00:26:23
Was it really possible to
represent other functions,
454
00:26:23 --> 00:26:27
maybe even including a step
function, in terms of
455
00:26:27 --> 00:26:31
sines or maybe cosines?
456
00:26:31 --> 00:26:33
And Fourier said
yes, go with it.
457
00:26:33 --> 00:26:34
So let's do it.
458
00:26:34 --> 00:26:42
OK, so and he turned out
to be incredibly right.
459
00:26:42 --> 00:26:43
How do I find b_2?
460
00:26:44 --> 00:26:48
Do you remember how to, I don't
want to know the formula.
461
00:26:48 --> 00:26:50
I want to know why.
462
00:26:50 --> 00:26:55
What's the step to find
the coefficient b_2?
463
00:26:56 --> 00:27:02
Well, the step is,
the key point.
464
00:27:02 --> 00:27:03
Which makes
everything possible.
465
00:27:03 --> 00:27:08
Why don't I identify the
key point without which we
466
00:27:08 --> 00:27:11
would be in real trouble.
467
00:27:11 --> 00:27:17
The key point is that all these
sine functions, sin(2x),
468
00:27:17 --> 00:27:22
sin(3x), sin(4x),
are orthogonal.
469
00:27:22 --> 00:27:27
Now, what do I mean by two
functions being orthogonal?
470
00:27:27 --> 00:27:31
Somehow my picture in function
space, so my picture in
471
00:27:31 --> 00:27:38
function space is that here is,
this is the sine x coordinate.
472
00:27:38 --> 00:27:41
And somewhere there's a sin(2x)
coordinate and it's 90 degrees
473
00:27:41 --> 00:27:44
and then there's a sin(3x)
coordinate, and then there's
474
00:27:44 --> 00:27:47
a sine, I don't know
where to point now.
475
00:27:47 --> 00:27:51
But there is a sin(4x), and
we're in infinite dimensions.
476
00:27:51 --> 00:27:56
And the sine vectors are
an orthogonal basis.
477
00:27:56 --> 00:27:58
They're orthogonal
to each other.
478
00:27:58 --> 00:28:00
What does that mean?
479
00:28:00 --> 00:28:03
Vectors we take
the dot product.
480
00:28:03 --> 00:28:07
Functions, we take, we don't
use the word dot product
481
00:28:07 --> 00:28:09
as much as inner product.
482
00:28:09 --> 00:28:12
So let me take the inner
product of, the whole
483
00:28:12 --> 00:28:13
point is orthogonality.
484
00:28:13 --> 00:28:15
Let me write that word down.
485
00:28:15 --> 00:28:17
Orthogonal.
486
00:28:17 --> 00:28:19
The sines are orthogonal.
487
00:28:19 --> 00:28:21
And what does that mean?
488
00:28:21 --> 00:28:27
That means that the integral
over our 2pi interval, or any
489
00:28:27 --> 00:28:34
2pi interval, of one sine,
sin(kx), let's say, multiplied
490
00:28:34 --> 00:28:42
by another sine, sin(lx), the x
is, you can guess the answer.
491
00:28:42 --> 00:28:47
And everything is
depending on this answer.
492
00:28:47 --> 00:28:49
And it is?
493
00:28:49 --> 00:28:51
Zero.
494
00:28:51 --> 00:28:53
It's just terrific.
495
00:28:53 --> 00:28:55
If k is different
from l, of course.
496
00:28:55 --> 00:29:00
If k is equal to l then I
have to figure that one out.
497
00:29:00 --> 00:29:01
I'll need that one.
498
00:29:01 --> 00:29:08
What is it if sine, if k=l
so I'm integrating sine
499
00:29:08 --> 00:29:12
squared of kx, then it's
certainly not zero.
500
00:29:12 --> 00:29:16
I getting like, the
length squared of the
501
00:29:16 --> 00:29:18
sin(kx) function.
502
00:29:18 --> 00:29:25
If k=l, what is it?
503
00:29:25 --> 00:29:27
It has some nice formula.
504
00:29:27 --> 00:29:28
Very nice.
505
00:29:28 --> 00:29:28
Let's see.
506
00:29:28 --> 00:29:32
Sine squared, do I need to
think about sine squared kx?
507
00:29:33 --> 00:29:37
Sine squared kx,
what does it do?
508
00:29:37 --> 00:29:39
Well, just graph
sine squared x.
509
00:29:39 --> 00:29:45
What would the graph of
sine squared x look like,
510
00:29:45 --> 00:29:48
from minus pi to pi?
511
00:29:48 --> 00:29:51
So it goes up, right?
512
00:29:51 --> 00:29:52
Doesn't it go up?
513
00:29:52 --> 00:29:54
And then it goes back down.
514
00:29:54 --> 00:29:55
OK.
515
00:29:55 --> 00:30:00
Sorry, I made that
a little hard.
516
00:30:00 --> 00:30:03
Is that right?
517
00:30:03 --> 00:30:05
And then it keeps it up.
518
00:30:05 --> 00:30:06
Right.
519
00:30:06 --> 00:30:08
So, what's the
integral of that?
520
00:30:08 --> 00:30:12
I'm not seeing quite why.
521
00:30:12 --> 00:30:16
The answer is its
average value is 1/2.
522
00:30:16 --> 00:30:23
The integral of sine squared
is 1/2 of the length.
523
00:30:23 --> 00:30:29
The whole interval is of length
2pi, and we're taking the
524
00:30:29 --> 00:30:31
area under sine squared.
525
00:30:31 --> 00:30:34
I may have to come back to
it, but the answer would be
526
00:30:34 --> 00:30:36
half of 2pi, which is pi.
527
00:30:36 --> 00:30:37
Yeah, yeah.
528
00:30:37 --> 00:30:41
So you could say the length
of the sine function
529
00:30:41 --> 00:30:46
is square root of pi.
530
00:30:46 --> 00:30:48
So these are integrals.
531
00:30:48 --> 00:30:51
You told me the
answer was zero.
532
00:30:51 --> 00:30:55
And I agreed with you, but
we haven't computed it.
533
00:30:55 --> 00:30:57
And nor have we
really got that.
534
00:30:57 --> 00:31:00
So a little bit to fix, still.
535
00:31:00 --> 00:31:09
But the crucial fact, I mean,
those are highly important
536
00:31:09 --> 00:31:12
integrals that just
come out beautifully.
537
00:31:12 --> 00:31:16
And beautifully
really means zero.
538
00:31:16 --> 00:31:19
I mean, that's the beautiful
number, right, for an integral.
539
00:31:19 --> 00:31:23
OK, so now how do I use that?
540
00:31:23 --> 00:31:24
Again, I'm looking for b_2.
541
00:31:26 --> 00:31:32
How do I pick off b_2, using
the fact that sin(2x) times any
542
00:31:32 --> 00:31:37
other sine integrates to zero.
543
00:31:37 --> 00:31:38
Ready for the moment?
544
00:31:38 --> 00:31:39
To find the coefficient b_2?
545
00:31:40 --> 00:31:44
I should, let me start this
sentence and if you finish it.
546
00:31:44 --> 00:31:50
I'll multiply both sides of
this equation by sin(2x).
547
00:31:51 --> 00:31:56
And then I will integrate.
548
00:31:56 --> 00:32:00
I'll multiply both sides by
sin(2x), so I take S(x)sin(2x).
549
00:32:00 --> 00:32:04
550
00:32:04 --> 00:32:07
And on the right hand, I
have b_1*sin(x)sin(2x).
551
00:32:07 --> 00:32:12
552
00:32:12 --> 00:32:13
And then I have b_2.
553
00:32:14 --> 00:32:16
Now, here's the one that's
going to live through
554
00:32:16 --> 00:32:17
the integration.
555
00:32:17 --> 00:32:20
It's going to survive, because
it's the sin(2x) times
556
00:32:20 --> 00:32:26
sin(2x) sin(2x) squared.
557
00:32:26 --> 00:32:30
And then comes the b_3 guy,
would be b_3*sin(3x)sin(2x).
558
00:32:37 --> 00:32:40
Everybody sees what I'm doing?
559
00:32:40 --> 00:32:44
As we did with the weak form in
differential equations, I'm
560
00:32:44 --> 00:32:47
multiplying through
by these guys.
561
00:32:47 --> 00:32:51
And then I'm integrating
over the interval.
562
00:32:51 --> 00:32:55
And what do I get?
563
00:32:55 --> 00:32:57
Integrate everyone dx.
564
00:32:59 --> 00:33:02
And what's the result?
565
00:33:02 --> 00:33:06
What is that integral?
566
00:33:06 --> 00:33:08
Zero.
567
00:33:08 --> 00:33:09
It's gone.
568
00:33:09 --> 00:33:11
What is this integral,
the integral of
569
00:33:11 --> 00:33:12
sin(3x) times sin(2x)?
570
00:33:14 --> 00:33:15
Zero.
571
00:33:15 --> 00:33:21
All those sines integrate to
zero, and I have to come
572
00:33:21 --> 00:33:27
back and see it's a simple
trig identity to do it.
573
00:33:27 --> 00:33:29
To see why that's zero.
574
00:33:29 --> 00:33:32
Do you see that everything is
disappearing, except b_2.
575
00:33:33 --> 00:33:36
So we finally have the
formula that we want.
576
00:33:36 --> 00:33:42
Let me just with put
these formulas down.
577
00:33:42 --> 00:33:46
So b_k, b_2 or b_k, yeah tell
me the formula for b_k.
578
00:33:47 --> 00:33:50
Let me go back, here.
579
00:33:50 --> 00:33:53
What did b_2 come out to be?
580
00:33:53 --> 00:33:56
So I have b_2, that's a number.
581
00:33:56 --> 00:33:59
It's got this right-hand side.
582
00:33:59 --> 00:34:01
That's the integral
that I mentioned.
583
00:34:01 --> 00:34:04
You'd have to compute
that integral.
584
00:34:04 --> 00:34:07
And then what about this stuff?
585
00:34:07 --> 00:34:10
This sin(2x) squared?
586
00:34:10 --> 00:34:13
I've integrated that.
587
00:34:13 --> 00:34:16
And what did I get for that?
588
00:34:16 --> 00:34:20
This is b_2, and then
this is some number.
589
00:34:20 --> 00:34:22
And it's pi.
590
00:34:22 --> 00:34:26
So this is b_2, and
multiplying, right?
