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Algebra 2 - 03 edition
Summary: Applications with "Real" Data Since the graphics calculator is recommended, students experience excitement as they use "real" data in Algebra 2. Students investigate and extend relevant applications through engaging activities, examples, and exercises. Graphics Calculator Technology In Algebra 2, the graphics calculator is an integral tool for presenting, understanding, and reinforcing concepts. To assist student...show mores in using this tool, a detailed keystroke guide is provided for each example and activity at the end of each chapter. Functional Approach Algebra 2 examines functions through multiple representations, such as graphs, tables, and symbolic notation. Working with transformations (investigating how functions are related to each other and their parent functions) prepares students for advanced courses in mathematics by developing an extensive, workable knowledge of functions. ...show less
2003 Hardcover Brand New 2003 Copyright. Book number written on inside front cover. Book has never been used. Multiple copies available. For quick service, please consider Expedited shipping since s...show moretandard delivery may range from 4-18 business days. Thank you. ...show less
$54.13 Minor wear on covers. Best buy. Shipped promptly and pa...show moreckaged carefully
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Elementary Algebra : Graphs And Models - 05 edition
Summary: Geared toward helping students visualize and apply mathematics, Elementary Algebra: Graphs and Models uses illustrations, graphs, and graphing technology to enhance students' mathematical skills. This is accomplished through Interactive Discoveries, Algebraic/Graphical Side-by-Sides, and the incorporation of real-data applications. In addition, students are taught problem-solving skills using the Bittinger hallmark five-step problem-solving process coupled with Connec...show moreting the Concepts and Aha! exercises. And, as you have come to expect with any Bittinger text, we bring you a complete supplements package that now includes an Annotated Instructor's Edition and MyMathLab, Addison-Wesley's online course solution
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many teaching aspects and experiences to her work with an engineering degree,
tutoring since 1989, homeschooling since 1997, and classroom experience.
This Study Guide provides a thorough treatment of expectations as outlined in the "Numbers and Numeration" and the
"Measurement" strands of the curriculum, including topics such as, integers, fractions, decimals, expanded form, percentages,
ratios, decimals, sales tax, discount, simple interest, rates, circles, right prisms, and cylinders.
Please note that this is Book 1, and Book 2 is needed to complete the course of studies for Grade 8.
Retail Price $25.95
many teaching aspects and experiences to her work with an engineering
degree, tutoring since 1989, homeschooling since 1997, and classroom
experience.
Math
8 Workbook 1 is a companion to Study Guide 1This Workbook provides a
thorough treatment of expectations as outlined in the "Numbers
and Numeration" and the "Measurement" strands of the curriculum,
including topics such as, integers, fractions, decimals, expanded
form, percentages, ratios, decimals, sales tax, discount, simple
interest, rates, circles, right prisms, and cylinders.
Please
note that this is Book 1, and Book 2 is needed to complete the
course of studies for Grade 8. It is recommended that the Workbooks
are used on a daily basis to build skills in a consistent manner,
and to enhance classroom learning.
Math
8 Workbook 2 is a companion to Study Guide 2In
Ontario, the overall expectations outlined by the Ministry of
Education are very similar for both Math 9 Applied
and Math 9 Academic courses, so this book can
be used for either course. While the overall stress of this study
guide is towards helping the Applied student succeed,
we have also included some multi-step questions and calculations
of prisms and cylinders to comply with all of the specific expectations
of the Academic Program of Studies. Throughout the book are helpful
hints to jog your memory, methods to make learning concepts much
easier, and a series of exercises, pop quizzes and final exams.
This means you will become increasingly confident and prepared
when it comes to school tests. Supplementary exercises are included
for students who want extra reinforcement. This book was recently
revised by a 39 year Ontario math professional teacher as an added
benefit to students of his many years of teaching experience including
the new Ontario math curriculum. All of these supplementary exercises
compliment the present Math 9 Text Ontario edition of
Addison-Wesley Mathematics 9 (ISBN 0-201-614737- 5).
Exercises
in the Math 9 Workbook compliment the Math 9by both a retired professional math teacher and an honours mathematics
graduate who is also involved in tutoring students in the new
Ontario Math curriculum. Once again, this will give you an excellent
overview of any type of question you may be tested on during your
school year. The drill and practice questions in this workbook
were derived from a combination of questions received from a number
of Ontario schools and from those created by both authors. Students
receive a great insight into the variety of questions and exercises
presently used by teachers in Ontario. We recommend that you use
the EZ Learning Solutions workbooks on a daily basis to build
skills and enhance your classroom learning.
Exercises
in the Math 9 Workbook 2, Fermi Math Problem Solving, follow new
recommendations from the Ministry of Education for Math 9 in Ontario
schools. The Fermi Math 9 problem solving workbook provides unique
step by step problem solving procedures that combine math and
science calculations. Teachers, students, and parents will appreciate
not only following the scientific methodology but the ability
to integrate many disciplines in real life applications following
the Enrico Fermi method of estimation used to develop strong analytical
and problem solving skills. Fermi incredible estimation skills
included activities like estimating the number of pianos in Chicago
or the number of grains of sand on the beaches of the world. These
labs were designed by a retired professional math who has taught
in Ontario Schools for over 38 years and presented by your favourite
author of EZ Learning Solutions study guides and workbooks.
The directions for these labs recommend the TIPS (Targeted Implementation
& Planning) Think/Pair/Share format on how to complete these
mathematical problem-solving labs. TIPS activities are designed
to include the subjects of math, science, geography, business
and probability while making observations and calculations with
decimals, integers, fractions, percents, rations, algebra and
formulas.
This is not a high-tech solution but a "back to basics",
student-oriented method of directed learning designed to enable
students to meet the requirements of the difficult new curriculum.
Our approach is "Get to the point and get on with the lesson".
This guide covers concepts taught in the Ontario Math Academic Curriculum
such as quadratic functions, linear systems, multi-step problems,
geometric figures, trigonometry, and factoring. It contains helpful
hints to jog your memory, methods to make learning concepts much
easier, and a series of exercises, quizzes and final exams. The
"Chalk Talk" areas provide the student with useful memory
aids and a helpful overview of what was just learned. This all means
that students will be more confident and better prepared at exam
time. Supplementary exercises are included for students who want
extra practice. All of these supplementary exercises compliment
the present Ontario edition of Nelson Mathematics 10 (ISBN
017-615704-2)and Principles of Mathematics 10 by Addison
Wesley (ISBN 0-201-71122-2).
Exercises
in the Math 10 Workbook compliment the Math 10to give you an excellent overview of any type of question you
may be tested on during your school year. The drill and practice
questions in this workbook were derived from a combination of
questions received from a number of Ontario schools to those created
by an Ontario Professional Teacher with 39 years of experience.
Never again will you get such an insight into the variety of questions
and exercises presently used by teachers in Ontario. We recommend
that you use the EZ Learning Solutions workbooks on a daily basis
to build skills and enhance your classroom learning.
Exercises
in the Math 10 Workbook 2 complete the second half of the school
year following the Math 10 Academic Workbook 1. All EZ Learning
workbooks follow the Math 10 Study Guide and the program of studies
providing both much needed summative questions as well as many
classic drill and practice questions. Once again you not only
have a great variety of questions but also complete solutions
to give you that extra edge to excel beyond the average student.
The questions were designed to give you an excellent overview
of any type of question you may be tested on during your school
year. The drill and practice questions in this workbook were derived
from a combination of questions received from a number of Ontario
schools to those created by an Ontario Professional Teacher with
39 years of experience. Never again will you get such an insight
into the variety of questions and exercises presently used by
teachers in Ontario. We recommend that you use the EZ Learning
Solutions workbooks on a daily basis to build skills and enhance
your classroom learning.
The
unique aspect of this book is that it was written by both a professional
teacher who has continually chosen to work with students having
the greatest difficulty with the new Ontario math curriculum and
by a teacher who was working with an ADD child. This is not a
high-tech solution but a "back to basics", student-oriented
method of directed learning designed to enable students to be
successful with the requirements of this difficult new curriculum.
Our approach is "Get to the point and get on with the lesson".
This study guide covers concepts taught in the new Ontario Math
Applied Curriculum including proportional reasoning: ratios/rates/percent,
introduction to Trigonometry, linear functions and linear systems,
and quadratic functions and factoring. It contains helpful hints
to jog your memory, methods to make learning concepts much easier,
and a series of exercises and quizzes, and a final exam for home
and school students. The "Chalk Talk" areas provide
the student with useful memory aids and a helpful overview of
what was just learned. This all means that students will be confident
and better prepared at exam time. Supplementary exercises are
included for students who want extra practice. All of these supplementary
exercises compliment the present Ontario edition of Mathematics,
Applying the Concepts by McGraw-Hill Ryerson (ISBN 0-07-086490-X)
and Foundations of Mathematics 10 by Addison Wesley (ISBN 0-201-68484-5).
In
this book you will find a very thorough review of grade 9, 10,
and 11 academic mathematics to help the University/College and
University student with the fundamentals in a straightforward
style. Both professionals felt
a great need to provide a spiral learning situation building on
past skills due to the length of time between semesters and the
need to strengthen these past skills in order to master the new
Ontario Math 11 curriculum.This study guide covers all key concepts
taught in the new Ontario Math University/College and University
Curriculum including a review of essential skills, rational expressions
and complex numbers, reciprocal functions, trigonometric ratios,
modeling periodic functions, trigonometric functions and radians,
and trigonometric graphs and transformations. Please note that
this is only the first book of the University/College and University
program of studies and you will need Book 2 to complete these
courses. The order of the topics covered in this book are in the
usual order that most Math professionals teach these courses in
Ontario schoolsThis
is Book 2 of the complete Math 11 University/College and University
high school programs. In this book you will find a very thorough
review of grade 9, 10, and 11 academic mathematics to help the
University/College and University level student with the fundamentals
in a straightforward style completing the second half of this
course but with that academic edge
with university ready skills. Both professionals felt a great
need to provide a spiral learning situation building on past skills
due to the length of time between semesters and the need to strengthen
these past skills in order to master the new Ontario Math 11 curriculum.
This study guide covers all key concepts taught in both the new
Ontario University/College and University Math 11 Curriculum including
solving trigonometric equations, trigonometric identities, and
solving quadratic trigonometric equations, Sequences and Series,
and ConicsMath
12 Advanced Functions and Introductory Calculus, University Preparation Book 1 - Available September 30th, 2005
Following
the Ontario program of studies this study guide builds on students
past abilities using functions then introduces the basic concepts
and skills of calculus. In this book you will find a very thorough
step by step understanding to help the University preparation
student with the fundamentals of Calculus after investigating
and applying the properties of polynomials and exponential and
logarithmic functions in a straightforward style. This book was
written by Bruce Mullens, after 40 years of teaching Mathematics
in Ontario and stands as a testament to his ability to understand
the type of problems that constantly evolve throughout this difficult
course. Bruce consulted with many colleagues and review tests,
quizzes, and final exams reflect the level of understanding taught
in Ontario schools and required to have success in University.
The first half of the school term covered in Book 1 thoroughly
completes all required expectations by presenting lessons in the
following: investigating the graphs of Polynomial functions, manipulating
algebraic expressions, understanding the nature of exponential
growth and decay, applying logarithmic functions, understanding
rates of change, understanding the graphical definition of the
derivative and connecting derivatives and graphs.
Please note that this is only the first book of the University
program of studies and you will need Book 2 to complete this course.
The order of the topics covered in this book are in the usual
order that most Math professionals teach these courses in Ontario
schools
Math
12 Advanced Functions and Introductory Calculus, University Preparation
Book 2 - Available February 15, 2006
Following
the Ontario program of studies this study guide builds on students past abilities using functions then
introduces the basic concepts and skills of calculus. In this book you will find a very thorough step by step
understanding to help the University preparation student with the fundamentals of Calculus after investigating
and applying the properties of polynomials and exponential and logarithmic functions in a straightforward style.
This book was written by Bruce Mullen, after 40 years of teaching Mathematics in Ontario and stands as a testament
to his ability to understand the type of problems that constantly evolve throughout this difficult course.
Bruce consulted with many colleagues and review tests, quizzes, and final exams reflect the level of understanding
taught in Ontario schools and required to have success in University.