591
00:34:26 --> 00:34:29
That b_2 comes out, and then
I have the integral of
592
00:34:29 --> 00:34:32
sine squared 2x, and
that's what's pi.
593
00:34:32 --> 00:34:37
So that's b_2 times pi here,
and I just divide by the pi.
594
00:34:37 --> 00:34:43
So I divide by pi and I get the
integral from minus pi to pi
595
00:34:43 --> 00:34:52
of my function times my sine.
596
00:34:52 --> 00:34:59
That's the model for all
the coefficients of
597
00:34:59 --> 00:35:02
orthogonal series.
598
00:35:02 --> 00:35:04
That's the model.
599
00:35:04 --> 00:35:11
Cosines, the complete ones,
the complex coefficients.
600
00:35:11 --> 00:35:15
The Legendre series, the Bessel
series, everybody's series
601
00:35:15 --> 00:35:18
will follow this same model.
602
00:35:18 --> 00:35:23
Because all those series are
series of orthogonal functions.
603
00:35:23 --> 00:35:26
Everything is hinging
on this orthogonality.
604
00:35:26 --> 00:35:31
The fact that one term
times another gives zero.
605
00:35:31 --> 00:35:34
What that means, really.
606
00:35:34 --> 00:35:43
I want to say it with a
picture, too. so let me draw
607
00:35:43 --> 00:35:46
two orthogonal directions.
608
00:35:46 --> 00:35:53
I intentionally didn't make
them just x and y axes.
609
00:35:53 --> 00:35:58
This might be the direction
of sin(x), and this might be
610
00:35:58 --> 00:35:59
the direction of sin(2x).
611
00:36:00 --> 00:36:05
And then I have a function.
612
00:36:05 --> 00:36:08
And I'm trying to find
out how much of sin(2x)
613
00:36:08 --> 00:36:09
has it got in it?
614
00:36:09 --> 00:36:12
How much of sin(x) has it got
in it, and then of course
615
00:36:12 --> 00:36:16
there's also a sin(3x) and
all the other sin(kx)'s.
616
00:36:17 --> 00:36:25
The point is, the point of this
90 degree angle there is, that
617
00:36:25 --> 00:36:34
if I can split this S(x),
whatever it might be, I can
618
00:36:34 --> 00:36:39
find its sin(x) piece directly.
619
00:36:39 --> 00:36:44
By just projecting it,
it's the projection of my
620
00:36:44 --> 00:36:48
function on that coordinate.
621
00:36:48 --> 00:36:51
If you don't like sin(x),
sin(2x), S(x), write
622
00:36:51 --> 00:36:54
v_1, v_2, whatever.
623
00:36:54 --> 00:36:56
To think of it as vectors.
624
00:36:56 --> 00:36:57
What's the sin 2?
625
00:36:57 --> 00:36:59
So that is b_1*sin(x).
626
00:37:01 --> 00:37:04
That's the right
amount of sin(x).
627
00:37:04 --> 00:37:11
And the whole point is that
that calculation didn't
628
00:37:11 --> 00:37:14
involve b_2 and b_3
and all the other b's.
629
00:37:14 --> 00:37:19
When I'm projecting onto
orthogonal directions, I
630
00:37:19 --> 00:37:22
can do them one at a time.
631
00:37:22 --> 00:37:25
I can do one one-dimensional
projection at a time.
632
00:37:25 --> 00:37:35
This b_ksin(kx) is the, so I'm
just saying this in words,
633
00:37:35 --> 00:37:42
is the projection of my
function onto sin(kx).
634
00:37:44 --> 00:37:49
And the point is, I could
do this and get this
635
00:37:49 --> 00:37:52
answer because of
that 90 degree angle.
636
00:37:52 --> 00:37:54
If I didn't have 90
degrees, do you see that
637
00:37:54 --> 00:37:55
this wouldn't work?
638
00:37:55 --> 00:38:02
Suppose my two basis functions
are at some 40 degree angle.
639
00:38:02 --> 00:38:05
Then I take my function.
640
00:38:05 --> 00:38:08
Can I project that
onto this guy?
641
00:38:08 --> 00:38:13
And project that onto this guy,
so the projections are there?
642
00:38:13 --> 00:38:14
And there?
643
00:38:14 --> 00:38:20
Do they add back to the
function that I started with?
644
00:38:20 --> 00:38:22
The given function?
645
00:38:22 --> 00:38:23
No way.
646
00:38:23 --> 00:38:26
I mean, these are
much too big, right?
647
00:38:26 --> 00:38:30
If I add that one to this one
I'm way out here somewhere.
648
00:38:30 --> 00:38:34
But over here, with 90
degrees, these are the two
649
00:38:34 --> 00:38:36
projections, project there.
650
00:38:36 --> 00:38:37
Project there.
651
00:38:37 --> 00:38:42
Add those two pieces and
I got back exactly.
652
00:38:42 --> 00:38:48
I just want to emphasize the
importance of orthogonality.
653
00:38:48 --> 00:38:52
It breaks the problem down into
one-dimensional projections.
654
00:38:52 --> 00:38:55
So here we go with b_k*sin(kx).
655
00:38:56 --> 00:38:59
OK, let me do the
key example now.
656
00:38:59 --> 00:39:01
This example.
657
00:39:01 --> 00:39:07
Let me find the coefficients of
that particular function S(x).
658
00:39:08 --> 00:39:13
This is the step function,
the square wave, S(x), let's
659
00:39:13 --> 00:39:15
find its coefficients.
660
00:39:15 --> 00:39:17
I'll just use this formula.
661
00:39:17 --> 00:39:22
OK, maybe I'll erase so that
I can write the integration
662
00:39:22 --> 00:39:23
right underneath.
663
00:39:23 --> 00:39:24
OK.
664
00:39:24 --> 00:39:26
Oh, one little point here.
665
00:39:26 --> 00:39:30
Well, not so little,
but it's a saving.
666
00:39:30 --> 00:39:35
It's worth noticing.
667
00:39:35 --> 00:39:42
The reward for picking off the
odd function is, I think that
668
00:39:42 --> 00:39:46
this integral is the same from
minus pi to zero
669
00:39:46 --> 00:39:48
as zero to a pi.
670
00:39:48 --> 00:39:51
In other words, I think that
for an odd function, I get
671
00:39:51 --> 00:39:57
the same answer if I just do
the integral from zero to
672
00:39:57 --> 00:40:03
pi, that I have to do.
673
00:40:03 --> 00:40:05
And double it.
674
00:40:05 --> 00:40:11
So I think if I just double
it, I don't know if you
675
00:40:11 --> 00:40:14
regard that as a saving.
676
00:40:14 --> 00:40:18
In some way, the work
is only half as much.
677
00:40:18 --> 00:40:21
It'll make this particular
example easy, so let
678
00:40:21 --> 00:40:23
me do this example.
679
00:40:23 --> 00:40:27
What are the Fourier
coefficients of
680
00:40:27 --> 00:40:29
the square wave?
681
00:40:29 --> 00:40:34
OK, so I'll do this integral.
682
00:40:34 --> 00:40:39
So from zero to pi,
what is my function?
683
00:40:39 --> 00:40:42
My N from the graph?
684
00:40:42 --> 00:40:44
Just one.
685
00:40:44 --> 00:40:47
This is going to be
a picnic, right?
686
00:40:47 --> 00:40:50
The function is one here.
687
00:40:50 --> 00:40:58
So S(x) is one, so I want 2/pi,
the integral from zero to pi
688
00:40:58 --> 00:41:03
of just sin(kx)dx, right?
689
00:41:03 --> 00:41:10
Which is, so I've got 2/pi,
now I integrate sin(kx), I
690
00:41:10 --> 00:41:15
get minus cos(kx), right?
691
00:41:15 --> 00:41:18
Between zero and pi.
692
00:41:18 --> 00:41:20
And what else?
693
00:41:20 --> 00:41:21
What have I forgotten?
694
00:41:21 --> 00:41:23
The most important point.
695
00:41:23 --> 00:41:27
The integral of sin(kx) k
x is not minus cos(kx).
696
00:41:28 --> 00:41:34
I have to divide by k.
697
00:41:34 --> 00:41:36
It's the division by k
that's going to give me
698
00:41:36 --> 00:41:41
the correct decay rate.
699
00:41:41 --> 00:41:41
2/(pi*k).
700
00:41:42 --> 00:41:45
Alright, now I've got a
little calculation to do.
701
00:41:45 --> 00:41:49
I have to figure out what
is cos(kx) at zero,
702
00:41:49 --> 00:41:51
no problem, it's one.
703
00:41:51 --> 00:41:54
And at the other
point, at x=pi.
704
00:41:55 --> 00:41:57
So what am I getting, then?
705
00:41:57 --> 00:41:58
I'm getting 2/(pi*k).
706
00:41:58 --> 00:42:09
707
00:42:09 --> 00:42:13
With that minus sign, I'll
evaluate it at x=0, I have
708
00:42:13 --> 00:42:17
one minus whatever I get
at the top. cos(k*pi).
709
00:42:17 --> 00:42:21
710
00:42:21 --> 00:42:22
That's b_k.
711
00:42:22 --> 00:42:24
712
00:42:24 --> 00:42:32
So there's a typical, well not
typical but very nice, answer.
713
00:42:32 --> 00:42:35
Now let's see what
these numbers are.
714
00:42:35 --> 00:42:39
So let me take a 2/pi out here.
715
00:42:39 --> 00:42:44
And then just list
these numbers.
716
00:42:44 --> 00:42:47
So k is one, two, three,
four, five, right?
717
00:42:47 --> 00:42:51
Tell me what these numbers are
for, let me put the k in here
718
00:42:51 --> 00:42:55
because that's part of it.
719
00:42:55 --> 00:42:56
So it's a constant, 2/pi.
720
00:42:56 --> 00:42:59
721
00:42:59 --> 00:43:03
At k=1, what do I get?
722
00:43:03 --> 00:43:03
At k=1?
723
00:43:04 --> 00:43:09
This is the little bit
that needs the patience.
724
00:43:09 --> 00:43:15
At k=1, the cos(pi) is?
725
00:43:15 --> 00:43:16
Negative one.
726
00:43:16 --> 00:43:20
So I have net minus
minus one, I get a two.
727
00:43:20 --> 00:43:25
I get a two over
a one. k is one.
728
00:43:25 --> 00:43:30
Alright, that is the
coefficient for k=1.
729
00:43:30 --> 00:43:33
Now, what's b_2, the
coefficient for k=2?
730
00:43:33 --> 00:43:37
I have 1-cos(2pi),
what's cos(2pi)?
731
00:43:39 --> 00:43:40
One.
732
00:43:40 --> 00:43:43
So they cancel,
so I get a zero.
733
00:43:43 --> 00:43:43
There is no b_2.
734
00:43:45 --> 00:43:45
What about b_3?
735
00:43:47 --> 00:43:49
So now b_3, I have 1-cos(3pi).
736
00:43:49 --> 00:43:51
737
00:43:51 --> 00:43:52
What's the cos(3pi)?
738
00:43:54 --> 00:43:56
It's negative one again.
739
00:43:56 --> 00:43:57
Right, same as the cos(pi).
740
00:43:58 --> 00:44:01
So that gives me a two, and
now I'm dividing by three.
741
00:44:01 --> 00:44:08
2/3, alright, let's do two
more. k=4, what do I get?
742
00:44:08 --> 00:44:12
Zero, because the cos(4pi)
has come back to one.
743
00:44:12 --> 00:44:13
So I get a zero.
744
00:44:13 --> 00:44:15
And what do I get from k=5?
745
00:44:15 --> 00:44:18
746
00:44:18 --> 00:44:25
1-cos(5pi), which is?
cos(5pi) is back to negative
747
00:44:25 --> 00:44:29
one, so 1--1 is a two.
748
00:44:29 --> 00:44:31
You see the pattern.
749
00:44:31 --> 00:44:39
And so let me just copy the
famous series for this, S(x).
750
00:44:40 --> 00:44:44
This S(x) is, let's see.
751
00:44:44 --> 00:44:48
The twos, I'll make
that 4/pi, right?
752
00:44:48 --> 00:44:50
I'll take out all those twos.
753
00:44:50 --> 00:44:51
So I have 4/pi*sin(x).
754
00:44:51 --> 00:44:55
755
00:44:55 --> 00:44:58
I have no sin(2x), forget that.
756
00:44:58 --> 00:45:02
Now I do have some sin(3x)'s,
how much do I have?
757
00:45:02 --> 00:45:12
4/pi*sin(3x)'s, But
divide by three, right?
758
00:45:12 --> 00:45:14
And then there's no
4x's, no sin(4x)'s.
759
00:45:16 --> 00:45:19
But then there will be
a 4/pi*sine, what's
760
00:45:19 --> 00:45:22
the next term now?
761
00:45:22 --> 00:45:23
Are you with me?
762
00:45:23 --> 00:45:27
So this is a typical nice
example, an important example.
763
00:45:27 --> 00:45:29
Sine of what?
764
00:45:29 --> 00:45:33
5x divided by five.
765
00:45:33 --> 00:45:35
OK.
766
00:45:35 --> 00:45:37
That's a great example,
it's worth remembering.
767
00:45:37 --> 00:45:40
Factor the 4/pi out
if you want to.
768
00:45:40 --> 00:45:47
4/pi time sin(x), sin(3x)/3,
sin(5x)/5, it's a beautiful
769
00:45:47 --> 00:45:49
example of an odd function.
770
00:45:49 --> 00:45:53
OK, and let's see.
771
00:45:53 --> 00:46:01
So what do you think, MATLAB
can draw this graph far
772
00:46:01 --> 00:46:02
better than we can.
773
00:46:02 --> 00:46:07
But let me draw enough so
you see what's really
774
00:46:07 --> 00:46:09
interesting here.
775
00:46:09 --> 00:46:12
Interesting and famous.
776
00:46:12 --> 00:46:15
So the leading term is 4
over pi sine x, that would
777
00:46:15 --> 00:46:18
be something like that.
778
00:46:18 --> 00:46:21
That's as close as
sin(x) can get, 4/pi
779
00:46:21 --> 00:46:23
is the optimal number.
780
00:46:23 --> 00:46:24
The optimal coefficient.
781
00:46:24 --> 00:46:29
The projection, this
4/pi*sin(x) is the best, the
782
00:46:29 --> 00:46:32
closest I can get to one.
783
00:46:32 --> 00:46:34
On that integral.
784
00:46:34 --> 00:46:35
With just sin(x).
785
00:46:36 --> 00:46:40
But now when I put in
sin(3x), I think it'll do
786
00:46:40 --> 00:46:44
something more like this.
787
00:46:44 --> 00:46:46
Do you see what's
happening there?
788
00:46:46 --> 00:46:49
That's what I've got with
sin(3x), and of course
789
00:46:49 --> 00:46:50
odd on the other side.
790
00:46:50 --> 00:46:52
What do you think it
looks like with sin(5x)?
791
00:46:54 --> 00:46:59
It's just so great you have
to let the computer draw
792
00:46:59 --> 00:47:00
it a couple of times.
793
00:47:00 --> 00:47:04
You see, it goes up here.
794
00:47:04 --> 00:47:08
And then it's sort of, you
know, it's getting closer.
795
00:47:08 --> 00:47:13
It's going to stay
closer to that.
796
00:47:13 --> 00:47:18
But I don't know if you can see
from my picture, I'm actually
797
00:47:18 --> 00:47:19
proud of that picture.
798
00:47:19 --> 00:47:22
It's not as bad as usual.
799
00:47:22 --> 00:47:27
And it makes the crucial
point, two crucial points.
800
00:47:27 --> 00:47:31
One is, I am going to get
closer and closer to one.
801
00:47:31 --> 00:47:38
These oscillations, these
ripples, will be smaller.
802
00:47:38 --> 00:47:42
But here is the great fact
and it's a big headache
803
00:47:42 --> 00:47:45
in calculation.
804
00:47:45 --> 00:47:51
At the jump, the first
ripple doesn't get smaller.
805
00:47:51 --> 00:47:57
The first ripple gets thinner,
the first ripple gets thinner.
806
00:47:57 --> 00:47:59
You see the ripples moving
over there, but their
807
00:47:59 --> 00:48:01
height doesn't change.
808
00:48:01 --> 00:48:04
Do you know whose name is
associated with that,
809
00:48:04 --> 00:48:06
in that phenomenon?
810
00:48:06 --> 00:48:07
Gibbs.
811
00:48:07 --> 00:48:15
Gibbs noticed that the ripple
height as you add more and more
812
00:48:15 --> 00:48:20
terms, you're closer and closer
to the function over more
813
00:48:20 --> 00:48:23
and more of the interval.
814
00:48:23 --> 00:48:25
So the ripples get
squeezed to the left.
815
00:48:25 --> 00:48:29
The area under the ripples
goes to zero, certainly.
816
00:48:29 --> 00:48:32
But the height of the
ripples doesn't.
817
00:48:32 --> 00:48:36
And it doesn't stay constant,
but nearly constant.
818
00:48:36 --> 00:48:39
It approaches a famous number.
819
00:48:39 --> 00:48:43
And of course we'll have the
same odd picture down here.
820
00:48:43 --> 00:48:47
And it'll bump up again, the
same thing is happening
821
00:48:47 --> 00:48:49
at every jump.
822
00:48:49 --> 00:48:52
In other words, if
you're computing shock.
823
00:48:52 --> 00:48:56
If you're computing air flow
around shocks, with Fourier
824
00:48:56 --> 00:49:00
type method, Gibbs is
going to get you.
825
00:49:00 --> 00:49:02
You'll have to deal with Gibbs.
826
00:49:02 --> 00:49:09
Because the shock has
that extra ripple.
827
00:49:09 --> 00:49:12
OK, that's a lot
of Section 4.1.
828
00:49:12 --> 00:49:15
Energy, we didn't get
to, so that'll be the
829
00:49:15 --> 00:49:17
first point on Friday.
830
00:49:17 --> 00:49:19
And I'll see you this
afternoon and talk about the
831
00:49:19 --> 00:49:21
MATLAB or anything else.
832
00:49:21 --> 00:49:22
OK. | 677.169 | 1 |
Unfortunately App will not even open on my phone, it will simply stop responding before it's even done loading.
A Google UserA Google User
Dosent even deserve a star Just not helpful
A Google UserFortunately for you, the brains at Education Flow have already thought about this and bring to you the first ever application to help you learn or revise. You only need an Internet connection to download additional content packages. The following topics are covered in this app:
Differentiation is all about finding rates of change (derivative) of one quantity compared to another. We need differentiation when the rate of change is not constant. Derivative Calculator computes a derivative of a given function with respect to a given variable using analytical differentiation. In calculus, the subtraction rule in differentiation is a method of finding the derivative of a function that is the subtraction of two other functions for which derivatives exist. The subtraction rule in integration follows from it. The rule itself is a direct consequence of differentiation. In calculus, the product rule of derivatives is a method of finding the derivative of a function that is the multiplication of two other functions for which derivatives exist. The product rule in integration follows from it. The rule itself is a direct consequence of differentiation. In calculus, the quotient rule of derivatives is a method of finding the derivative of a function that is the division of two other functions for which derivatives exist. The quotient rule in integration follows from it. The rule itself is a direct consequence of differentiation. In calculus, the power rule of derivatives is a method of finding the derivative of a function that is the multiplication of two other functions for which derivatives exist. The rule itself is a direct consequence of differentiation In calculus, the chain rule of derivatives is a method of finding the derivative of a function that is the composition of two functions for which derivatives exist. The rule itself is a direct consequence of differentiation.. In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. The sum rule in integration follows from it. The rule itself is a direct consequence of differentiation. Trapezoidal / Trapezium Rule is a method of finding an approximate value for an numerical integral, based on finding the sum of the areas of trapezia. Trapezium rule is also known as method of approximate integration. A slight underestimate will often be cancelled by a similar slight overestimate from another trapezium. Using narrower intervals will improve accuracy. Simpson's 1/3 Rule Numerical Integration is used to estimate the value of a definite integral. It works by creating an even number of intervals and fitting a parabola in each pair of intervals. Simpson's rule provides the exact result for a quadratic function or parabola. Romberg's Method Numerical Integration is based on the trapezoidal rule, where we use two estimates of an integral to compute a third integral that is more accurate than the previous integrals. This is called Richardson's extrapolation.
A perfect tool for college going students and mathematicians.
Disclaimer: This app used web Api of calculation from with Due permission
Easy drawer and slider navigation. Common, hard-to-remember formulas all in one place. Equations are professionally typeset in LaTeX and rendered with MathJax, not scanned or pasted screenshots. This means: * No blurry or jagged images .. these equations scale smoothly to all zoom levels. * Relatively small installation size for information of this type * Faster updates in response to user feedbackYou will use it from high school all the way to graduate school and beyond.