The second half of the school term covered in Book 2 thoroughly completes all required expectations by presenting lessons
in the following: understanding the first-principles definition of the derivative, determining derivatives, determining
the derivatives of exponential and logarithmic functions, using differential calculus to solve problems, sketching the
graphs of polynomial, rational, and exponential functions, and using calculus techniques to analyse models of functions.
Please note that this is only the second book of the University program of studies and you will need Book 1 to complete
the first half of this course. The order of the topics covered in this book are in the usual order that most Math professionals
teach these courses in Ontario schools
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A significant portion (my observation was about 20-30% at Berkeley, which means it must approach 100% at some schools) of first year students in the US do not understand multiplication. They do understand how to calculate $38 \times 6$, but they don't intuitively understand that if you have $m$ rows of trees and $n$ trees in each row, you have $m\times n$ trees. These students had elementary school teachers who learned mathematics purely by rote, and therefore teach mathematics purely by rote. Because the students are very intelligent and good at pattern matching and at memorizing large numbers of distinct arcane rules (instead of the few unifying concepts they were never taught because their teachers were never taught them either), they have done well at multiple-choice tests.
These students are going to struggle in any calculus course or any discrete math course. However, it is easier to have them all in one place so that one instructor can try to help all of them simultaneously. For historical reasons, this place has been the calculus course.
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Home
Welcome to the home page for Topics in Algebra, Trigonometry and Pre-calculus. This course is designed to fill in the holes in your mathematical training, and prepare you to take more advanced courses, such as Calculus or Satistics, or to study for the GRE. There is no official textbook for the course. Instead, each unit has a set of notes explaining the material, and a set of exercises. In addition, you might find the following useful:
Algebra and Trigonometry Review: notes made by the Vanderbilt University Mathematics department. It has many exercises and answers for the odd numbered problems.
An Algebra RefresherFrom the foreword: The material in this refresher course as been designed to enable you to prepare for your university mathematics programme. When your programme starts you will find that your ability to get the best from lectures and tutorials, and to understand new material, depends crucially upon having a good facility with algebraic manipulation. We think that this is so important that we are making this course available for you to work through before you come to university
Math review for the GRE:From the introduction: The Math Review is designed to familiarize you with the mathematical skills and concepts likely to be tested on the Graduate Record Examinations General Test. This material, which is divided into the four basic content areas of arithmetic, algebra, geometry, and data analysis, includes many definitions and examples with solutions, and there is a set of exercises (with answers) at the end of each of these four sections. Note, however, this review is not intended to be comprehensive. It is assumed that certain basic concepts are common knowledge to all examinees. Emphasis is, therefore, placed on the more important skills, concepts, and definitions, and on those particular areas that are frequently confused or misunderstood. If any of the topics seem especially unfamiliar, we encourage you to consult appropriate mathematics texts for a more detailed treatment of those topics.
The course is organized into 43 units. For each credit you register for, you will have to complete 6 of those units. The units are further grouped into streams, which contain the units you should attempt if your goal is to prepare for Calculus, Statistics of the GRE.
Feel free to request other units if you need help on topics which are not listed here. I will be happy to look for or produce material for them. I encourage you to work together as much as possible on the units. There is no better way to internalize this material than by trying to explaining it to someone else.
INFORMATION
If you need help understanding the units, come by my office on Wednesdays between 9:00 and 12:00, or send me an e-mail so we can set up a time to meet that is convenient for you.
I encourage you to work together on the material of the units as much as possible. However, the work you turn in must be your own.
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Calculators
for college folk. a primo calculator. hi-res display, which is *really* good news for entering equations & graphing. does everything you could want, even functions as a word processor. even 3d graphs, and I think they can even rotate. solves equations on-screen. Bulky, so put big stick-on rubber feet on the bottom if it doesn't already have them - so it won't slide off your desk or book. This is not a balance-on-one-leg calculator: too wide & too heavy - like a brick. I think it's 1-1.5 inches thick If I remember right. TI programming languages are easy, NOT RPN (yay!), full-featured, and resemble a BASIC function with arguments and uses keywords like IF and WHILE. programming: flowcharts are a good idea if you are getting complicated. This is an algebraic-mode calc. only HP's use RPN. You can download new applications into this calc.
for college folk and especially the professional. a primo calculator. 2300 built-in functions. normal resolution display (average for a graphing calc). does 3d graphs. HP's equivelant to the TI voyage-200 but with a lot more beef (and a bit more fixed), seems mostly for the scientific professional with serious business in mind. The programming language (RPL) can detect math-object types (such as strings or integers) & branch on that condition. but remember this is an RPN calculator, and programming is achieved the RPN way (ugh! well, sometimes anyway. write programs out with your favorite text editor on your PC/MAC first...), arguments on the stack... RPL is a *LOT* easier on the eyes & fingers than was the HP-33s and looks like a calculator language (ugh) that uses keywords like IF and WHILE and a number of special symbols (the calculator language look). If you can afford a calculator, get one of these or the TI voyage, depending on your needs. I would go for this one personally for general use, but if you need to do an amount of programming and can't handle programming stack-based languages like FORTH, LOGO, and RPN/RPL (glance though the advanced HP49+manual to test yourself), I would choose the TI. Programming: flowcharts are a good idea if you are getting complicated. the RPN (Reverse-Polish Notation) will very probably throw you off in programming because you have to keep track of what's in the stack, at least in your head. Some say not as sturdy as the 48g.
it does a fair amount and has a lot of nice features, the standard scientific calc stuff including fractions in any form you like, a dandy decimal-to-fraction-to-decimal converter (worth the price, and it tells you if it's over or under the exact conversion), °F/°C conversion, lb/kg conversion, number-base conversion for binary and other number systems, cm/in conversion, time/map (HR/HMS) coordinate conversion, random number generator, has integer dividd & remainder, sums, x!, percent, and has lots of memory-store locations, at least as many as there are letters on the keyboard. Works in algebraic or RPN mode. says it's programmable but as a programmer I think it's a tough as nails to program. don't bother. Casio makes easier programming to work with. You can find it on ebay for $20. supposedley can solve equstions, but only if everything is pretty much known - as it's one of those programming jobs (same with integrals), don't bother. Still a nice calc to have around though. For a serious calc, you should probably try the HP-50G if you can afford it.
prorammable calculator software for mathematitians & numerical analysisprecision to N digits. want pi to 1000 digits? just do \p 1000 and then Pi does matrices, eigenvalues, does for loops, etc. Built on cygwin.
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A Simple Math DictionaryA critical and often overlooked part of any math program is the use of proper mathematical language. Did you know that by 5th grade there are 53 geometry terms your students need to comprehend?
This 30 page math dictionary uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference. When students are uncertain of a word or its meaning, this simple and straightforward dictionary is a good starting place. Make copies for your students so that they can place it in their math folder at the beginning of the year.
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1607.91 KB | 33You are welcome. If you find a word missing, let me know. I'm always updating this product.
September 23, 2012
carolblaisdell
Love this! I keep it on my thumb drive and it's always ready to pull up and use when my GED students have a question about math terminology. Only one disappointment - I see that it was included in the free items last week, and I had already purchased it! It is a great resource - thanks!
Oh my goodness! I've hit the Jackpot today, for sure. This is so thorough and clear... I'm sure it's going to set me on the road to curing my "math phobia" for sure! Will definitely come in handy when tutoring, and providing extra help to kids who are just using my room to catch up on other subjects! THANKS!
I currently teach remedial math students on the college level. I found their textbook was lacking on definitions that weren't written in "mathtese" so that is why I took the time to write this dictionary. All my students keep it in their math folders so it easily accessible.
I created this when I found the glossary of our math book was inadequate. I use it in both of my remedial college math classes.
February 1, 2012
jdrobb
Using your dictionary as a model, the kids are hyped about creating their own personal math dictionary.
Thanks big time.
October 27, 2011
Marlington
I wish I had been given something like this when I was in school. The language is so simple, no matheze. The illustrations are uncluttered and compliment the definitions perfectly. The best thing I have ever seen. So easy to use. Great job Scipi
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NZ 'pilot' School :
Who is this Aussie anyway? :
Who is this Aussie anyway? I'm not a salesman
I currently teach at a large co-educational, independent, metropolitan school in Melbourne.
My school currently use TI89's but are up-dating soon to . . .
Who do we have here today? :
Who do we have here today? RED – What is this 'CAS' you speak of.
AMBER – I know what it is but I haven't really used it in the classroom
GREEN - I use CAS with my classes already
What is CAS? :
What is CAS? Computer Algebra System
A CAS has the ability to perform symbolic manipulations in much the same way as we might do ourselves with pen and paper.
For example, expand and factorise algebraic expressions.
CAS has also powerful numerical computational capabilities and the ability to represent and analyse mathematical problems graphically and in spreadsheets.
But CAS can also be used as a learning tool
Why do I think that CAS is good for 14 – 16 year olds? :
Why do I think that CAS is good for 14 – 16 year olds? For learning rules from pattern recognition.
For scaffolding of Algebra.
For multiple representations and making connections between them.
Can use parameters to explore graphs or equations to find generalised solutions to big questions. And all on the one portable piece of plastic
1. For learning rules from pattern recognition :
1. For learning rules from pattern recognition How would you normally teach
Solving quadratic equations
The 'Null factor law'
2.For scaffolding of Algebra. :
2. For scaffolding of Algebra. :
2. For scaffolding of Algebra. Training wheels for solving equations
Allows a 'safe environment' for students to make mistakes and learn from the effects.
Undo the mistake and try something else.
Particularly good for weaker students.
Demonstrates that there isn't just one algorithm to solve something.
How many ways can you solve it?
Can work backwards to generate own questions.
3. Multiple representations and making connections. :
3. Multiple representations and making connections. How do you currently teach Simultaneous Equations?
Substitution?
Elimination?
Graphically?
3. Simultaneous Equations – what we did with year 10 :
3. Simultaneous Equations – what we did with year 10 Tell me a story
Worded problems
Table of values
Graphically
Home screen
Substitution and elimination by CAS
Saved 'by hand' techniques for extension and until year 11.
3. Tell me a story :
3. Tell me a story A picture is worth a thousand words
In small groups, students were given a theme and asked to decide what was happening in the graph and then present their story to the rest of the class.
Themes included: Cyclists, Cars, Planes, Bushwalkers, Mobile phones, Filling a beaker, Taxi fare, movies, goal scoring, Chinese characters
3. Tell me a story :
3. Tell me a story Hints
Decide what each of the axes represent
What is happening
At the start
Before the lines cross
When the lines cross
After the lines cross
Slide 17:
E.g. Two groups went to the movies. The first group included 5 adults and 5 kids and paid a total of $115. The second group included 2 adults and 7 kids and paid a total of $107. If ticket prices were the same for each group, find the cost of each type of ticket.
Examine the following screen from a CAS calculator, which has been used to find a solution to the simultaneous equations x + y = 5 and 3x + 2y = 11. :
Examine the following screen from a CAS calculator, which has been used to find a solution to the simultaneous equations x + y = 5 and 3x + 2y = 11. Explain how the CAS has been used to find a solution
Use this method to check the solutions you obtained earlier
3. Multiple representations and making connections. :
3. Multiple representations and making connections. Next generation CAS Nothing new CAS Sketch-pad add on
4. Can use parameters to explore graphs or equations. :
What happens to the volume of a sphere if the radius is doubled?
Beyond most kids algebra, but raises questions so we can then explore why. 4. Can use parameters to explore graphs or equations.
Belt around the Earth :
Belt around the Earth Consider a belt that is placed to fit around the equator of the earth. If 6m is then added to the belt circumference, can you:
A slip a piece of paper under it?
B slide your hand under it?
C crawl under it?
D walk under it?
CAS as a learning tool :
CAS as a learning tool Represents a move away from algorithms, providing opportunities to develop thinking and a deeper understanding.
Moving away from compartmentalised Mathematics
Instead of skill, skill, skill, application
Now start with a 'real-life' problem as a hook and learn the skills because we need them
Also represents a move to less 'contrived' Mathematics
Provides Motivation
CAS the 'black box' :
CAS the 'black box' Great way to get answers
Great way to Generate Questions
But won't it mean my student will lose their algebra skills? :
But won't it mean my student will lose their algebra skills? Yes CAS gives the Answer
Want to know why? Must do by hand
Still need algebra skills, in fact more of them to interpret CAS output as it doesn't always come up as expected. (e.g. transposing some formulae)
Issues :
Issues Cost
Theft
Class sets
Syntax – Can be a pain to start with
i-pods and mobiles are here to stay
Qualified staff - Some staff prefer the algorithmic approach
Assessment
Assessment :
Assessment :
Assessment Don't ask traditional type questions
Assessment of students understanding of mathematical concepts
Not assessing how well students have memorised algorithms
Benefits :
Benefits Top of the tree analogy (Tony McRae)
We can see the destination, where we are headed.