Features
Includes both Calculus I and II Clear and concise explanations Difficult concepts are explained in simple terms Illustrated with graphs and diagrams Search for the words or phrases Access the guide anytime, anywhere - at home, on the train, in the subway. Use your down time to prepare for an exam. Always have the guide available for a quick reference. Table of Contents
Introduction: Functions
Limits and Continuity: Limit of a Sequence | Limit of a Function | Limit of a function at infinity | Continuity | Classification of Discontinuities substitute TOP precalculus math game to help you get ready for college calculus. Join thousands of precalculus students who love to use games + learning to prepare for algebra, trigonometry, geometry and calculus.
Designed by expert educators, master mathematicians and amazing game designers, ThUMP is The Ultimate Math Practice for helping you improve every skill in high school mathematics. From numbers to exponents to equations and graphs, all precalculus math topics are covered here.
Mathtoons works closely with leading colleges and universities to combine cutting edge educational research with vibrant game features to make precalculus practice hugely successful.
THEWIREDHOMESCHOOL.COM: "What a great app! Many math concepts are covered in this colourful and fun bundle of electronic goodness!"
APPSFORHOMESCHOOLING.COM: "A math practice app for older children - a pre-calculus math game for high schoolers? Rare find!"
Students use ThUMP to achieve: * Faster recall of math facts * Increased understanding of algebra * Better preparation for college and university * Quick problem solving skills
Kymmee12, "Such a great interactive tool for learning math!" Em Rivers, "I love using it!" CSogers, "I wish this app was around when I was in school!"An application to help you find the formula in math that you need. It covers from Algebra to Calculus. Feel free to contact us with any problems, as we will mostly likely never sign in to the Gmail account again! This is a student project | 677.169 | 1 |
Video Summary: This learning video presents an introduction to graph theory through two fun, puzzle-like problems: "The Seven Bridges of Königsberg" and "The Chinese Postman Problem". Any high school student in a college-preparatory math class should be able to participate in this lesson. Materials needed include: pen and paper for the students; if possible, printed-out copies of the graphs and image that are used in the module; and a blackboard or equivalent. During this video lesson, students will learn graph theory by finding a route through a city/town/village without crossing the same path twice. They will also learn to determine the length of the shortest route that covers all the roads in a city/town/village. To achieve these two learning objectives, they will use nodes and arcs to create a graph and represent a real problem. This video lesson cannot be completed in one usual class period of approximately 55 minutes. It is suggested that the lesson be presented over two class sessions | 677.169 | 1 |
...Algebra 2 is a lot harder than it used to be. It's also more important than it used to be because algebra 2 concepts are included on the new SAT. Math education has improved in recent decades while grammar education has suffered. | 677.169 | 1 |
Synopsis
This DVD-ROM for PC and Mac contains hundreds of friendly, step-by-step video tutorials that clearly explain Higher Level GCSE Maths (for the AQA, Edexcel, OCR and WJEC exam boards). It also includes printable exam-style questions, with fully-worked answers on video. There's grading information to show which topics are the most challenging, plus a full search function that makes it easy to find what you're | 677.169 | 1 |
Cinnamon Hillyard,Statway® and Quantway® Senior Associate
The Carnegie Post-Bacs,Post Baccalaureate Fellows
The Course
In this short course, you will see four lessons from the Quantway® course developed by the Carnegie Foundation. These lessons will develop your understanding of common numbers often found in the news, on advertisements, and online. You will see how numbers play important roles in arguments you hear about daily like issues such as gun control, smoking, pollution, and heart attacks. And by the end of this course, you'll be able to use numbers to communicate your ideas.
The format of this course is probably not like other math courses you've taken. Instead, the Quantway® course was developed using principles that have been shown in research studies to be important in better developing students' understanding of the material. One of these key principles is Productive Struggle. In short this means that to learn something you must try it out AND stick with it even when it gets challenging!
The goal of the Quantway® is to help you learn things that you can actually use in life--not so you can memorize it for a test and then forget it. In fact, by the end of this course, you'll be able to create a final project that uses numbers to prove a point to anyone who sees it. Given the large number of students enrolled in this class, that's a lot of people! We are all learning together and helping each other along the way.
More Information
This course takes 5 weeks. Each week you will watch a new video and do a new assignment.
Workload
You can expect to spend 3-5 hours a week on this course. Most of your time will be spent doing assignments
Prerequisites
A willingness to learn math in a new way.
The Instructors
Karon Klipple
Managing Director of Programs and Strategic Partnerships
Karon Klipple directs the Community College Pathways program. She comes to the Carnegie Foundation for the Advancement of Teaching from San Diego City College, where she was associate professor of mathematics. Her focus has been on implementing innovative approaches to improving student performance through contextualized discovery-based learning. She has taught statistics and mathematics at the high-school level and at Texas A & M University. She has a B.A. in mathematics from Trinity University and holds a Ph.D. in statistics from Texas A&M University. | 677.169 | 1 |
Most
Description: This accelerated course prepares students for transfer-level Statistics. It covers core concepts from elementary algebra, intermediate algebra, and descriptive statistics. Topics include ratios, rates, and proportional reasoning; arithmetic reasoning using fractions, decimals and percents; evaluating expressions, solving equations, analyzing algebraic forms to understand statistical measures; use of linear, quadratic, absolute value, exponential, and logarithmic functions to model bivariate data; graphical and numerical descriptive statistics for quantitative and categorical data. This course is designed for students who do not want to major in fields such as math, science, computer science, and business. Note: This course is NOT intended for students who plan to study science, technology, engineering, math, as well as business and other non-STEM majors. Please see your counselor. | 677.169 | 1 |
Math Modeling is all about word problems which is my weakness. If you excel at this, this is the course for you.
No rating
Reviewed on
May 5, 2013
Is this helpful?
YesNo
Very easy, class. If you pay attention at least half of the time you'll still get a b.
No rating
Reviewed on
May 4, 2013
Is this helpful?
YesNo
I recommend math modeling for students who excel at word problems. Me, I do not. Therefore, this class was challenging for me.
No rating
Reviewed on
May 3, 2013
Is this helpful?
YesNo
Levi is fantastic. The class was web based and the program would tell you if your answer was right or wrong, and have you three chances to correct it. She would give you an extra 10% on assignments turned in at least a day before they were due. Math is not my strong suit, but I got an A in the class.
No rating
Reviewed on
Feb 5, 2013
Is this helpful?
YesNo
easy A
Reviewed on
Jan 8, 2013
Is this helpful?
YesNo
Really easy class, just make sure that you don't try to over explain it! watch your decimals and such.
(The reason I ranked it so low is because I hate math, the book is basically brand new.)
No rating
Reviewed on
Dec 14, 2012
Is this helpful?
YesNo
Course was great!
Reviewed on
Dec 4, 2010
Is this helpful?
YesNo
Waters was great but she teaches high school now.. maybe that's why she was so easy! hah | 677.169 | 1 |
ALGEBRA 1AB SYLLABUS
CLASS REQUIREMENTS AND EXPECTATIONS
Basic Objectives
Algebra 1 is considered a high school level course. This is a course that is required for a student to be
accepted into college and is very important in establishing the student's math foundation. The goals of this
class are to develop proficiency with mathematical skills, to expand understanding of mathematical concepts,
to improve logical thinking, and to promote success. We will be using Algebra: Structure and Method this
year as your textbook. You will be expected to have the correct book in class, unless told differently.
Classroom Rules and Policies
Expected Materials to have everyday: Expectations for Classroom Behavior
Three ring binder with paper (Spiral Binders are not Rules
recommended) Follow Directions
Agenda for keeping track of homework No conversations during lectures.
Pencils with erasers No yelling or swearing
Pens
Calculator(expected but not required) Rewards
Praise
Attendance Class rewards (no homework and others)
Regular attendance is important since new material Positive comments in the agendas and phone calls to
will be taught every day. parents
If you are absent you are responsible for making up
all homework missed. Consequences
Homework is listed on the sideboard and should be 1. First offense: Warning
made up by the following Friday. 2. Second offense: Stay 1 minute after bell
Failure to make up homework will affect both letter 3. Third offense: Stay 1 minutes after the bell
and work habits grade. and Parent is contacted after school
Being late to class will affect both Cooperation and 4. Fourth offense: Send to Dean
Work Habit grades.
Contacting Me
If you need to get in touch with me, you can call the Main Office at 818-920-2050 and I will return your call.
You can also write me a note in the agenda and I will either call you or give you a written answer. One of the
ways that worked well for some parents was to e-mail me. I am very good about answering e-mail and
parents and students find that this very convenient. My e-mail address is swarland@lausd.net.
Web Site
I also have a web site at which I have a place for students to get the upcoming homework if they are absent. I
will also post grades every 5 weeks and will have example of projects that we are doing. You can also get
copies of class rules and the class project there. The site is You can also e-mail me
from that site
- - - - - - - - - - - - - -
Student's Name______________________________________________________
Please sign and return this portion of this letter.
I have read and understand the above and agree to follow the instructions.
______________________________________ _______ ____________________________________ ________
Student's Signature Date Parent's Signature Date
Grading and Marking Policies
Letter Grade Work Habits
Quizzes Your Work Habits grade is based upon your
There will be a quiz every Friday. Homework percentage:
These quizzes will be worth 20% of your grade. 85% and greater is an E
There will be no make up on quizzes. (See below) 70% to 84% is an S
Homework 69% and less is Unsatisfactory.
There will be homework at least four times a week. Cooperation
These will be worth 20% of your grade. Your cooperation grade will be based upon
Failure to turn in homework will result in a "U" in classroom behavior and your ability to follow
Work Habits and can lower your letter grade. directions.
Homework is to be turned in daily. If you are absent
you are responsible for making up all homework Grading Policy
missed. To show mastery of the subject, students that receive
Homework is listed on the sideboard and should be a D or lower really need to retake Algebra 1. My
made up by the following Friday. grading is set on the following scale:
Late Homework will not be accepted more than one
week after the due date. A ........................................................ 100-90%
Tests
There will be unit tests and a final for each semester B ....................................................... 89.9-82%
These will be worth 40% of your grade. C ....................................................... 81.9-74%
Tests will be made up.