Allows you to quickly see where you are going.
F1 race car analogy (Tony McRae)
Safety car holds back all the cars.
Ear piece, can give instruction but let your better students fly ahead at their own pace.
First golf game analogy (Peter Fox)
Driving range first to get skills? Or . . .
Play the game first, then want to learn skills .
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This geometry book is written foremost for future and current middle school teachers, but is also designed for elementary and high school teachers. The book consists of ten seminars covering in a rigorous way the fundamental topics in school geometry, including all of the significant topics in high school geometry. The seminars are crafted to clarify and enhance understanding of the subject. Concepts in plane and solid geometry are carefully explained, and activities that teachers can use in their classrooms are emphasized. The book draws on the pictorial nature of geometry since that is what attracts students at every level to the subject. The book should give teachers a firm foundation on which to base their instruction in the elementary and middle grades. In addition, it should help teachers give their students a solid basis for the geometry that they will study in high school. The book is also intended to be a source for problems in geometry for enrichment programs such as Math Circles and Young Scholars.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
Readership
Undergraduate students interested in secondary education, particularly the teaching of geometry, and current middle school teachers teaching geometry.
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The important math your child
will work on in Don's books(note,
all of Don's materials come from work he has done with young people!):
Counting how many pieces of a size make the whole cake,
to name
a fraction of the cake in chapter
1, or cookie in chapter
2. This is a key idea which many students are not aware of and
causes
difficulty in all their math courses!
quadratic--by guessing!!, sum and
product of roots, by iteration many ways, by graphing, by quadratic
formula, using a calculator to hone in on the answer
cubic--by iteration, by computer
Iterating functions:
like 5 + x/2, if you start with 4 in
for x, what happens to the sequences?
to solve equations
to find compound interest
to find the square root of a number
to find the square root of the square
root..of a number
The use of computers and calculators: computer programs are in
almost
every chapter, with an appendix in the worksheet book on how to write
programs
to get infinite sequences and series. Don uses Derive to zoom in on a
curve to
get the slope of the tangent leading to the derivative. He uses
Mathematica to
show 100 iterations of a function. (See the use of computers page).
Probability: The area under the normal
curve is 1 and is related to probability
Trig functions: sine is used in finding the perimeter of
polygons
inscribed in a circle to get to Pi and is shown as an infinite series.
See also Trig
for young people.
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Claas Röver's Teaching Pages - Study Advice
Here I list a few points which when followed, I believe, make studying
mathematics easier and even enjoyable. Most of these points are also
relevant for computer science courses. The colors mean nothing.
1. Try to understand the logic and main ideas behind the concepts taught.
This reduces the amount of stuff you need to learn by heart.
2. Learn continuously, mostly definitions and technical terms, along with the lectures.
This will help you to actually understand what the lecturer is saying.
Studying involves a certain amount of learning vocabulary.
3. Do exercises regularly, i.e. at least once a week.
This way you get practice which is needed for exams and you may find things
you want to ask. It is also a great feeling when you solve a problem.
Both mathematics and programming can only be learned by doing.
Lectures are there to help you to move on in an organised fashion.
In short: solve problems/exercises or write programs.
4. Do ask questions whenever you have problems.
You can ask fellow students as well as the lecturer or tutor, or even a book.
Ask during the lectures, not afterwards. This way the whole class can benefit.
Usually, there are others with the same problem, but don't rely on them to aks. Go for it!
5. Read about the material in books.
Repetition or a slight change of viewpoint or different explanation can help a lot.
Everybody has different ideas and preferences about style, so choose a book you like.
6. Discuss or solve problems together with other students.
Discussions often help to solidify knowledge and streamline thoughts.
What you find easy others may find difficult and vice versa.
7. If there are tutorials/labs on offer, then do take part regularly.
Tutorials/labs are often less formal and hence it may feel easier to ask questions.
There you learn how one writes up solutions to problems.
|
Synopsis
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 just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with limits, continuity, curve-sketching, natural logarithms, derivatives, integrals, infinite series, and more! 100s of Problems! Step-by-step answer sets clearly identify where you went wrong (or right) with a problem The inside scoop on calculus shortcuts and strategies Know where to begin and how to solve the most common problems Use calculus in practical applications with confidence
Found In
eBook Information
ISBN: 97804717627
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This study examined the effectiveness of four types of review sessions given the day before a unit exam. Over a three week period, four Algebra 1 classes were taught the same unit by the principal investigator. At the end ...
Due to steady increases in students being diagnosed with disabilities, schools have transitioned to becoming more inclusive. As a result, children with disabilities are receiving more instruction within the general education ...
In this experiment two classes received instruction on integer operations. The first received instruction with the use of technology and the second class was instructed through a traditional approach. The study progressedBone Morphogenetic Protein 1 (BMP 1) functions in normal embryological development. The goal of this research was to obtain the sequence of salamander BMPl. Following sequence determination, an in situ probe for BMPJ ...
We used radio telemetry to determine the distribution and movements of paddlefish Polyadon spathula in the Allegheny Reservoir. Thirty-one adult and subadult paddlefish collected from spring congregation areas in the ...
This paper discusses a study that solicited data from teachers within two small city school districts. The study resulted from a five year federal education grant whose main objective was to provide intensive training and ...
This study explores the connection between student understanding of arithmetic and algebra through the evaluation of numeric expressions and the simplification of structurally comparable algebraic expressions. It isThis study examines the types of mistakes that students make solving multi-step linear equations. During this study, students completed a 15-problem test containing different types of multi-step linear equations appropriate ...
Understanding the concept of mathematical variables gives an opportunity to expand and work on high-level mathematics. This study examined college students' comprehension of variables as well as variable use in well-known ...
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Abstract Algebra
Provides an introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by ...Show synopsisProvides an introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. This text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings Contemporary Abstract Algebra
This book is very well written, but I advise you that it is not for the weak brained! I am taking this class as an independent study, so I am working almost enturely independently, and it is very hard. Great book though
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Calendar
Math
The New Mathematics Program The 2010-2011 school year saw the implementation of the new high school mathematics program witht the addition of two new courses at the grade ten level. Students entering grade ten will need to decide which mathematics sequence will best suit their needs and abilities. The following provides a brief background on the new mathematics course sequences at the high school level, but parents and students should seek advice from the counselling department and the mathematics department head if they are unsure of which sequence best meets the needs and abilities of the student.
The "-1" sequence is designed for students who are planning on pursuing post-secondary programs that require the study of calculus.
The "-2" sequence is designed for students who are planning on pursing post-secodnary studies in programs that do not require the study of calculus.
The "-3" sequence is designed for students who are planning on entry into the majority of trades and direct entry into the work force.
Grade ten students planning to complete either the "-1" or "-2" sequence will enrol in Mathematics 10-C (10-Combined), while students planning to complete the "-3" sequence will enrol in 10-3.
The Old Mathematics Program WCHS offered Mathematics 30 Pure & Mathematics 30 Applied in the 2011-2012 school year, but it was the final year WCHS will be offering these courses. The 2012-2013 school year will see the implementation of Mathematics 30-1 and Mathematics 30-2.
Math 10-C5 credits A combined mathematics course for students planning on completing the "-1" or "-2" sequences. Students need to have successfully completed Mathematics 9 in order to enrol in Mathematics 10-C.
Math 10-3 5 credits The introductory course for students pursuing the "-3" sequence. Units of study in this course include measurement (systems of measurement, surface area, volume) geometry (Pythagorean theorum, 2-dimensional geometry, trigonometry), numbers (mathematics of income and finance), and algebra (algebraic manipulations and application).
Math 20-1 5 credits Math 20-1 is the grade eleven course in the -1 high school mathematics sequence. Students need to have successfully completed Mathematics 10-C to be enrolled in this course. Topics of study include radical arithmetic and equations; rational expressions and equations; coordinate trigonometry and the sine and cosine laws; polynomial functions and inequalities; absolute value functions; quadratic functions and equations; and arithmetic and geometric sequences. A TI-83+ or TI-84+ calculator is required for this course.
Math 20-2 5 credits Math 20-2 is the grade eleven course in the -2 high school mathematics sequence. Students need to have successfully completed Mathematics 10-C to be enrolled in this course. Topics of study include rate, ratios, and scale factors; trigonometric proofs and the sine and cosine laws; inductive and deductive reasoning; radical arithmetic and equations; statistics; and quadratic functions and equations. A TI-83+ or TI-84+ calculator is required for this course
Math 20-3 5 credits Mathematics 20-3 is the grade eleven course in the -3 high school mathematics sequence. Students need to have successfully completed Mathematics 10-3 to be enrolled in this course. Topics of study include surface area and volume; right angle trigonometry; numerical reasoning; finance; slope and unit analysis; and statistical graphing.
Math 30-1 5 credits Math 30-1 is the grade 12 course in teh -1 high school mathematics sequence. Students need to have successfully completed mathematics 20-1 to be enrolled in this course. Topics of study include trigonometric functions, equations and identities: function composition and transformations: logarithmic and exponential functions; polynomial, rational and radical functions; and combinatorics. A TI-83+ or TI84+ graphing calculator is required for this course.
Math 30-2 5 credits Mathematics 30-2 is the grade twelve course in the -2 high school mathematics sequence. students need to have successfully completed Mathematics 20-2 to be enrolled in this course. Topics of study include logic and reasoning; probability and combinatorics; rational expressions and equations; logarithmic and exponential functions; and exponential , logarithmic, polynomial and sinusoidal data. A TI-83+ or TI84+ graphing calculator is required for this course.
Math 30-3 5 credits Mathematics 30-3 is the grade twelve course in the -3 high school mathematics sequence. students need to have successfully completed Mathematics 20-3 to be enrolled in this course. Topics of study include measurement; applied trigonometry; two and three dimensional transformations; vehicle and small business finance; linear relations, statistics; and probability.
Math 31 5 credits Math 31 is a course in basic differential and integral calculus, and is designed for those students planning on enrolling in post-secondary programs such as engineering, commerce, and science. Completion of this course provides students with significant preparatory skills for their first year calculus courses in post-secondary programs. Topics of study include limits, rules of differentiation, applications of derivatives, calculus of trigonometric, exponential, & logarithmic functions, anti-differentiation, techniques of integration & differentials, and applications of integration. Math 30 Pure is a prerequisite/ co-requisite for Math 31.
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Stock Status:In Stock Availability: Usually Ships in 5 to 10 Business Days
Product Code:9781741250107
About this book
Author Information
This book has been specifically designed to help Year 12 students thoroughly revise all topics in the HSC Mathematics course and prepare for class assessments, trial HSC and HSC exams. Together with the Year 11 Preliminary Revision & Exam Workbook, the whole senior Mathematics course is covered.
The book includes: - topics covering the complete HSC Mathematics course. - 200 pages of practice exercises, with topic tests for all chapters. - cross-references to relevant pages in the HSC Mathematics study guide. - topic tests for all chapters. - two sample examination papers. - answers to all questions.
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MA-209-G
Mathematics 3
Learning outcomes
After completing the course, the student is expected
to be able to:
characterise the properties
with quadratic curves and surfaces
master
vector multiplications and applications associated with lines
and planes
put up and solve equations for
motion of particles and bodies in the gravitational
field
master curve, surface and volume
integrals in different systems of
coordinates
calculate work, circulation and
flux
apply the theorems of Green, Stokes
and Gauss
solve differential equations by
means of power series and Fourier
series
Course contents
Conic section and quadratic curves, polar
coordinates. Vectors, lines and planes in three dimensions,
cylinders and quadratic surfaces. Two and three dimensional
position vectors, tangent and normal vectors. Newton's laws,
particle motion in the gravitational field, planet and
satellite motion. Double and triple integrals. Area, substance,
momentum and radius of inertia. Cylindrical and spherical
coordinates. Curve, surface and volume integrals of scalar and
vector functions. Work, circulation and flux. Divergence and
curl. The theorems of Green, Stokes and Gauss. Fourier series:
sine series and cosine series. Solving partial differential
equation by means of power series and Fourier
series.
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Introductory Physics With Algebra Mastering Problem-solving
9780471762508
ISBN:
0471762504
Pub Date: 2006 Publisher: Wiley & Sons, Incorporated, John
Summary: Get a better grade in Physics! Physics may be challenging, but with training and practice you can come out of your physics class with the grade you want! With Stuart Loucks' Introductory Physics with Algebra as a Second Languagea?: Mastering Problem-Solving, you'll get the practice and training you need to better understand fundamental principles, build confidence, and solve problems. Here's how you can get a better ...grade in physics: Understand the basic language of physics Introductory Physics with Algebra as a Second Languagea? will help you make sense of your textbook and class notes so that you can use them more effectively. The text explains key topics in algebra-based physics in clear, easy-to-understand language. Break problems down into simple steps Introductory Physics with Algebra as a Second Languagea? teaches you to recognize details that tell you how to begin new problems. You will learn how to effectively organize the information, decide on the correct equations, and ultimately solve the problem. Learn how to tackle unfamiliar physics problems Stuart Loucks coaches you in the fundamental concepts and approaches needed to set up and solve the major problem types. As you learn how to deal with these kinds of problems, you will be better equipped to tackle problems you have never seen before. Improve your problem-solving skills You'll learn timesaving problem-solving strategies that will help you focus your efforts and avoid potential pitfallsBloomingdale
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TI-84+ Material
Buy a TI-84+ (or
a calculator from the TI-84+ line or calculators, including the TI-83+, TI-84+,
TI-84+ silver edition, TI-Nspire or TI-Nspire CAS) calculator.A calculator from the TI-84+ line of calculators is an integral part of the course. It is
used very heavily throughout the course. You will be at a disadvantage if
you do not purchase this calculator.
You
can use any calculator you want, but be forewarned the course (Attendance
Workbook, homework assignments and quizzes) is designed for the TI-84+ (and only
the TI-84+) line of calculators. I support the
TI-84+
line of calculators only.
The courses listed below use programs I have created for the TI-84+ line of
calculators only.
If you are a class student, I will distribute the programs during class time. If you are a local internet student, drop by my office so that I can load these
programs onto your calculator. If you are a distance learning student, buy
the cable to connect your computer to your calculator, to allow you to
take these programs from this web site and install them on your calculator.
Do this immediately, at the beginning of the semester. Previous students
have commented how time consuming this is to do. Do not wait until half
way through the semester to load these programs onto your calculator!
A special cable that connects your calculator to your computer,
called a TI-Graph
Link, can access these programs.These cables are available at most computer or office supply stores.In addition, you must install special software on your computer that
instructs your computer on how to talk to your calculator through the
TI-Graph Link cable.
Instructions on how to transfer programs from one
calculator to another are given here: instructions
Other calculators are not supported.
Although the operating system of a TI-86 (TI-89, TI-89 Titanium, TI-92,
Voyage 200 and any other calculator which is not in the TI-84+ line of calculators) can be altered somewhat to have
"similar" statistical capabilities as the TI-84+ line of calculators, these other
calculators are simply too different from the TI-84+ line of calculators to be
useful for homework assignments, quizzes or any other classroom activity
requiring their use. In particular, there are list-like functions in a TI-86,
but they are so different than the lists in the TI-84+ line of calculators, they
are very difficult to use in any meaningful way. The important programs written for
the TI-84+ line of calculators for the courses I teach cannot be run on any other
line of calculators. Students who use a TI-86 (TI-89, TI-89 Titanium,
TI-92, Voyage 200 and so on)
calculators will be at a serious disadvantage to students who use a calculator
from the TI-84+ line of calculators.
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Re: Why is quiz part often harder than content in a maths textbook?
It seems a convention that maths teachers leave the hardest part in problems.
I am forced to agree. I have seen some where the problems are undoable even if you are familiar with the chapter
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INTERMEDIATE ALGEBRA F/COLL STUD comprehensive coverage of intermediate algebra to help students prepare for precalculus as well as other advanced math.
The material will also be useful in developing problem solving, critical thinking, and practical application skills. Real World Data and Visualization is integrated. Paying attention to how mathematics influences fine art and vice versa, the book features works from old masters as well as contemporary artists.
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Classroom Rules:
1. Be in your seat and ready to work when class begins.
2. Be prepared. Bring your book, calculator, paper, pencil and any
other materials with you to class. Locker passes will not be issued.
3. No hats can be worn in the building.
4. No beverages in the hallways or classrooms.
5. Respect others when they are talking, this includes both the
instructor and other students.
6. Do your best! I expect you to put forth a good effort, have a good attitude and attend class regularly.
Grading:
Points will be assigned for homework assignments, quizzes, tests, projects, and group work. Points will vary for each of these items.
Grades will be assigned according to the points earned out of the total points possible.The following grade scale will be used:
Late Work:
Assignments will not be accepted late, unless it is due to an excused absence. Every student is allowed 2 days of make-up for the first day missed, and one day for every day thereafter. If you are absent the day of a quiz or test, you will be expected to take the quiz/test the day you return to school. It is your responsibility to make up any work on a timely basis.
Extra Help:
If you need extra help, please come to my room before (7:45 - 8:10 a.m.) and after school (3:00 - 3:45 p.m.). Let me know ahead of time if possible, but never be afraid to ask for help.
Benchmarks:
You will be responsible for passing all benchmarks during the course. All benchmarks must be passed with 70% unless otherwise specified by the instructor. Individual students need to arrange times with the instructor for reteaching and retaking of any benchmarks. Any benchmark retakes must be taken in a timely manner, preferably within one week of the original benchmark.
If you are planning to purchase a calculator for high school, the West Delaware High School Math Department recommends purchasing a TI-84 Plus calculator. The TI-83 or TI-83 Plus calculators would be sufficient as well. These calculators will be used for all high school math course except Math Applications, and possibly some college courses.
Please be responsible for your calculator! Keep it locked in your locker! The library has an engraver that you may use to engrave your name on the cover of your calculator.
If you feel you are unable to purchase a graphing calculator you may talk with Mr. Nordass, H.S. Principal, about making arrangements to rent or loan a school calculator.
If you haven't purchased a graphing calculator yet, please bring the calculator you used in middle school (TI-34) with you to class.
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Organizational Links
Founded in 1888 to further mathematical research and scholarship, the
society fulfills its mission through programs and services that promote
mathematical research and its uses, strengthen mathematical education,
and foster awareness and appreciation of mathematics and its
connections to other disciplines and to everyday life.
The largest professional society that focuses on undergraduate
mathematics education. Members include university, college, and high
school teachers; graduate and undergraduate students; pure and applied
mathematicians; computer scientists; statisticians; and many others in
academia, government, business, and industry.
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Book Review: Bond Math
This book is the opposite of the book Interest Rate Markets, where bond markets were described, but there was no math. This book was written by an academic who has done many seminars for bond professionals so that they could understand the math behind bonds.
The math rarely transcends algebra, except where he used calculus to briefly explain duration and convexity. Perhaps he could have consulted with actuaries who use discrete approximations.
One more virtue of the book is that if you use Bloomberg, which is common for bond professionals, the book explains the nuances of how Bloomberg does many of its detailed bond calculations. It even explains why you have to interpret some of what you get from Bloomberg with caution, because it may use different assumptions than you would expect.
So if you want to learn the bond math, this book is a congenial way to do so. I recommend it highly.
Quibbles
Now, the writer is an academic who has never managed bonds. As such, he can't help a great deal with bond selection or portfolio management issues. But that's not the main goal of the book… he's here to teach us the math, and nothing more.
Who would benefit from this book:
Anyone who wants to learn the bond math would benefit from this book. Go learn and conquerIt has examples of duration and convexity for callable bonds, but lacks detailed calculations there. The discussion is more qualitative and relies on the idea that you will have a Bloomberg terminal to help you.
If I rewrite this review, I think I should add that this is an introductory book, and for the most part does relatively simple securities, though it does do floaters and linkers. Concepts like partial duration are touched on but glossed over.
I think the best way to understand the book is that it is aimed at people who trade/manage bonds that lack a strong sense of the mathematics behind the bonds, so that can use the mathematical tools that they have (like a Bloomberg Terminal) better
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Math Workbook For The New Sat
(Paperback) by Lawrence S. Leff Linda Math Workbook For The New Sat
Review and practice exercises in this workbook are specifically designed to prepare students for the math section of the SAT I. Special math strategies show students how to approach test questions and get correct answers that at first glance might seem too difficult or time consuming to tackle. Exercises include hundreds of multiple-choice questions, quantitative comparisons, and "grid-in" questions. All questions are answered with worked-out solutions. Review topics cover all question types found on the SAT I. An appendix reviews key SAT I math facts
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In article <6h19t5$934@nntp02.primenet.com> "James" <dolphin49@geocities.com>
writes:
>What are some of the problems?
>
>Blue wrote in message <3533c79f.2138627@news.megsinet.net>...
>>I've just started tech math 1 in College. I'm not familiar with
>>graphing calculators much. My professor told me to go get a TI-85.
>>When I went to the store, they didn't have any, but cut me a deal on a
>>TI-86 which said on the package it was an enhanced 85. Now I'm
Basically, it is. The 86 has more memory, more plotting options, some
more stat functions or something. It is basically an enhanced 85.
>>finding some compatibility problems. The things the professor says to
>>do, doesn't work on my calculator. So far I've been able to figure it
>>out, or get help from someone who knows. My thirty days isn't up on
>>this calculator. I'm wondering if I should take it back, and get a
>>85?
Odd. I had an 85 for years before it stopped working on me and I upgraded
to the 86. I had no problems making the switch. Sure, there were a few
slight differences (doing one var stats, I think, was a tad different),
but there were no major problems or differences that I can remember.
Unless the prof just happens to be using a lot of those functions and
stuff that are slightly different on the two calcs, I don't know why you
should be having problems. I'd try to stay with the 86 if you can,
because it does have more stuff than the 85
|
When understanding math is a problem, ModuMath is the solution.
Created by professional educators, ModuMath's approach is tailored for the needs of students who have had little success with classroom/textbook instruction.
Effective, engaging content for active learning.
ModuMath Basic Math and Algebra includes 83 interactive video lessons designed for learners who struggle with math. The lessons feature an engaging audio/visual presentation, contextual tutorials and continual comprehension checks that adapt the content and pace to the needs of each learner. This same series of adaptive lessons is at the heart of two packages.
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Math Calculations for Pharmacy Technicians
A Worktext
Designed — more than any other calculations book for pharmacy technicians— so readers can immediately apply what they are learning to realistic and challenging calculation problems. Other key concepts include conversions between the various measurement systems, reconstituting liquid medications, and calculating medications for special populations.
10. Calculation of Medications for Special Populations Based on Body Weight and Patient Age 11. Calculation of Medications Measured in Units, Milliequivalents and Percents of Concentration 12. Calculation of Medications for Intravenous Uses 13. Calculation of Mixtures from Stock Medications 14. Interpreting Physician's Orders for Dosages
|
First three weeks of class: Goldstein, et al., ``Brief Calculus and its
Applications'' (chapter 7, which is bundled with the main text)
The remainder of the term: Goldstein, et al., ``Finite mathematics
and its applications''
Calculator: A graphing calculator is required. Recommended:
TI-83. Actually, most graphing calculators will work. Sharing of
calculators during exams is not permitted.
Prerequisites: A grade of C-- or better in MATH 214.
Web page:
Email list: math215@yahoogroups.com (when you fill out your information
you may elect to be on this list)
Objectives: The student should be able:
To maximize or minimize a function of many variables, even including
constraints;
To compute probabilities for some discrete random processes;
To use Bayes' Theorem to compute conditional probabilities;
To compute and interpret mean and standard deviation;
To solve a system of linear equations to find all solutions, or to
show when no solution exists.
Goals: The student should develop:
An ability to translate a problem which is well-suited to mathematical
solutions into mathematical language;
A familiarity with functions of several variables;
A practical knowledge of probability that will be used in BA
216;
An ability to manipulate matrices and use them when it is well-suited
to the problem;
A confidence about doing mathematics to overcome ``math anxiety'';
An ability to think and reason in a structured logical manner.
Homework: Homework will be assigned daily, and due two class times later,
at the beginning of class. Some homework may be due the next class period,
and this will be specified on the homework.