D ....................................................... 73.9-50%
Projects
There will be class assignments and special projects F ......................................................... 49.9-0%
throughout the year. Agenda
These will be worth 20% of your grade. I would advise all students to use the agenda for
Marking Policy writing down their assignments. I write the
Every 5 weeks I will throw out one homework homework for the week on the sideboard. You may
grade or one quiz grade. If you missed a quiz, or did copy it down as soon as you walk in so that if you
not turn in a homework assignment, it will not be are absent you can already have the homework.
counted. This is why I will not accept late To help students keep track of how they are doing in
homework or allow you to make up a quiz. This class I will post the grades on a weekly basis. This
will reward the students that come every day and will be done with an ID that I give you and not by
turn in all their work because I will still throw out name. You may want to write down your grade and
their lowest grade. show it to your parents every Monday. (Parents-You
can ask for it every Monday.)
Algebra 1AB is a High School level class. Failure to achieve the required standards at the midterm or final
report cards can result in transfer out of this class to a an Algebra Readiness class..
Mr. Worland's Algebra 1AB Syllabus 2008 | 677.169 | 1 |
Matrices
The matrices section of QuickMath allows you to perform arithmetic operations on matrices. Currently you can add or subtract matrices, multiply two matrices, multiply a matrix by a scalar and raise a matrix to any power.
What is a matrix?
A matrix is a rectangular array of elements (usually called scalars), which are set out in rows and columns. They have many uses in mathematics, including the transformation of coordinates and the solution of linear systems of equations.
Here is an example of a 2x3 matrix :
1 2 3
4 5 6
Arithmetic
The arithmetic suite of commands allows you to add or subtract matrices, carry out matrix multiplication and scalar multiplication and raise a matrix to any power.
Matrices are added to and subtracted from one another element by element. For instance, when adding two matrices A and B, the element at row i, column j of A is added to the element at row i, column j of B to give the element at row i, column j of the answer. Consequently, you can only add and subtract matrices which are the same size.
Matrix multiplication is a little more complicated. Suppose two matrices A and B are multiplied together to get a third matrix C. The element at row i, column j in C is found by taking row i from A and multiplying it by column j from B. Two matrices can only be multiplied together if the number of columns in the first equals the number of rows in the second.
Multiplying a matrix by a scalar simply involves multiplying each element by that scalar, whilst raising a matrix to a positive integer power can be achieved by a series of matrix multiplications.
There is currently no advanced arithmetic section, though this may be introduced in the future. | 677.169 | 1 |
This popular Physical Chemistry text book is now available in electronic format. We have preserved much of the material of the former hard copy editions, making changes to improve understanding of the concepts in addition to including some of the recent discoveries in physical chemistry. Many chapters have new sections and the coverage of several chapters has been greatly expanded. The chapter on statistical mechanics, 15, has been completely rewritten.
The eBook has also been divided into smaller modules that are appropriate for specific courses in Physical Chemistry.
Easy to use Clebsch-Gordan coefficient solver for adding two angular momentums in Quantum Mechanics. This tool is created for my Quantum Mechanics II course offered by Dr. Thompson in Summer of 2007.
[Instruction]
Execute "GUI.m" script by invoking "GUI".
Inspired by a discussion with my father on how to solve sudokus, I decided to implement a GUI for MATLAB and play around with automatic solving. The result can be found here: You can use the GUI just for playing sudoku and having an online check or you may turn on the solving aids: Display tooltips showing all valid numbers so far, or have a semiautomatic or a automatic solver which evaluates the logical constraints. On top of that, a branching algorithm is implemented, which solves any arbitrary sudoku very fast.
Math Solver Free for Windows 8 is a handy tool for performing frequently used operations used for solving math problems. You can use this tool for solving quadratic equations or calculating the angles of a triangle.
The app also includes a unit converter and other useful tools for dealing with math problems by using your Windows 8 device.
Worksheet Generator for Chemistry is a handy and reliable software that helps you to easily and quickly create and customize your personal chemistry worksheets.
The application provides you with various exercise templates that allow you to adjust your worksheets. You are able to insert various chemistry exercises of different areas such as units and chemical formulae, thermochemistry, chemical kinetics, Redox reactions and organic chemistry | 677.169 | 1 |
Should College Classes Ditch the Calculator?
Should College Classes Ditch the Calculator?
According to Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research and Development Center, using calculators in college math classes may be doing more harm than good. In a limited study conducted with undergraduate engineering students and published in the British Journal of Educational Technology, King has determined that our use of calculators may be serving as an alternative to an actual, deep understanding of mathematical material.
"We really can't assume that calculators are helping students," says King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
King, along with co-author and director of the Mathematics Education Centre at Loughborough University, Carol Robinson, conducted the study by interviewing 10 second-year undergraduate students who were enrolled in a competitive engineering program. The students were given a number of mathematical questions dealing with sine waves, which are mathematical curves that describe a smooth repetitive oscillation. To help solve the problems, the students were given the option of using a calculator instead of completing the work entirely by hand. Over half of the students questioned opted to utilize their calculators in order to solve the problems and plot the sine waves.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," says King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the work, King and Robinson interviewed the students about how they approached the material. One student who used the calculator stated that she had trouble remembering the rules for how sine waves operate, and found it generally easier to use a calculator instead. In contrast, however, a student who opted to complete the work without a calculator stated that they couldn't see why anyone would have trouble completing the question, but did admit that it would likely be easier with a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," says King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
Given the small sample size used in the study, it is entirely possible that King's findings are largely anecdotal in how our usage of calculators and understanding of mathematical concepts may positively or negatively correlate. However, King does stress that while all the evidence may not be in, his study does raise important questions regarding how, when and why students choose to use calculators, and in doing so, we may develop a more holistic approach to math instruction | 677.169 | 1 |
Enter your mobile number or email address below and we'll send you a link to download the free Kindle Reading App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
Presents some common problems in mathematics and how they can be investigated using the Mathematica computer system. Problems and exercises include the calendar, sequences, the n-Queens problems, digital computing, blackjack and computing pi. This book is for those that would like to see how Mathematica is applied to real-world mathematics work is best suited for math lovers and programmers that already have Mathematica on their computer (version 1.2 will work, though the book presumes version 2.0 or above). Programmers: If you want to practice writing Function based programming techniques and get away from linear based styles like C, Pascal, etc. this book is great. It is more fun than other books on Mathematica programming since it is oriented around puzzles rather than lessons. Math Lovers: If you love Number Theory and want to see beutiful explorations, this book is great. Certain chapters are more challenging than others, but there is plenty here for hours of exploration!
The book is a mixture of (1) Mathematica programming insights I hadn't seen elsewhere --- taking my skills to a radically new level, (2) broad mathematical insights that, as a career mathematician, I really enjoyed, and (3) extremely narrow-focus material on primes and calendars, etc., that will be of interest to a select few. I was amazed to see that one computation --- finding the first two Wieferich primes --- was said to take 10.5667 seconds on a Sun workstation (in 1991 or before, I assume). On my Pentium II notebook PC, it took 0.04 seconds! | 677.169 | 1 |
The student workbook is 669 pages with 119 lessons, while the four CD-ROMs provide step-by-step audiovisual solutions to every homework and test problem. The CD-ROM's digital gradebook grades answers as soon as they are entered and calculates percentages for each assignment; a softcover answer booklet is also provided. Windows 2000. Teaching Textbooks Grade 4.
CBD Price: $39.95
( In Stock ) Have additional students using Teaching Textbooks Math 4? This student workbook and answer booklet will allow extra students to complete the course in their own book. Perfect for co-ops or siblings! Workbook is 669 pages, softcover, spiral-bound and answer booklet is included. Math 4 CD-ROMs are NOT included; this book is not designed to be used without the CDs. Teaching Textbooks Grade 4.
Perfect for families that already own the Teaching Textbooks Teaching Textbooks Math 4 Extra Workbook & Answer Key, this set includes four CD-ROMs that contain step-by-step audiovisual solutions to each homework and test problem. Topics covered include operations with whole numbers, fractions, and decimals, simple geometry concepts, units of measure, and percents. A digital gradebook grades answers as soon as they are entered and calculates percentages for each assignment. Though this CD-ROM set may technically be used without the workbook, students will then have to write out each problem; won't be able to work away from the computer; and won't receive the written summaries available in the textbook. Teaching Textbooks Grade 4. | 677.169 | 1 |
Enter your mobile number or email address below and we'll send you a link to download the free Kindle Reading App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
Using national and state standards to guide your math program is just a start. You still have to decide how to apply the standards in your curriculum, determine when students should learn different content, and decide which programs and textbooks will help you make math come alive in the classroom. That's where this new ASCD resource comes in. Priorities in Practice: The Essentials of Mathematics Grades 7-12 explores how educators--from classroom teachers to central office administrators--are tackling these major challenges in math education: *Emphasizing algebraic thinking, problem solving, and communication *Relying on research to guide the implementation of new teaching practices *Connecting math activities to larger purposes and everyday experiences *Differentiating instruction based on students' learning styles, interests, and readiness levels *Helping teachers use classroom assessment to guide instruction *Improving math teaching practices through teacher professional development and analysis of student work.
Whether you're working with an established math curriculum or rethinking your whole approach, here's an opportunity to see where your program stands in the context of current trends.
$2395,"ASIN":"1416604138","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":22.75,"ASIN":"1416603697","moqNum":1,"isPreorder":0}],"shippingId":"1416604138::wdcw0FcC6QFXxTrsFzlRZOAkatxSe5xrzxKd6Vc08ADPiKMC6UBoBE8e%2Fg3llrQTFgruXA66QsJMUWS%2FmgB441cvJBlFkIo7p%2BR4653rkEY%3D,1416603697::hXbLsvfTsB9HjU%2BedQ9DEApOLMxY%2FLIoGTNpcIse4A9qGLdxYeU4HvH1PyItfqCfdZIg1mwC4cjsAKMkhMFoxzOAVsTY9u3EqYJ4Diyk5u0aHgyNgZyxFor America to survive and thrive in a constantly evolving global market place it's grade school and high school students must be well grounded in a basic science curriculum. Over the past several decades, science curriculums at the secondary level have been founded upon textbooks, lectures, rote memorization, and lab demonstrations. A direct result of the diminishing global stature of America with respect to science and technology that the demand (as exemplified by the onset of NCLB-mandated science testing) grew for improved educational tools and student performance scores in the sciences. One result is that secondary school science teachers are confronted with an array of challenges that must be met and mastered if recent trends of the downward trend of student and graduate science scores is to be reversed. That is the rationale behind "The Essentials Of Science, Grades 7-12: Effective Curriculum, Instruction, And Assessment" by Rick Allen. Part of the ASCD 'Priorities in Practice' series, "The Essentials of Science, Grades 7-12" instructs the instructors how they can make use of research within the cognitive sciences to foster critical thinking skills and provide students with a deeper understanding and appreciation of scientific concepts and their applications; using 'backward design' to create greater coherence in their curriculums; initiate innovative ideas for implementing scientific inquiry within the classroom environment; apply holistic strategies when addressing complex problems with respect to the problems related to such issues as an achievement gap, equity, and resources of the science classroom; as well as create and implement strategies for dealing with day-to-day as well as NCLB assessments.Read more › | 677.169 | 1 |
The Hindu-Arabic numeral system (1, 2, 3, …) is one of mankind's greatest achievements and one of its most commonly used inventions. How did it originate? Those who have written about the numeral system have hypothesized that it originated in India; however, there is little evidence to support this claim. This book provides considerable... more...