Each assignment will include both problems not to be submitted, and
problems to be submitted. The problems not to be submitted are those
which have answers in the book. You are to do these problems to best
prepare you for the rest of the assignment, and if you find that you
understand how to do these problems before you finish them all, you
may skip to the problems to be submitted.
Not all homework problems will be graded--only a select few, representative
of the different kinds of problems. Those seeking solutions to past problem
sets may request them after the homework is due. Homework will be graded
primarily on your ability to complete the problem, and secondarily on its
correctness.
The five lowest homework scores will be disregarded.
Remember that the primary purpose of the homework is to prepare for the
exams, so treat it primarily as a training program for yourself, and only
secondarily as something you need to score highly on.
Late assignments: No late homework is accepted. Exceptions can
be granted, if you must give me notice that you are going to turn in
an assignment late at least the class before the assignment is due.
You must also have a good reason. These reasons will be treated on a
case-by-case basis. When you obtain permission to turn in an
assignment late, we will discuss a new due date for that homework.
Extra credit: If you find a way to apply any of the material to
something you are interested in, you can discuss with me the possibility of
getting extra credit. It should involve some research outside the material
in the book and be written in a professional manner.
Collaboration: You are encouraged to collaborate on all homework
assignments, unless otherwise specified. This means you work on it
independently before discussing it with each other, and it means you
must thoroughly understand how to do the problem before writing it up.
You must write up your answers separately; you cannot turn in one
homework for more than one person, nor can you simply include
photocopies of other students' work. There is no limit to the size
of a group for collaboration, although 3-5 people tends to be an efficient
size.
You should also use these groups to ask questions of each other to
better understand the material. If you do not see each other
frequently, you should set up a regular time and place to meet to work
on assignments. If you do not have a group, talk to me and I can
place you in a group. If you do not wish to work in a group, that is
your prerogative but this will be a disadvantage to you.
Comments: You should include comments about the class at the top
of your homework assignments. These comments can be ``You go too
fast'', ``You say `um' too often'', ``I like this chapter'', ``This is
too easy/hard'', ``Can we have more applications to Marketing'',
``Everything's okay'', and so on. You will not be graded on these
comments, but they will affect how I teach the class, and may make the
class more enjoyable for you.
Class participation: You are expected to actively participate in
class. Many students view learning as a passive act, where the
teacher takes the only active role, and the student simply listens, or
at most takes notes. This view is not advisable in this class. Here,
you will need to take an active role in learning the material. {\em
You} are in charge of your education, and {\em you} should take
responsibility to learn the material as thoroughly as you can. Part
of this involves asking questions in class, even questions that may
sound ``stupid''. A question clearing up a point you do not
understand is, by definition, not stupid. Similarly, when I ask the
class questions, you should try to answer them, even if you're not
sure of the answer. Your best guess is, by definition, not stupid.
The effect of class participation on your grade is noted under ``quizzes''
below.
Pre-class preparation: You are expected to read through the section
of the book we are covering before you come to class. If you don't
understand something, write down specific questions you have to ask in class.
Quizzes: There will be no regular quizzes, but to ensure you have
read through the section beforehand, I will, from time to time, give out
pop quizzes at the beginning of class. These will be short and only test
a superficial knowledge of the material. In this way, they are not useful
for indicating what an exam will be like. They will be used to decide
borderline cases in the final grade, as will class participation.
Remember that since there are 12 grades (counting +'s and --'s), almost
everyone in the class will be a borderline case. There is no make-up for
quizzes.
Attendance: Attendance is important simply due to the difficulty
of the course. Missing one class may have the effect of
your not being able to follow any of the classes for the rest of the
term. Furthermore, those who do not attend classes will have poor
scores on class participation and cannot take quizzes, and these will also
affect grades. In short, skip class at your peril.
Exams: There will be three midterms, and one final. Each
midterm counts for 20% of your grade, and the final counts for 30%.
The remaining 10% is your homework grade. The final exam grade will
substitute for your lowest midterm grade if this is to your advantage.
Note that borderline cases will be resolved by quiz grades and class
participation, as noted above.
Midterms will be in-class, and the final will be at a separate time as
listed below. All final exams follow the schedule listed at
There are no make up quizzes or exams. If you must miss an exam due
to a major emergency, you must make arrangements with me beforehand,
and exceptions may be granted on a case-by-case basis. If granted, your
final exam score will be used to calculate the score for the missed exam.
Midterm 1
February 3
during class
Midterm 2
February 24
during class
Midterm 3
March 31
during class
Final (sec. 3)
April 25
1:30 p.m. -- 4:00 p.m.
Final (sec. 4)
April 26
1:30 p.m. -- 4:00 p.m.
I will hold review sessions before each, at a time that is popular with the
class.
Holidays:
Conference
Jan. 12--13
Martin Luther King, Jr. Day
Jan. 16
Spring break
Feb. 27--Mar. 3
Grading: A grade of C indicates an ability to do homework-like
problems, and memorization of all techniques and definitions. In
order to receive a B, a student must demonstrate a deeper knowledge of
the material, being able to apply the course material to new
circumstances where applicable. An A student must demonstrate this
kind of deep understanding in all of the covered topics, as well as be
able to draw new conclusions from known facts in a logical manner, and
must also demonstrate persistence and dilligence. In the other
direction, a grade of D shows only superficial understanding of the material,
and shows inconsistency to do straightforward problems. An F
grade indicates that the student has severe gaps in even superficial
understanding of the material in the course.
Although this is the philosophy, grading will be done by counting points
received on each problem, as usual. But the difficulty level of the problems
will be arranged in order to achieve the above grading scale.
Christian attitude: Although not part of the grading for this
course, you are expected to approach this class with a Christian
attitude, being willing to help your fellow classmates to understand
the material outside of class, being willing to be corrected by your
fellow classmates when you see they are right, but firm in your
conviction otherwise, being bold to ask questions without feeling
ashamed of looking foolish, encouraging one another in love, being
patient with those who are asking questions, and preferring a grasp of
the material, which is enduring and becomes part of you, over a grade,
which is transient, external, and shallow. You should diligently
devote the time you spend on this class as to the Lord. As cheating
harms both the cheater and the rest of the class (though in different
ways), you should not cheat, nor should you provide temptations for
others to cheat.
For my part, I commit to approaching this class with a Christian
attitude, viewing my role as that of a servant, being concerned first
for your personal, especially intellectual, development. I
will also seek to produce an environment of encouragement and love,
that fosters a sense of community and understanding. I commit to reporting
grades that accurately and honestly reflect the level of work done in the
class, as described in the paragraphs above. I also commit the time I
spend preparing for this class as to the Lord, and I will pray for all
individuals in the class on a regular basis, understanding that even as I
may seek to educate, God provides the true transformation.
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Search Course Communities:
Course Communities
Illustrating the Washer Method
Course Topic(s):
One-Variable Calculus | Integration, Applications
Java applet allows the user to visualize the washer method for finding the volume of a solid formed by revolving a specific region about the \(x\)-axis. The region is formed by two fixed curves of functions in terms of the \(x\) variable. The applet allows the user to dynamically interact with the total volume, a specific washer, or a discrete number of washers. The user interacts with the 3D rendering with fluid mouse motions.
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Humongous Book of Statistics Problems - 09 edition
Most math and science study guides are dry and difficult, but this is the exception. Following the successful The Humongous Booksin calculus and algebra, bestselling author Mike Kelley takes a typical statistics workbook, full of solved problems, and writes notes in the margins, adding missing steps and simplifying concepts and solutions. By learning how to interpret and solve problems as they are presented in statistics courses, ...show morestudents prepare to solve those difficult problems that were never discussed in class but are always on exams.
''With annotated notes and explanations of missing steps throughout, like no other statistics workbook on the market
'' An award-winning former math teacher whose website (calculus-help. com) reaches thousands every month, providing exposure for all his books
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This is a decision that should be made by students and their parents. It should be based on a realistic view of the student's skills and aspirations.
How many units of mathematics should you study?
This will usually be 2 to 8.
Which units of mathematics should you study?
It is recommended that students should choose a "Pathway" in VCE mathematics. Suggested Pathways are shown below.
Accelerated Mathematics Yr 10
What is this unitUnitsSemester 1: Acceleration Mathematics
Real and Complex Number Systems
Matrices
Sequences and Series
Variation
Semester 2: Acceleration Mathematics
Non-linear Graphs
Trigonometric Ratios
Non linear Relations and Equations
Data
What type of things will I do?
Work with numbers and surds
Substitute, Transpose and Solve Equations
Plot and sketch graphs
Use technology to help with learning
Application of Matrices
Display and Summaries data
Correlations and Regression of data
Applications of Sequences and Series such as financial arithmetic
Minimization in problems of time and distance
What can this lead to?
Specialist Mathematics 3 and 4
Mathematical Methods (CAS) 3 and 4
Further Mathematics 3 and 4
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Accelerated Mathematics
Year 11
Further Mathematics (3 and 4)
Year 12
Mathematical Methods CAS
Specialist Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
CAS
Foundation Mathematics Units 1-2
What's it all UnitsUnit 1: Foundation Mathematics
Space and Shape
Patterns in Number
Handling Data
Measurement and Design
Unit 2: Foundation Mathematics
Pattern and Number
Space and Shape
Measurement and Design
Handling Data
What type of things will I do?
Two dimensional plans
Diagrams incorporating scales
Practical problems involving decimals, fractions and percentages
Formulas and their use
Reading roads maps, time tables, flowcharts
Metric measurement problems
Recording and analyzing instrument readings
Ordering and weighing food items
Interpreting financial information
What can this lead to?
VCAL
VET
Apprenticeships
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
Foundation Mathematics
Year 12
VCAL and VET
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Finance
General Mathematics (Further) Units 1-2
What's it all about?
This study is designed to provide access to worthwhile and challenging mathematical learning in a way which takes into account the needs and aspirations of a wide range of students Further Mathematics Units 3 and 4. A Computer Algebra System (CAS) will be used by students to assist them in their learning and understanding.
Assessment for satisfactory completion of Units 1 and 2 is by tests, analysis tasks, and students work during the year.
What will I learn?
Unit 1: General Mathematics (Further)
Number Theory
Number Patterns and Applications
Relations in Linear Equations
Linear Graphs
Unit 2: General Mathematics (Further)
Represent and Interpret types of Data
Describe and use Networks
Matrices and their Applications
What type of things will I do?
Work with schedules, time zones
Applications of Sequences and Series such as financial arithmetic
Formulate Equations
Plot and sketch graphs
Display and Summarise data
Correlations and Regression of data
Minimisation in problems of time and distance
Eulerian Paths and Circuits
Use a Computer Algebra System
What can this lead to?
Further Mathematics 3 and 4
VCAL
VET
Apprenticeships
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
General Mathematics (Futher)
Year 12
Futher Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
CAS
General Mathematics (Methods) Units 1-2
What's it all about?
This course is designed for students intending to do tertiary studies that will involve complex and/or specialized mathematical calculations and skills. Students selecting these units should be able to manipulate algebraic expressions and solve equations. These skills are further developed in this course Mathematical Methods (CAS) and/or Specialist Maths Units 3 and 4. A Computer Algebra System will be used by students to assist them in their learning and understanding.
Assessment for satisfactory completion of Units 1 and 2 is by tests, analysis tasks, and student's work during the year.
What will I learn?
Unit 1: General Mathematics (Methods)
Matrices
Number Systems
2D and 3D Geometry
Linear Graphs and Relations
Unit 2: General Mathematics (Methods)
Trigonometry
Non-Linear Graphs and Relations
Co-ordinate Geometry
Vectors
What type of things will I do?
Work with numbers and surds
Substitute, Transpose and Solve Equations
Apply geometry to applications
Plot and sketch graphs
Use trig ratios, pythagoras and geometry to solve problems
Use technology to help with learning
Application of Matrices
What can this lead to?
Specialist Mathematics 3 and 4
Mathematical Methods(CAS) 3 and 4
Further Mathematics 3 and 4
VET
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 2 Mathematics
Year 11
General Mathematics (Methods)
Mathematical Methods
Year 12
Mathematical Methods(CAS)
Specialist Mathematics
Further Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Applications of Geometry
Interpreting Graphs
Matrices
CAS
Further Mathematics Units 3-4
What's it all about?
This course is designed for those students whose employment and/or further study aspirations do not require heavily algebra based mathematical skills.
Students will develop their mathematical knowledge and skills to be able to investigate, analyse and solve problems. They will be required to communicate mathematical ideas clearly and concisely.