With a foreword by Tim Rice, this book will change the way you see the world. Why is it better to buy a lottery ticket on a Friday? Why are showers always too hot or too cold? And what's the connection between a rugby player taking a conversion and a tourist trying to get the best photograph of Nelson's Column? These and many other fascinating questions... more...
Study Guide for College Algebra and Trigonometry is a supplement material to the basic text, College Algebra and Trigonometry. It is written to assist the student in learning mathematics effectively. The book provides detailed solutions to exercises found in the text. Students are encouraged to use these solutions to find a way to approach a problem.... more... | 677.169 | 1 |
Welcome to the Superstring Store
The
Universe and the Teacup: The Mathematics of Truth
and Beauty by K. C. Cole "Pure
mathematics," Albert Einstein once remarked, "is, in its way, the poetry
of logical ideas." In her book The Universe and the Teacup, Los Angeles
Times science writer K. C. Cole discusses some of the ways this "poetry"
can be used to look at science and other realms of experience. Without
relying on a single equation, Cole's gently humorous prose helps make
mathematics unthreatening to laypeople, enabling them to better understand
the world in which they live.
Overcoming
Math Anxiety by Sheila Tobias The
book that made math anxiety a household word when it was originally published
in 1978 is here updated to reflect new findings of the last 15 years,
including new research demonstrating how little is actually known about
sex differences in brain organization and function. Tobias presents strategies
for math mental health and explains her view that math anxiety is a political
issue and that math competence doesn't have to be determined by gender
or class.
Bob
Miller's Calc for the Clueless: Precalc With Trigonometry by
Robert Miller, Bob Miller The
book presents the topics a high school or college student needs as preparation
for undertaking calculus, vital topics often poorly understood. They are
explained as an encouraging teacher would, in clear, easy-to-understand
terms, answering the questions that most often crop up, anticipating students'
difficulties and confusions like an experienced teacher, eliminating math
anxiety and bridging the "understanding gap" between student and standard
text.
Calc I (Bob Miller's Calc for the
Clueless) by Robert Miller, Bob Miller The
book presents the material in the first semester of a standard college
calculus sequence as an encouraging teacher would, explaining all topics
and techniques in clear, easy-to-understand terms.
Calc II (Bob Miller's Calc for the
Clueless) by Robert Miller, Bob Miller The
book presents the material in the second semester of a standard calculus
sequence as an encouraging teacher would, explaining all topics and techniques
in clear, easy-to-understand terms.
Calc III (Bob Miller's Calc for
the Clueless) by Robert Miller, Bob Miller The
book presents the material in the third semester of a standard calculus
sequence as an encouraging teacher would, explaining all topics and techniques
in clear, easy-to-understand terms.
A
Beginner's Guide to Constructing the Universe: The
Mathematical Archetypes of Nature, Art, and Science by
Michael S. Schneider Schneider,
an education writer and computer consultant, combines science, philosophy,
art, and common sense to reaffirm what the ancients observed: that a consistent
language of geometric design underpins every level of the universe, from
atoms to galaxies, cucumbers to cathedrals. He discusses numerical and
geometric symbolism through the ages, and concepts such as periodic renewal
and resonance.
E : The Story of a Number by Eli Maor Until
about 1975, logarithms were every scientist's best friend. They were the
basis of the slide rule that was the totemic wand of the trade, listed
in huge books consulted in every library. Then hand-held calculators arrived,
and within a few years slide rules were museum pieces. But e remains,
the center of the natural logarithmic function and of calculus. Eli Maor's
book is the only more or less popular account of the history of this universal
constant
Innumeracy: Mathematical Illiteracy and Its Consequences by John
Allen Paulos This
is the book that made "innumeracy" a household word, at least in some
households. Paulos (mathematics, Temple U.) examines many aspects of
popular culture, from stock scams and newspaper psychics to diet and
medical claims to demonstrate the popular misperceptions resulting
from the inability to deal with large numbers, probability, ratios. | 677.169 | 1 |
DU's new maths foundation course goes beyond the regular
Studying games such as lotto and Khul Ja Sim Sim that use the concept of probability, comparing the monthly weather for two successive years and doing a survey on classmates' spending habits is what Delhi University's foundation course called Building Mathematical Ability, is going to teach the first year students from the coming academic year.
The course is part of the new four-year baccalaureate with honours programme where students will have the option of exiting after two years (associate baccalaureate degree), three years (baccalaureate degree) or four years (baccalaureate with honours).
Under the new system, all students will have to take this course as part of the 11 compulsory foundation courses. This includes students from all disciplines irrespective of the degree that they want.
The move has increased apprehension among students as well as some teachers, who are worried that the imposition of the course will create problems for students who are weak in mathematics. The Vice Chancellor, who is overseeing the preparation of the course, believes, however, that the course will equip students with important skills without being very difficult. The penultimate structure of the course is ready. As per this structure, students will primarily focus on familiarity with numbers (mainly properties and patterns of prime numbers and their application), data and patterns, which will include discussions on risk assessment by insurance firms and weather predictions from atmospheric data.
According to statisticians and career councellors, mathematical ability is important to understand trends and use information effectively. | 677.169 | 1 |
Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Five hundred twenty Linear Algebra textbooks are available for sale on ValoreBooks.com, sixteen used from the cheapest price of $53.80, or buy new starting at $161 | 677.169 | 1 |
The theory of uniform distribution began with Hermann Weyl's celebrated paper of 1916. In later decades, the theory moved beyond its roots in diophantine approximations to provide common ground for topics as diverse as number theory, probability theory, functional analysis, and topological algebra. This book summarizes the theory's development from... more...
Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic. In this book the author solves the problem of maintaining the interest of students at both... more...
Ideal for a first course in number theory, this lively, engaging text requires only a familiarity with elementary algebra and the properties of real numbers. Author Underwood Dudley, who has written a series of popular mathematics books, maintains that the best way to learn mathematics is by solving problems. In keeping with this philosophy, the text... more...
"A very stimulating book ... in a class by itself." ? American Mathematical Monthly Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions... more...
Self-contained and comprehensive, this elementary introduction to real and functional analysis is readily accessible to those with background in advanced calculus. It covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, and much more. 350 problems. 1970... more...
This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given ? making the book self-contained in this respect. The author begins with an introductory chapter... more...
This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational... more...
The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development. This volume's great... more...
The three-volume series History of the Theory of Numbers is the work of the distinguished mathematician Leonard Eugene Dickson, who taught at the University of Chicago for four decades and is celebrated for his many contributions to number theory and group theory. This second volume in the series, which is suitable for upper-level undergraduates... more...
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary... more... | 677.169 | 1 |
The student performing at the proficient level demonstrates the ability to recognize, understand, and apply basic mathematical concepts, skills, and terminology to theoretical and real world situations. The student will be able to perform basic computational procedures, apply geometric properties and spatial relationships, interpret data and graphs, apply the concepts and methods of discrete mathematics, and use basic algebraic concepts and processes. The student will also be able to infer, reason, and estimate while problem solving. The student will demonstrate flexibility in selecting a successful process or strategy. The student will demonstrate basic understanding of mathematical concepts through written expression.
Advanced Proficient
The student performing at the advanced proficient level will consistently demonstrate the qualities outlined for proficient performance. In addition, the student will analyze methods for appropriateness, synthesize processes, and evaluate mathematical relationships. The student will demonstrate conceptual understanding by consistently providing clear and complete explanations. The student abstracts concepts for use in other applications and successfully forms conjectures. | 677.169 | 1 |
A lesson designed to teach students to define and demonstrate the six trigonometric functions; and to facilitate using the three most frequently used trigonometric functions. From the Algebra and Trigonometry section of a collection of almost 200 single concept lessons by the Science and Mathematics Initiative for Learning Enhancement. | 677.169 | 1 |
21Introductory Algebra Everyday Explorations
Summary
Alice Kaseberg's respected Introductory Algebra: Everyday Explorations, Fourth Edition, helps students build confidence in algebra. This text's popularity is attributable to the author's use of guided discovery, explorations, and problem solving, all of which help students learn new concepts and strengthen their skill retention. Known for an informal, interactive style that makes algebra more accessible to students while maintaining a high level of mathematical accuracy, Intermediate Algebra includes a host of teaching and learning tools that work together for maximum flexibility and a high student success rate. With the Fourth Edition, instructors have access to an Instructor's Annotated Edition that provides additional examples, as well as a robust Instructor's Resource Manual, algorithmic computerized testing, and an extensive online homework system.