A Computer Algebra System will be used by students to assist them in their learning and understanding.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS).
What will I learn?
Unit 3: Further Mathematics
Networks and Decision Mathematics
Statistics
Unit 4: Further Mathematics
Number Patterns and Applications
Matrices
What type of things will I do?
Use statistical techniques
Model relationships between data
Matrix representation and arithmetic
Predicting ahead in situations involving number patterns
Correlations and Regression of data
Minimisation in problems of time and distance
Features of networks and their applications
Use a Computer Algebra System
What can this lead to?
Tertiary Education
Apprenticeship
General Employment
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 1 Mathematics
Year 11
General Mathematics (Further)
General Mathematics (Methods)
Year 12
Further Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Numbers
Uses of Data
Interpreting Graphs
Matrices
Networks
Mathematical Methods (CAS) Units 1-4
What's it all about?
Mathematical Methods consists of study in the areas of 'Co-ordinate Geometry', 'Trigonometric Functions', 'Calculus', 'Algebra', and 'Statistics and Probability'.
There are no prerequisites for entry to Mathematical Methods (CAS) Units 1 and 2. However, students attempting Mathematical Methods (CAS) are expected to have a sound background in number, algebra, function, and probability. Students wanting to do Mathematical Methods (CAS) Units 3 and 4 should have completed Mathematical Methods (CAS) Units 1 and 2 and General Mathematics (Methods) Units 1 and 2.
The appropriate use of CAS technology to support and develop the teaching and learning of mathematics, and in related assessments, is to be incorporated throughout the course.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS) and exams.
What will I learn?
Unit 1: Maths Methods(CAS)
Functions and Graphs
Algebra
Probability
Rates of Change
Unit 2: Maths Methods(CAS)
Functions and Graphs
Algebra
Probability
Calculus
Unit 3: Maths Methods(CAS)
Functions and Graphs
Differential Calculus
Unit 4: Maths Methods(CAS)
Integral Calculus
Probability
What type of things will I do?
Problem solving
Substitute, Transpose and Solve Equations
Apply geometry to applications
Plot and sketch graphs
Calculate and Interpret Probabilities
Apply Algebra, Logarithmic and Trigonometric properties
Use CAS to assist with learning
What can this lead to?
Tertiary Education
Apprenticeship
General Employment
Possible Pathways @ TLSC
Year
Studies Offered
Year 10
Pathway 2 Mathematics
Acceleration Mathematics
Year 11
General Mathematics (Methods)
Mathematics Methods (CAS)
Year 12
Mathematics Methods (CAS)
Specialist Mathematics
Why choose this study?
Choose this study if you are interested in learning about:
Calculus
Geometry
Functions
Probability
CAS
Specialist Mathematical Units 3-4
What's it all about?
Specialist Mathematics consists of the following areas of study: 'Functions, relations and graphs' 'Algebra', 'Calculus', 'Vectors' and 'Mechanics'. Students are expected to be able to apply techniques, routines and processes, involving rational, real and complex arithmetic, algebraic manipulation, diagrams and geometric constructions, solving equations, graph sketching, differentiation and integration related to the areas of study, as applicable, both with and without the use of technology.
Enrolment in Specialist Mathematics Units 3 and 4 assumes a current enrolment in Mathematical Methods (CAS) Units 3 and 4.
Assessment is by satisfactory completion of three outcomes, judged by the student's results in Student Assessed Coursework (SACS).
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Research, consultation and experience have helped us select math materials excellent for their organization and presentation of college prep coursework. These texts provide step-by-step instruction and plenty of practice exercises along with periodic reviews and tests.
Star Academics Mathematics Courses
Fundamentals of Mathematics (grade 6 – 7)Pre-Algebra (grade 8)Algebra 1 (grade 9)Algebra 2 (grade 10 – 11)It is essential that parents consistently score math assignments. If parents are not personally able to provide this academic support, we recommend purchasing optional materials and/or obtaining private instruction. Parents should also ensure that students read and complete the practice problems before attempting the exercises.
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Teacher Resources
The School of Mathematics and Statistics is keen to develop links with teachers, to improve the quality of both secondary and tertiary mathematics education and ease students' transition between them.
The School operates a Visiting Teaching Fellow program,which allows teachers to spend a semester or a year in the School in a lecturer/tutor capacity. For information, contact the Director of First Year.
The School of Mathematics and Statistics endows maths prizes for local high schools. For information, please contact the Head of School.
Undergraduates, postgraduates and alumni of the School of Mathematics and Statistics are available to visit high schools to speak at careers fairs and similar events. For information on this, contact James Franklin.
The School appreciates the difficulties school students and teachers face. It is of course a trying time for students doing the HSC, and it is also difficult for teachers who are trying to prepare the students for the HSC - and not just prepare them to do well for the HSC, but also to adequately prepare students for their further tertiary studies. Mathematics, in particular, is somewhat special. Mathematics is an important subject that lies at the basis of many other subjects in Science, such as Engineering and Materials Science, Physics, Chemistry, Biology, and others. Mathematics is also one of the only fields where the knowledge is cumulative and somewhat hierarchical, so it is important to adequately prepare students.
Good mathematics skills and knowledge are essential for ensuring success in these other fields. How, then, can teachers help their students maximise their mathematics skills and knowledge?
First of all, it is important for students to know that they should take maths. Guidance for students who want to end up studying some of the Science subjects mentioned above should stress the importance of mathematics. Only through taking mathematics can an understanding of mathematics be fostered.
Even though first year University-level mathematics courses do not have any formal high school prerequisites, students may struggle without at least 3 units of HSC-level mathematics. Many of the first year subjects assume knowledge of this level of mathematics. For students who did not take this level of mathematics in high school, the School of Mathematics and Statistics offers bridging courses through January and February.
However, for especially talented students who have an interest in mathematics, a talented students program is available. Run by the School of Mathematics and Statistics, the Talented Students program is geared to a faster pace, and has an emphasis on harder problems. A guideline for entrance into this program is approximately 180 for 4 unit mathematics in the HSC level, or a high 80 mark in MATH1141. Entrance details are given in the first lecture.
There are other avenues to encourage students in Mathematics, such as the University of New South Wales Schools Mathematics Competition with the chance to win prizes and certificates of recognition for work in solving a set of problems designed to test mathematical ingenuity. If you need more information, Bruce Henry of the School of Mathematics and Statistics will be glad to answer any queries you may have.
Teachers can also encourage students by distributing our information on careers in mathematics. Some role models can be found among our alumni.
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Book of Algebra
This book is essentially "The most complex looking equations reduced to the most simplest of ideas " . It starts with an introductory chapter on what is algebra using a jargon free language and all major concepts like , Equations , Inequalities , Maxima Minimas , Graphs , Functions are all linked together in one cohesive whole and ends with an assignment which will be toughest examination of your ability to think in the world of algebraic notation. In between these two you will find a treasure trove of ideas and techniques making tough concepts very easy to grasp and aiding you in learning fundamentals using which the most esoteric looking equations can be solved in matter of seconds rather than minutes .
The book is devoid of any shortcuts which are not explained in great detail . The hows and whys of working of rules are taught with great emphasis , since without understanding what really is going on behind the equations you will always struggle with how a given rule works not in the standard scenarios but in the real world CAT problems (which are by definition twisted ) .
The Book is a Conversation with you , it talks to you points out the pitfalls , not only it teaches you the coolest tips and tricks of the trade but it also explains with great detail the workings of those same tricks and tips so that you own them and use them as needed.
The book shows more than telling , Algebra is a world of terrific visuals and since a picture is worth a thousand words the book with with more than 200 visual aids is an invaluable resource which aids and enhances comprehension of key concepts.
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Computational Mathematics
The study of computational mathematics has grown rapidly over the past 15 years and has allowed mathematicians to answer questions and develop insights not possible only 20-30 years ago. Modern computational methods require an in-depth knowledge of a variety of mathematical subjects which include linear algebra, analysis, ordinary and partial differential equations, asymptotic analysis, elements of harmonic analysis, and nonlinear equations. Since computers are invaluable tools for an applied mathematician, students are expected to attain a highly professional level of computer literacy and gain a substantial knowledge of operating systems and hardware. Computational mathematics courses include the study of computational linear algebra, optimization, numerical solution of ordinary and partial differential equations, solution of nonlinear equations as well as advanced seminars in wavelet and multi-resolution analysis.
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Matrix Methods: Applied Linear Algebra, 3e, as a textbook, provides a unique and comprehensive balance between the theory and computation of matrices. The application of matrices is not just for mathematicians. The use by other disciplines has grown dramatically over the years in response to the rapid changes in technology. Matrix methods is the essence of linear algebra and is what is used to help physical scientists; chemists, physicists, engineers, statisticians, and economists solve real world problems.
* Applications like Markov chains, graph theory and Leontief Models are placed in early chapters * Readability- The prerequisite for most of the material is a firm understanding of algebra * New chapters on Linear Programming and Markov Chains * Appendix referencing the use of technology, with special emphasis on computer algebra systems (CAS) MATLAB less
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Arithmetic Sequences
In the first lesson of Sequences and Series, Dr. Eaton begins with Arithmetic Sequences. She starts with the general form of sequences and moves into the common difference between each term of the sequence. Then she will teach you the formula and equation for the nth term before finishing with arithmetic means. Four video examples round out this first lesson.
This content requires Javascript to be available and enabled in your browser.
Arithmetic Sequences
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
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I'd call this book perfect for middle school on up if you plan to use it as is. For many younger children (lower than 4th grade)to teach the material, you'd have to be willing to present the concepts in examples you've rewritten. Examples: [In Chapter 1] 160 divided by 1/3 = 53 1/3. In Chapter 3, Level 1 problem 1 is simple subtraction...but if a student doesn't already know that a straight line = 180 degrees, he or she will just get frustrated that a number is "missing". And later in the book there are some variables and exponents. My son is entering 5th; his math level is just out of pre-algebra/getting into "real" algebra (I've supplemented at home)...if I were relying only upon what children learn through fourth grade at school, he would be too distracted by the computation to get the problem solving benefit out of it. I don't think the book is presented as being for elementary students. But if you purchase this book to use [as is] for someone comfortable with fractions and some basic pre-algebra concepts...you'll have no regrets.
I have a 2nd grader (who is advanced for his age) and this book is written in such a way that he can grasp the concepts without feeling overwhelmed. The "cartooning" style is easy to follow and entertaining while still being educational. We were looking for something for him to use to develop his probleming solving skills. This was an excellent place to start.
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Short description
Presents MATLAB both as a mathematical tool and a programming language, giving an introduction to its potential and power. This book illustrates the fundamentals of MATLAB with many examples from a wide range of familiar scientific and engineering areas. It includes coverage of Symbolic Math and SIMULINK. It highlights common errors and pitfalls.
Long description
This is the essential guide to MATLAB as a problem solving tool. This text presents MATLAB both as a mathematical tool and a programming language, giving a concise and easy to master introduction to its potential and power. The fundamentals of MATLAB are illustrated throughout with many examples from a wide range of familiar scientific and engineering areas, as well as from everyday life. The new edition has been updated to include coverage of Symbolic Math and SIMULINK. It also adds new examples and applications, and uses the most recent release of Matlab. Features of this title include: new chapters on Symbolic Math and SIMULINK provide complete coverage of all the functions available in the student edition of Matlab; more exercises and examples, including new examples of beam bending, flow over an airfoil, and other physics-based problems; a bibliography that provides sources for the engineering problems and examples discussed in the text; and, a chapter on algorithm development and program design. In this book, common errors and pitfalls are highlighted. It provides extensive teacher support on: solutions manual, extra problems, multiple choice questions, PowerPoint slides. It features companion website for students providing M-files used within the book.
Product details
Publisher:
Academic Press
ISBN:
9780123748836
Publication date:
October 2009
Length:
236mm
Width:
191mm
Thickness:
20mm
Weight:
821g
Edition:
4th edition
Pages:
391
Illustrated:
True
Review
This book provides an excellent initiation into programming in MATLAB while serving as a teaser for more advanced topics. It provides a structured entry into MATLAB programming through well designed exercises. - Carl H. Sondergeld, Professor and Curtis Mewbourne Chair, Mewbourne School of Petroleum and Geological Engineering, University of Oklahoma This updated version continues to provide beginners with the essentials of Matlab, with many examples from science and engineering, written in an informal and accessible style. The new chapter on algorithm development and program design provides an excellent introduction to a structured approach to problem solving and the use of MATLAB as a programming language. - Professor Gary Ford, Department of Electrical and Computer Engineering, University of California, Davis For a while I have been searching for a good MATLAB text for a graduate course on methods in environmental sciences. I finally settled on Hahn and Valentine because it provides the balance I need regarding ease of use and relevance of material and examples. - Professor Wayne M. Getz, Department Environmental Science Policy & Management, University of California at Berkeley This book is an outstanding introductory text for teaching mathematics, engineering, and science students how MATLAB can be used to solve mathematical problems. Its intuitive and well-chosen examples nicely bridge the gap between prototypical mathematical models and how MATLAB can be used to evaluate these models. The author does a superior job of examining and explaining the MATLAB code used to solve the problems presented. - Professor Mark E. Cawood, Department of Mathematical Sciences, Clemson University This has proved an excellent book for engineering undergraduate students to support their first studies in Matlab. Most of the basics are covered well, and it includes a useful introduction to the development of a Graphical User Interface. - Mr. K
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Linear Algebra
This course is an introduction to linear algebra. It has been argued that linear algebra constitutes half of all mathematics. Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between.
Linear algebra can be viewed either as the study of linear equations or as the study of vectors. It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it. Learning to connect the facts with their geometric interpretation will be very useful for you.
The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra. As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts in this subject. Additionally, you will learn some standard techniques in numerical linear algebra, which allow you to deal with matrices that might show up in applications. Toward the end, the more abstract notions of vector spaces and linear transformations on vector spaces will be introduced.
In college algebra, one becomes familiar with the equation of a line in two-dimensional space: y = mx+b. Lines can be generalized to planes and "hyperplanes" in many-dimensional space; these objects are all described by linear relations. Linear transformations are ways of rotating, dilating, or otherwise modifying the underlying space so that these linear objects are not deformed. Linear algebra, then, is the theory and practice of analyzing linear relations and their behavior under linear transformations. According to the second interpretation listed above, linear algebra focuses on vectors, which are mathematical objects in many-dimensional space characterized by magnitude and direction. You can also think of them as a string of coordinates. Each string may represent the state of all the stocks traded in the DOW, the position of a satellite, or some other piece of data with multiple components. Linear transformations change the magnitude and direction of vectors—they transform the coordinates without changing their fundamental relationships with one another. Linear transformations are often written in a compact and easily-readable way by using matrices.
Linear algebra may at first seem dry and difficult to visualize, but it is one of the most useful subjects you can learn if you wish to become a business-person, a physicist, a computer-programmer, an engineer, or a mathematician.
Remember, the prerequisite of this course is one variable calculus course and a reasonable background in college algebra.
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete assignments for Unit 1 through Unit 6 as well as the final exam.
Note that you will only receive an official grade on the Final Exam. However, in order to adequately prepare for this exam, you will need to work through the assignments and all the reading material in the 135.75 25.5 hours. Perhaps you can sit down with your calendar and decide to complete all of Unit 0 (a total of 1.25 hours) and half of subunit 1.1 (a total of 3 hours) on Monday night; the rest of subunit 1.1 (a total of 3 hours) on Tuesday; half of subunit 1.2 (a total of 3 hours) on Wednesday; etc.
Tips/Suggestions: It will likely be helpful to have a calculator on hand for this course.
Make sure you have a solid understanding of the pre-requisite topics outlined in Unit 0 before moving on to other units in the course. As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, and definitions that stand out to you. It will be useful to use this "cheat sheet" as a review prior to completing the Final Exam.
Preliminary Information
Open Textbook Challenge Winner: Elementary Linear Algebra
Elementary Linear Algebra was written and submitted to the Open Textbook Challenge by Dr. Kenneth Kuttler of Brigham Young University. Dr. Kuttler wrote this textbook for use by his students at BYU. 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." A solutions manual for the textbook is included.
To learn linear algebra, you must have knowledge of some topics from elementary algebra. In this unit, you will review these topics. Specifically, the unit begins with a brief review of set notation and then moves to a review of complex numbers. This is especially important in linear algebra, since polynomial equations with real coefficients often have complex numbers as roots. The quadratic formula, which is the next topic you will review, is a good example of this phenomenon.
Instructions: Please click on the link above, and read the indicated sections on pages 11–15. Section 1.1 briefly discusses set notation; Sections 1.2 and 1.3 provide a brief review of functions. This reading should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
This unit begins with a review of vectors. You will learn the geometric meaning of vectors, which is especially significant in R2 and R3. Next, you will learn the geometric meaning of vector addition and scalar multiplication. Finally, you will apply study vectors in the context of physics to model force and other physical vectors like velocity.
In the next chapter, you will begin learning about vector products. There are two ways of multiplying vectors, both of which are of great importance in applications. The first type of a product is called the dot product, also called the scalar product or the inner product. You will then study the geometric significance of the dot product and applications of dot product by studying the concepts of work and projections. Next, you will begin the study of cross products. The cross product is the other way of multiplying vectors, and it is different from the dot product in fundamental ways. You will learn both the geometric meaning of the cross product and the description in terms of coordinates. Both descriptions of the cross product are important; the geometric description is necessary to understand the applications to physics and geometry while the coordinate description is necessary to actually compute the cross product. You will then learn techniques, which will allow you to discover vector identities and simplify expressions involving cross and dot products in three dimensions.
Next, you will begin exploring systems of linear equations. The basic idea is to study situations where there are several different variables that are related in multiple ways. These linear equations could describe budget constraints in a business, physical constraints in an engineering problem, or any number of other situations. The key is that these constraints can be described by linear equations. The geometric interpretation of these constraints is that each equation describes a line or plane where potential solutions to the problem must lie. The task then is to figure out what combination of variable values solves all of the different linear equations at the same time. Geometrically, this is where all of the lines or planes intersect. Just as is the case with the problems that the equations may be modeling, the system of equations will sometimes have no solution, will sometimes have a single solution, and will sometimes have an infinite number of solutions. Finally, you will learn about matrices andhow to write a system of linear equations as a matrix equation. While this may have at first appeared to be merely a way of putting your coefficients in a table, matrices in fact have many interesting (but not immediately obvious!) properties.
Instructions: Please click on the link above, and read the indicated sections on pages 23–27. Section 2.1 introduces you to algebraic operations done with elements of Fn. In Section 2.2, you will explore the geometric meaning of vectors, and in Section 2.3, you will study the geometric interpretation of vector addition. These readings should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read the indicated sections on pages 27–31. In Section 2.4, you will study how distance is defined between two points in Rn. In Section 2.5, you will explore the geometric meaning of scalar multiplication. These readings 31 to Section 2.6, and complete problems 1, 2, 4, and 5. Next, click on "Solutions" (PDF) and check your answers on pages 5–6. This assessment should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read Section 2.7 on pages 32–36. In this section, you will learn about the concept of force. This reading 36, and complete problems 2, 3, 7, 9, 10, and 11. Next, click on "Solutions" (PDF) and check your answers on pages 6–8. This assessment should take you approximately 2 hours to complete.
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Instructions: Please click on the link above, and read the indicated sections on pages 39–47. Section 3.1 will provide the definition and properties of the dot product. Section 3.2 will discuss the geometric meaning of the dot product and then apply the ideas to the concepts of work and projection. These readings 47, and work on problems 1, 2, 4, 6, 10, 14, 15, 17, 20, and 21. Next, click on "Solutions" (PDF) and check your answers on pages 8–10. This assessment should take you approximately 2 hours and 30 minutes to complete.
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Instructions: Please click on the link above, and read Section 3.4 on pages 48–54. Section 3.4 will provide the definition, properties, and the geometric meaning of the cross product. This reading should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read Section 3.5 on pages 54–56. Section 3.5 will introduce a technique that will allow you to discover vector identities and simplify expressions involving cross and dot products in three dimensions. This reading should take you approximately 30 minutes to complete.
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Instructions: Please click on the link above to open the PDF. Scroll down to page 56, and complete problems 1, 4, 6, 7, 8, 9, 10, 13, 16, 18, and 20. Next, click on "Solutions" (PDF) and check your answers on pages 11–14 4.1 on pages 59–61. Section 4.1 will explore how to find solution(s) for a system of linear equations by graphing. This reading should take you approximately 30 minutes to complete.
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Instructions: Please click on the link above, and read Section 4.2 on pages 61–72. Section 4.2 will explore how to find solution(s) for a system of linear equations using elementary operations, Gaussian elimination, and other algebraic procedures 72, and complete problems 1, 6, 12, 13, 17, 19, 22, 30, 35, and 36. Next, click on "Solutions" (PDF) and check your answers on pages 14–20 5.1 on pages 77–91. Section 5.1 will provide an overview of matrices and matrix arithmetic. This reading should take you approximately 3 hours to complete.
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Despite the complicated definition of determinants, they are very useful because they allow you to determine, using only one number, whether or not a matrix is invertible. They also are helpful for computing eigenvalues, which you will learn about later. In this unit, you will learn about ways to find the determinants and to use determinants to compute the inverse of a matrix.
You will learn about the rank of the matrix, a very important concept in linear algebra. You will begin by studying the row-reduced echelon form of a matrix and proving that the row-reduced echelon form for a given matrix is unique. This is useful, because you can logically deduce important conclusions about the original matrix by examining its unique row-reduced echelon form. You will then learn that the rank of a matrix is related to the number of linearly independent columns or rows of that matrix; it describes the dimensionality of the space. It is also very important in the use of matrices to solve a system of linear equations, because it tells you whether Ax = 0 has zero, one, or an infinite number of solutions.
Finally, you will learn how matrices also arise in geometry, especially while studying certain types of linear transformations. You will learn that a mxn matrix can be used to transform vectors in Fn to vectors in Fm via matrix multiplication. As you will see, these types of transformation arise quite naturally in linear algebra and are important for applications in mathematics, physics, and engineering.
Instructions: Please click on the link above, and read Section 6.1 on pages 97–104. Section 6.1 will provide an introduction to determinants and techniques for finding them. This reading should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read Section 6.2 on pages 104–109. Section 6.2 will introduce some applications, including Cramer's rule. This reading should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read Section 8.1 on pages 129–134. Section 8.1 will introduce the elementary matrices, which result from doing a row operation to the identity matrix2 on pages 135–139. Section 8.2 will review the description of the row-reduced echelon form3 on pages 139–142. Section 8.3 will define rank and explain how to find the rank4 on pages 142–152. Section 8.4 will introduce linear independence and bases. This reading should take you approximately 2 hours to complete.
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Instructions: Please click on the link above, and read Section 8.5 on pages 153–156. Section 8.5will introduce the Fredholm Alternative for the case of real matrices here. This reading 156, and complete problems 2, 5, 7, 10, 12, 16, 18, 25, 32, 34, 45, 50, and 54. Next, click on "Solutions" (PDF) and check your answers on pages 38–49. This assessment should take you approximately 4 hours to complete.
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Instructions: Please click on the link above, and read Sections 9.1 and 9.2 on pages 163–173. Section 9.1 and Section 9.2 will introduce linear transformations. These readings should take you approximately 3 hours to complete.
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Instructions: Please click on the above link to open the PDF. Scroll down to page 173, and work on problems 2, 7, 9, 14, 17, 20, 23, 28, 32, 47, and 51. Next, click on "Solutions" (PDF) and check your answers on pages 49–59. This assessment should take you approximately 4 hours to complete.
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We have seen that matrix A corresponds to a linear transformation, T, i.e. T(x) =Ax. If the matrix is square, then those nonzero vectors, any of which are transformed to a multiple of themselves, are called eigenvectors of the matrix, and the mutliples by which they are transformed are called eigenvalues (eigenis the German word for characteristic). The set of eigenvalues is called the spectrum of A and plays an important role in linear algebra. In this unit, you will study the spectrum of a square matrix in detail.
The first thing that we will see is that the eigenvalues of an nxn matrix correspond to the roots of an nth degree polynomial, called the characteristic polynomial of A. As you know from algebra, roots of a polynomial can have multiplicity larger than 1. For eigenvalues, the multiplicity of a root is called the algebraic multiplicity of the eigenvalue, while the dimension of the set of vectors for which it is an eigenvalue is called the geometric multiplicity of the eigenvalue. If these two multiplicities are the same for all eigenvalues, then you will see that A is similar to a matrix with nonzero entries only along the main diagonal, that is A=S-1DS for an invertible matrix S. In fact, the diagonal entries of D are the eigenvalues of A, and the columns of S are a basis of Rn consisting of eigenvectors. Since solving polynomial equations can be difficult, you will study methods of estimating eigenvalues just by looking at the matrix.