Table of Contents
Algebraic Representations
Problem-Solving Steps and Strategies
Numeric Representations
Verbal Representations
Symbolic Representations
Visual Representations: Rectangular Coordinate Graphs
Operations with Real Numbers and Expressions
Addition and Subtraction with Integers
Multiplication and Division with Positive and Negative Numbers
Properties of Real Numbers Applied to Simplifying Algebraic Fractions and Adding Like Terms
Exponents and Order of Operations
Unit Analysis and Formulas
Inequalities, Intervals, and Line Graphs
Solving Equations and Inequalities in One Variable
Linear Equations in One and Two Variables
Solving Equations with Algebraic Notation
Solving Equations with Tables, Graphs, and Algebraic Notation
Solving Linear Equations with Variables on Both Sides of the Equation
Solving Linear Inequalities in One Variable
Formulas, Functions, Linear Equations, and Inequalities in Two Variables
Solving Formulas
Functions and Graphs
Linear Functions: Slope and Rate of Change
Linear Functions: Intercepts and Slope
Linear Equations
Inequalities in Two Variables
Ratios, Rates, and Proportional Reasoning
Ratios, Rates, and Percents
Proportions and Proportional Reasoning
Proportions in Similar Figures and Similar Triangles
Averages
Writing Equations from Word Problems with Quantity-Rate Tables
Systems of Equations and Inequalities
Solving Systems of Equations with Graphs
Setting Up Systems of Equations
Solving Systems of Equations by Substitution
Solving Systems of Equations by Elimination
Solving Systems of Equations in Three Variables
Solving Systems of Linear Inequalities by Graphing
Polynomial Expressions and Integer Exponents
Operations on Polynomials
Multiplication of Binomials and Special Products
Factoring Trinomials
Factoring Special Products and Greatest Common Factors
Exponents
Scientific Notation
Squares and Square Roots: Expressions and Equations
Pythagorean Theorem
Square Root Expressions and Properties and the Distance Formula
Solving Square Root Equations and Simplifying Expressions
Graphing and Solving Quadratic Equations
Solving Quadratic Equations by Taking the Square Root or by Factoring
Solving Quadratic Equations with the Quadratic Formula
Range, Box and Whisker Plots, and Standard Deviation
Rational Expressions and Equations
Rational Functions: Graphs and Applications
Simplifying Rational Expressions
Multiplication and Division of Rational Expressions
Finding the Common Denominator and Addition and Subtraction of Rational Expressions | 677.169 | 1 |
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
The main objective of this work is to develop a thorough understanding of the structure of graphs and the techniques used to analyze problems in graph theory. Fundamental graph algorithms are also included. Examples and over 600 exercises - at various levels of difficulty - guide students.
Hardcover, ISBN 0132278286 Publisher: Pearson, 1995 Usually dispatched within 1-2 business days, NEW Book, unused. Sent Airmail from New York. Please allow 7-15 Business days for delivery. Excellent Customer Service.
Used books: 1 - 25 of 54
#
Bookseller
Notes
Price
1.
Anybookltduk via United Kingdom
Hardcover, ISBN 0132278286 Publisher: Prentice Hall College Div, 1995 No dust jacket.
Hardcover, ISBN 0132278286 Publisher: Prentice Hall College Div, 1995 Good. US Edition. May include moderately worn cover, writing, markings or slight discoloration. SKU:97801322782870132278287-4-0
Hardcover, ISBN 0132278286 Publisher: Prentice Hall College Div, 1995 Very Good. US Edition. Has minor wear and/or markings. SKU:97801322782879780132278287-3-0
Hardcover, ISBN 0132278286 Publisher: Prentice Hall, 1996 Used - Good Good . Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780132278287-4 College Div, 1995 Like New. US Edition. Almost new condition. SKU:9780132278287-2 Like New, Usually ships in 24 hours, Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Almost new condition. SKU:9780132278287-2-0
Hardcover, ISBN 0132278286 Publisher: Prentice Hall, 1996 Used - Very Good Very good . Hardcover. Has minor wear and/or markings. SKU: 9780132278287-3 Fine/Like New Fine . Hardcover. Almost new condition. SKU: 9780132278287-2 Poor. This is an ex-library book and may have the usual library/used-book markings inside.This book has hardback covers. In poor condition, suitable as a reading copy. No dust jacket. (OTHER)
Hardcover, ISBN 0132278286 Publisher: Prentice Hall College Div, 1995 Used - Good, Usually ships in 1-2 business days, This Book is in Good Condition. Used Copy With Light Amount of Wear. 100% Guaranteed. | 677.169 | 1 |
Calculus and Vectors: Content and Reporting Targets
Mathematical Processes across all strands: Problem Solving, Reasoning and Proving, Reflecting, Selecting Tools and Computational Strategies,
Connecting, Representing, and Communicating
Unit 1 Unit 2 Unit 3
Explore rate of "flow" problems Derivative Functions from First Derivative Functions: Properties and
using non-algebraic means Principles their Applications
Explore contexts and solve problems Recognize numerical and graphical Investigate properties of derivatives
where one needs to know rate of representations of increasing and (power rule, chain rule as change of
change at a specific point, using decreasing rates of change. scale and as patterning, no quotient
verbal and graphical representations Use patterning and reasoning to rule use product rule, Sample
of the function. Include examples determine that there is a function that Problem: Examine the relationship
where mechanical tools are not describes the derivative at all points. between the derivative of a function
readily available, e.g., income flow, For polynomial, rational and radical and the derivative of its inverse.
garbage accumulation rate. functions, determine, using limits, Generalize the power rule for all
Analyse rates of change and provide the algebraic representation of the rational powers).
qualitative solutions to problems, derivative at any point. Apply these properties to form
e.g., increase, decrease, tend towards Application of Derivatives of derivatives of functions and simple
something. Polynomial Functions combinations of functions (no
Standardize the process of finding Graph, without technology, the simplification of derivatives formed
instantaneous rate of change at a derivative of polynomials with given outside of problem-solving contexts).
particular point equations.
Apply a standard process for Given the graph of the derivative,
determining instantaneous rate of sketch the original polynomial.
change of a function at a specific Derivative Functions Through
point on its graph. Investigation
For polynomial, simple rational, and Through investigation, determine the
radical functions, form, evaluate, and algebraic representation of the
interpret the first principles definition derivative at any point for
of the derivative, using a fixed exponential, logarithmic and
(numerical) value of "a," e.g., What sine/cosine functions.
is the graphical significance of Applications of Derivatives, Given
f (4h ) f (4) Algebraic Representations
lim h
?
h0 Pose and solve problems that require
identifying conditions that result in a
desired rate of change.
Unit 4 Unit 5 Unit 6
Applications of Derivatives in Rate of Representing Vectors Representing Lines and Planes
Change and Optimization Problems, Introduce vectors in 2-D and 3-D. Parametric equations of functions.
Including Those Requiring Represent vectors geometrically and Represent lines and planes in a
Modelling algebraically. variety of ways.
Solve rate of change and Operate with vectors. Find intersections of two planes.
optimization problems given Solve problems involving vectors. Find intersections of three planes.
algebraic models.
Solve rate of change and
optimization problems requiring the
creation of an algebraic model (more
variety in problems to get at various
types of algebraic simplification and
analysis).
Solve problems calling for the
modelling of the rate of change flow
problems), not necessarily finding
the original function but just a
property of it, e.g., point of
inflection.
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 1
Rationale
Teaching Calculus before Vectors
Provides a natural flow from Advanced Functions to this course and students build on prior knowledge
Calculus problems are situated in a two-dimensional context while vector problems progress from two-
dimensions to three-dimensions.
The introduction of parametric equations can help make connections.
Focusing Unit 1 on rates of change problems:
Provides an opportunity for students to investigate a variety of real-world contexts involving change; develops
an appreciation of the need to analyse rates of change
Establishes a need for algebraic representations of rates of changes, e.g., the need for precision, for information
at many different data points
Separating Units 1, 2, and 3:
Introduces abstract concepts at a developmentally appropriate pace
Provides opportunities to connect each abstract concept to problem solving situations
Provides the time for students to investigate and consolidate conceptual understanding of rates of change,
derivatives and limits, prior to combining these concepts with algebraic procedures
Graph analysis within Unit 2
Curriculum revisions focus curve sketching on polynomials only.
Graph analysis can be one of the strategies students use to confirm the reasonableness of solutions to problems in
Unit 4.
Problems requiring modelling congregated in Unit 4
These problems require students to choose from amongst all possible function types when formulating a
mathematical model.
The problem solving in Unit 4 provides a segue from calculus to vectors.
Numbers of Units
2 1
It is recommended that calculus concepts be taught in of the time available, and that vectors be taught in of
3 3
the time available.
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 2
Calculus and Vectors
Year Outline – Planning Tool
P Number of pre-planned lessons (including instruction, diagnostic and formative
assessments, summative assessments other than summative performance tasks)
J Number of jazz days of time (instructional or assessment)
T Total number of days
SP Summative performance task
Cluster of Curriculum
Unit Overall Expectations P J T SP
Expectations
1 Explore rates of change in context to A1 demonstrate an understanding of rate of
consolidate understanding from change by making connections between
Advanced Functions average rate of change over an interval and
instantaneous rate of change at a point,
Connect instantaneous rates of change using the slopes of secants and tangents and
with the derivative the concept of the limit;
8 1 9
Connect the characteristics of the A2 graph the derivatives of polynomial,
instantaneous rate of change with the sinusoidal, and exponential functions, and
characteristics of the function make connections between the numeric,
graphical, and algebraic representations of
a function and its derivative.
2 Investigate connections graphical and A2 graph the derivatives of polynomial,
numerically between the graph of a sinusoidal, and exponential functions, and
function and its derivative make connections between the numeric,
graphical, and algebraic representations of
Determine, using limits, the algebraic a function and its derivative;
representation of derivatives
A3 verify graphically and algebraically the
Determine and apply the power, chain rules for determining derivatives; apply
and product rules these rules to determine the derivatives of
polynomial, sinusoidal, exponential,
Apply power, product and chain rules to rational, and radical functions; and simple
rational and radical functions combinations of functions; and solve 18 2 20
related problems.
Develop the derivatives of f x e x ,
f x sin x and f x cos x
Explore the relationship between
f x e x , and f x ln x
Solve problems involving instantaneous
rates of change
3 Examine the relationship between first B1 make connections, graphically and
and second derivatives and the original algebraically, between the key features of a
polynomial or rational function function and its first and second
derivatives, and use the connections in
Sketch curves of polynomial or rational curve sketching;
functions given information or equations
8 1 9
B2 solve problems, including optimization
Apply the properties of derivatives to problems that require the use of the
real-world problems concepts and procedures associated with
the derivative, including problems arising
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 3
Cluster of Curriculum
Unit Overall Expectations P J T SP
Expectations
from real-world applications and involving
the development of mathematical models.
4 Solve rate of change and optimization B2 solve problems, including optimization
problems in a wide variety of contexts problems that require the use of the
using properties of derivatives concepts and procedures associated with
the derivative, including problems arising 11 2 13
Collect data, create mathematical from real-world applications and involving
models and solve problems arising from the development of mathematical models.
real-world contexts
5 Introduce vectors in two-space and C1 demonstrate an understanding of vectors in
three-space two-space and three-space by representing
them algebraically and geometrically and
Represent vectors geometrically and by recognizing their applications;
algebraically
C2 perform operations on vectors in two-space
16 3 19
Determine vector operations and and three-space, and use the properties of
properties these operations to solve problems,
including those arising from real-world
Solve problems involving vectors applications.
6 Represent equations of lines in two- C3 distinguish between the geometric
space and three-space using a variety of representations of a single linear equation
forms or a system of two linear equations in two-
space and three-space, and determine
Investigate intersections of planes different geometric configurations of lines
and planes in three-space;
Solve problems involving planes arising
from real-world contexts C4 represent lines and planes using scalar,
vector, and parametric equations, and solve
problems involving distances and
intersections.
Summative Performance Tasks 12
Total Days 64 9 73 85
The number of prepared lessons represents the lessons that could be planned ahead based on the range of student readiness,
interests, and learning profiles that can be expected in a class. The extra time available for "instructional jazz" can be taken a few
minutes at a time within a pre-planned lesson or taken a whole class at a time, as informed by teachers' observations of student
needs.
The reference numbers are intended to indicate which lessons are planned to precede and follow each other. Actual day numbers
for particular lessons and separations between terms will need to be adjusted by teachers.
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 4
Appendix A: Electronic Learning Objects to Support MCV4U
E-Learning Ontario Web Site: MGA4U Unit 3 Vectors
Activity 2: Vector Laws
The last applet on Vector Laws allows the user to investigate the commutative, associative, distributive
properties of two-space vectors in geometric form.
At the bottom of Activity 2 is a link to the University of Guelph's Physics department where a tutorial for
vectors is provided.
Activity 3: Applications of Geometric Vectors
The second applet in the Velocity Java Applets allows the user to investigate the resultant vector for a
boat crossing a river. The user controls two-space vectors in geometric form for the boat's velocity and
the current.
Activity 5: Algebraic Vectors
The first applet allows users to interactively explore the connections between geometric and algebraic
forms of vectors in two-space.
At the end of this activity is a link to a three-space Graphing Tool that allows students to graph points,
lines, and planes in various forms.
Activity 6: Operations with Algebraic Vectors
There are four applets on addition of vectors, scalar multiplication, unit vectors, and position vectors.
They allow the user to interactively manipulate two-space vectors.
E-Learning Ontario Web Site: MGA4U Unit 5 Vector Methods with Planes and Lines
Activity 1: Equations of Lines in two-space
There are five guided and three interactive applets on forms of vector equations, how to convert between
forms, distance from a point to a line.
Activity 3: Intersection of Lines
There are two guided applets on intersection of lines in two-space and three-space.
Activity 5: Equations of Planes
There are four guided applets on the forms of equations of planes and how to convert between forms.
Activity 6: Intersection of a Line and a Plane
There is one guided applet.
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 5
Appendix A: Electronic Learning Objects (continued)
to Support MCV4U
Activity 7: Intersection of Planes
There is one guided applet on solving systems of planes algebraically.
Activity 8: Task: X, Y, and Z Factor
An open ended task using the three-space Graphing Tool allows students to consolidate vector concepts.
Vector Applets on the Web
NCTM
This site has two applets. The first illustrates the components of a vector to control a car. The
user interactively controls the speed and direction. The second illustrates vector addition for
an aircraft flying that is acted upon by wind. The user controls the speed and direction of both
the aircraft and wind.
Syracuse University
This applet demonstrates cross product of two vectors in three-space. It allows users to
interactively change the vectors and see the resulting cross-product. The two vectors are
limited to one plane but the plane can be moved to different viewing angles.
International Education Software
This Japanese site has a collection of applets that cover a wide variety of two-space and
three-space vector topics. The controls are not very user-friendly but there are topics covered
here like vector forms of lines in two-space and three-space that are not covered on other
sites.
Professor Bob's Physics Lab (Rob Scott)
This interactive site has flash applets on various Physics topics. Some topics such as Milliken
and Momentum labs allow students to apply vector concepts.
B.Surendranath Reddy (Physics Teacher in India)
This site has several applets that can be used in MCV4U. For vectors there are applets for
addition, cross product of vectors, converting between Cartesian and directed line segment
forms and several kinematics applets. For calculus there are applets for instantaneous speed
and velocity.
TIPS4RM: Calculus and Vectors (MCV4U) – Overview 2008 6 | 677.169 | 1 |
Algebra I Essentials for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
A helpful review guide for the 300,000 Texas high school freshmen who annually needand#160;to pass the exam in order to graduate Relevant to all Texas high school students needing to take the Algebra I end-of-course exam, this Quick Review includes and... (read more)
CliffsNotes Algebra I Practice Pack [With CDROM] (CliffsNotes)
by Mary Jane Sterling Publisher Comments
About the Contents: Pretest Helps you pinpoint where you need the most help and directs you to the corresponding sections of the book Topic Area Reviews Math Basics Numbers (Signed Numbers and Fractions) Linear Equations and Algebraic Fractions... (read more)
Essentials of College Algebra, Alternate Edition
by Lial Publisher Comments
Essentials of College Algebra , Updated Edition, 1/e, has been specifically designed to provide a more compact and less expensive alternative to better meet the needs of colleges whose algebra courses do not include the more advanced topics. The authors... (read more)
Modern Algebra 2 Volumes Bound As One
by Seth Warner Publisher Comments
This standard text, written for junior and senior undergraduates, is unusual in that its presentation is accessible enough for the beginner, yet its thoroughness and mathematical rigor provide the more advanced student with an exceptionally... (read more)
High School Pre-Calculus Tutor (High School Tutor Series)
by Rea Publisher Comments
REAs High School Tutors is a series of useful and practical study guides. Each High School Tutor provides practice and understanding of various math, science, and history subjects, making them challenging and interesting. Hundreds of solved problems... (read more)
Excursions in Modern Mathematics 5TH Edition
by Peter Tannenbaum Publisher Comments
This collection of "excursions" into modern mathematics is written in an informal, very readable style, with features that make the material interesting, clear, and easy-to-learn. It centers on an assortment of real-world examples and... (read more)
The Geometry of Art and Life
by Matila Costie Ghyka Publisher Comments
Is everything chaos and chance, or is there order, harmony, and proportion in human life, nature, and the finest art? Can one find a natural aesthetic that corresponds to a universal order? If so, what importance can it have for the scientist, artist | 677.169 | 1 |
Product description
This paperback edition is not available in the U.S. and Canada.This text offers a comprehensive presentation of the mathematics required to tackle problems in economic analysis. To give a better understanding of the mathematical concepts, the text follows the logic of the development of mathematics rather than that of an economics course. The only prerequisite is high school algebra, but the book goes on to cover all the mathematics needed for undergraduate economics. It is also a useful reference for graduate students. After a review of the fundamentals of sets, numbers, and functions, the book covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics. To develop the student's problem-solving skills, the book works through a large number of examples and economic applications. This streamlined third edition offers an array of new and updated examples. [Some of the lengthier proofs and examples have been moved to the book's Web site. This combination of formats allows the authors to add content without expanding the physical size of the book. The book and the Web material are integral to each other; examples and figures on the Web are cross-referenced in the text. A student solutions manual will be available in e-book form. Instructors will be able to access online instructor's material that includes Power Point slides.
Author information
Michael Hoy is a faculty member in the Economics Department at the University of Guelph. John Livernois is a faculty member in the Economics Department at the University of Guelph, Ontario. Chris McKenna is a faculty member in the Economics Department at the University of Guelph, Ontario. Ray Rees is a faculty member at the Ludwig Maximilians University, Munich. Thanasis Stengos is a faculty member in the Economics Department at the University of Guelph, Ontario.
Review quote
"Mathematics is the language of economics, and this book is an excellent introduction to that language." George J. Mailath , Walter H. Annenberg Professor in the Social Sciences and Professor of Economics, University of Pennsylvania "While there are many mathematics texts for economics available, this one is by far the best. It covers a comprehensive range of techniques with interesting applications, and the numerous worked examples and problems are a real bonus for the instructor. Teaching a course with this book is enjoyable and easy." Kevin Denny , University College Dublin | 677.169 | 1 |
9780201726343
ISBN:
0201726343
Edition: 5 Pub Date: 2003 Publisher: Pearson
Summary: This text is organised into 4 main parts - discrete mathematics, graph theory, modern algebra and combinatorics (flexible modular structuring). It includes a large variety of elementary problems allowing students to establish skills as they practice.
Ralph P. Grimaldi is the author of Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, published 2003 under ISBN 9780201726343 and 0...201726343. Seven hundred fourteen Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition textbooks are available for sale on ValoreBooks.com, ninety eight used from the cheapest price of $82.99, or buy new starting at $166 Binding is loose. Ships same day or next bu [more]
Book has signs of cover wear. Inside pages may have highlighting, writing and/or underlining. Used books may have stickers on them. Binding is loose | 677.169 | 1 |
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.
8th grade (U.S.)
8th grade is all about tackling the meat of algebra and getting exposure to some of the foundational concepts in geometry. If you get this stuff (and you should because you're incredibly persistent), the rest of your life will be easy. Okay, maybe not your whole life, but at least your mathematical life. Seriously, if you really get the equations and functions stuff we cover here, most of high school will feel intuitive, even relaxing.
(Content was selected for this grade level based on a typical curriculum in the United States.)
A strong contender for coolest symbol in mathematics is the radical. What is it? How does it relate to exponents? How is the square root different than the cube root? Learn all about square roots and cube roots in this tutorial.
The values of irrational numbers can't be written perfectly as decimals or fractions. However, we can approximate them, which is usually good enough. In this tutorial, we learn how to approximate and compare irrational numbers.
Learn all about negative exponents. (It's normally a bad idea to hang around with negative people or do negative things, but it's totally cool to associate with negative exponents. They're surprisingly useful!)
Scientists and engineers often deal with super huge numbers like 6,000,000,000,000,000,000,000 and super small numbers like 0.0000000000532. How can they do this without tiring their hands out? How can they look at a number and understand how large or small it is without counting the digits? The answer to both questions: scientific notation. | 677.169 | 1 |
9780883856Ingenuity in Mathematics (Anneli Lax New Mathematical Library)
The nineteen essays here illustrate many different aspects of mathematical thinking. The author is very well-known for his best-selling books of problems; in this volume he seeks to share his appreciation of the elegant and ingenious approaches used in thinking about even elementary mathematics. Standard high school courses in algebra and geometry furnish a sufficient basis for understanding each essay. Topics include number theory, geometry, combinatorics, logic and probability, and the methods used often involve an interaction between these disciplines. Some of the essays are easy to read, others more challenging; some of the exercises are routine, others lead the reader deeper into | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.