You will then consider situations where the eigenvalues are known to be real and the matrix S, which diagonalizes A, can take a special form. If the matrix A is symmetric, that is aij=aji for all i,j, then you will see that A can be diagonalized, that is, all roots of the characteristic polynomial are real numbers. Further, you will learn how to choose the basis of eigenvalues, which comprise S so that the rows and columns each are unit vectors and are mutually perpendicular. Such matrices are called orthogonal and have the property that S-1=ST, where ST is the matrix obtained from S by interchanging its rows and columns. Since not all square matrices are diagonalizable, you will need one of the most important theorems in the spectral theory of matrices: Schur's Theorem, which is useful for analyzing the structure of matrices.
In linear algebra, you usually want to see whether or not Ax = b has any solutions. Often Ax = b has no solution because there are more equations than unknowns, that is, the linear system is inconsistent and b is not in the column space of A. In this case, you can try to find an approximation using a very important technique of the least square approximation. Another thing you want to do is to characterize when Ax = b has a solution, that is, construct conditions for solvability of the system. One way of doing this is by using Fredholm's Alternative, which is discussed in this unit. Fredholm's Alternative is important, because it can be generalized to more general vector spaces, where the concept of rank of a determinant is not defined. Finally, this unit discusses an important result known as the singular value decomposition, which gives a factorization of a matrix.
Instructions: Please click on the link above, and read Section 12.1 on pages 215–231. Spectral Theory refers to the study of eigenvalues and eigenvectors of a matrix, which is introduced in Section 12.1. This reading should take you approximately 2 hours to complete.
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Instructions: Please click on the link above and read Section 12.3 on pages 236–237. Section 12.3 introduces Gerschgorin's Theorem, which provides a way to estimate where the eigenvalues are just from looking at the matrix. This reading should take you approximately 30 minutes to complete.
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Instructions: Please click on the above link to open the PDF. Scroll down to page 237, and complete problems 3, 5, 8, 11, 13, 20, 25, 28, 43, 48, and 54. Next, click on "Solutions" (PDF) and check your answers on pages 76–84. This assessment should take you approximately 4 hours and 30 minutes to complete.
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Instructions: Please click on the link above, and read Section 13.1 on pages 245–255. Section 13.1 will introduce symmetric and orthogonal matrices2 on pages 255–262. Sections 13.2 will discuss several results including the Gram-Schmidt process and the Schur's Theorem3 on pages 263–266. Sections 13.3 discusses a very important technique known as the Least Square Approximation. This reading should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read the indicated sections on pages 267–274. Sections 13.4–13.7 will introduce several important topics and results. Please work through these sections carefully 274, and work on problems 2, 5, 10, 14, 19, 24, 28, 32, 37, 39, 43, 48, 52, and 57. Next, click on "Solutions" (PDF) and check your answers on pages 88–106. This assessment should take you approximately 4 hours to complete.
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This unit discusses vector spaces and linear transformations. The first topic you will study in this unit is that of an abstract vector space. You will see that a vector space is a collection of vectors that satisfies a set of axioms. Next, you will study ideas such as subspaces, linear independence, and bases in the context of vector spaces. Remember to think of R or C if you are confused. Next, you will construct abstract fields and vector spaces. You will begin this by first reviewing some basic algebra relating to polynomials. This is both interesting and important, because it provides the basis for constructing different kinds of fields. Finally, you will look at inner product spaces, which are vector spaces that also have an inner product, before moving on to linear transformations, which is the second topic of study in this unit.
Linear transformations have many applications within mathematics as well as in fields outside of mathematics, such as physics. You will study the basic definitions of linear transformations and properties and relations between these and matrices.
Instructions: Please click on the link above, and read Section 16.1 on page 323 and section 16.2.1 on pages 325–330. Section 16.1 will provide the definition of a vector space, and Section 16.2.1 will discuss subspaces, linear dependence, and bases 330, and complete problems 1, 2, 3, and 4. Next, click on "Solutions" (PDF) and check your answers on pages 124–125. This assessment should take you approximately 2 hours to complete.
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Instructions: Please click on the link above, and read Section 16.3 on pages 330–343. Section 16.3 will discuss vector space and fields. This reading should take you approximately 3 hours and 30 minutes to complete.
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Instructions: Please click on the link above to open the PDF. Scroll down to page 243, and work on problems 2, 4, 10, 12, 16, 20, 24, and 29. Next, click on "Solutions" (PDF) and check your answers on pages 125–135. This assessment should take you approximately 4 hours to complete.
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Instructions: Please click on the link above, and read Section 16.5 on pages 348–361. Section 16.5 will discuss vector spaces with inner products. This reading should take you approximately 3 hours and 30 minutes to complete.
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Instructions: Please click on the above link to open the PDF. Scroll down to page 361, and complete problems 1, 4, 7, 12, 15, 17, and 22. Next, click on "Solutions" (PDF) and check your answers on pages 135–150. This assessment should take you approximately 4 hours to complete.
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Instructions: Please click on the link above, and read the indicated sections on pages 367 and 368. Sections 17.1 and 17.2 will discuss linear transformations and vector spaces. These readings should take you approximately 30 minutes to complete.
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Instructions: Please click on the above link, and read Section 17.3 on pages 369–373. Section 17.3 will discuss finding eigenvalues and eigenvectors of linear transformations. This reading should take you approximately 1 hour and 30 minutes to complete.
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Instructions: Please click on the link above, and read Section 17.4 on pages 373–377. Sections 17.4 will discuss linear transformations, which will result by multiplication by n × n matrices. This reading should take you approximately 1 hour to complete.
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Instructions: Please click on the link above, and read Section 17.5 on pages 377–386. Section 17.4 will discuss matrices of linear transformations. This reading should take you approximately 2 hours to complete.
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Instructions: Please click on the link above, and read Section 17.6 on pages 386–391. Section 17.6 will introduce fundamental matrices, matrices whose entries are differentiable functions 391, and work on problems 1, 3, 8, 12, 15, 18, and 21. Next, click on "Solutions" (PDF) and check your answers on pages 151–157. This assessment should take you approximately 4 hours to complete
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Certificate of Use of Maths
Introduction
The course is suitable for students who attained a grade D or lower at GCSE in their secondary school. It provides an ideal opportunity for students to improve their mathematical skills. The course has an emphasis on the everyday application of mathematics. It consists of two Free Standing Mathematics Qualifications (FSMQs) in Money Management and Using Data, as well as a broader Core unit.
Further Details
This is an alternative to the GCSE Mathematics re-sit course and, with its emphasis on using mathematics in real life, enables students to develop skills that are highly regarded by employers.
Progression Options
Successful students would be able to attempt the GCSE Mathematics course.
Additional Info
Qualification:AQA Certificate of Use of Mathematics and FSMQ (Level 2)
Entry Requirements:College entry requirements
Duration:1 year
Assesment:The course is taught and assessed in 3
modules with each being assessed by a
written calculator exam taken in June. Each
examination is based on some pre-release
material.level may wish to continue to the
AS level in their second year. This can well
lead to some further study of Spanish in
Higher Education. GCSE Spanish will help
support a career in a wide range of areas,
including those in the Business and
Travel sectors.
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Fundamentals Business Mathematics
Fundamentals of Business Mathematics
is geared towards the first-year college student and provides basic
skills in business mathematics. As in previous editions, the philosophy
of the book is still "learning by doing," and the method is the successful
explanation-example-exercise format. The book covers traditional topics in
business mathematics along with a wide selection of related topics such as
taxes, insurance, statistics, and the metric system. The text is designed
to be flexible for the instructor. The wealth of topics and the text design
permits the instructor to tailor his or her course for any term length and
level of student maturity.
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Stock Status:In Stock Availability: Usually Ships in 5 to 10 Business Days
Product Code:9781741252415
About this book
Author Information
This Excel Mathematics Study Guide is essential for all students studying Year 9-10 Intermediate Mathematics as a comprehensive guide to the topics covered at this level. It provides tools to explain important mathematical concepts and gives ample opportunity to practise at a variety of levels.
Excel Mathematics Study Guide Years 9-10 contains: - topics covered in Years 9 and 10 Mathematics courses in Australia. - explanations of important concepts with examples and worked solutions. - checklists at the end of each chapter with page references to explanations. - plenty of exercises in the "Practise Practise" sections at the end of each chapter with page references to explanations. Answers and worked solutions are provided. - three levels of timed tests (straightforward, average difficulty and challenging) at the end of each chapter. Worked solutions and a marking system that provides feedback are provided. - two levels of timed sample exam papers that cover the entire two years' work and provide preparation for the final Years 9 and 10 Advanced Mathematics examinations. Worked solutions are provided. - a comprehensive index.
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textbook represents an extensive and easily understood introduction to tensor analysis, which is to be construed here as the generic term for classical tensor analysis and tensor algebra, and which is a requirement in many physics applications and in engineering sciences. Tensors in symbolic notation and in Cartesian and curvilinear co-ordinates are introduced, amongst other things, as well as the algebra of second stage tensors. The book is primarily directed at students on various engineering study courses. It imparts the required algebraic aids and contains numerous exercises with answers, making it eminently suitable for self study.
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These
classes are special sections of Basic Math Skills, Elementary Algebra, and Intermediate
Algebra in which students will learn mathematics primarily using the computer
software rather than the traditional lecture/discussion method of instruction.
Tests will be done by hand in the usual manner and there may be (occasionally) short lecture/discussions
in classes.
The same materials are used in the computer-mediated and lecture sections (and distance learning sections) of the courses, except that some of the lecture teachers may not use the web software much and ALL of the computer-mediated classes and distance learning classes will use the web software. In the ACC bookstores, the shrink-wrapped package of materials for this course includes a folder with the access number for the web software called MyMathLab. If you purchase a shrink-wrapped package of materials, please don't open it until you talk with your instructor to be sure you are in the right course. Shrink-wrapped materials are NOT RETURNABLE after they are opened.
Used books will not contain access to MyMathLab and so students must purchase that separately for about $75. New books purchased in some way other than through the ACC bookstore may or may not include access to MyMathLab.
The computer-mediated course is taught in a computer lab with Internet
access and students use the computer software during the class. Students may
use the software outside of class in the Learning Labs at the campuses, and
students may use the software at home, provided the computer meets the minimum
requirements.
Students in these classes are in charge of their learning in a way that
is different from a traditional lecture class. The format of the course is somewhat
self-paced. This allows the student some freedom to set the speed at which he/she
works through the material, which means that he/she may be able to complete
the course before the end of the semester. It also means that students may spend
less time on topics with which they are already familiar and more time on topics
that are troublesome for them. Students will be provided with a weekly schedule
of topics to be covered and a schedule of exams. In order to complete the course
within the sixteen-week semester, students must generally keep up with the weekly
schedule and test schedule. In order to succeed in this class, students should
plan to spend about 9 to 12 hours each week working on the material, depending
on how much of the material is already review for them.
To determine whether a computer-mediated math
class is right for you, please complete the attached survey which is available
as either PDF
file or RTF file. Please be aware that
in order to view the PDF file, you must have Adobe Acrobat Reader. To download
it free of charge, please visit:
The RTF file should open with any text program, such as Microsoft Word.
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Product Description
Review
"This book will fill an interesting niche in a library collection...it should be used by browsing students interested in making sure that they are prepared for success in their graduate programs." Choice
"All the Mathematics You Missed...is a help for students going on to graduate school..Since many students beginning graduate school do not have the mathematical knowledge needed, All the Mathematics You Missed aims to fill in the gaps." Berkshire Eagle, Pittsfield, MA
"From the preface: 'The goal of this book is to give people at least a rough idea of many topics that beginning graduate students at the best graduate schools are assumed to know." Mathematical Reviews
"The writing is lucid mathematical exposition, at a level quite appropriate to beginning graduate students." The American Statistician
"Before classes began, I jump started my graduate career with the help of this book. Even though I didn't believe that I could have missed much math, it became clear that my belief was wrong during the first week of class. While proving a theorem, my professor asked if anyone remembered a previous result from calculus. While I did not remember it from my days as an undergraduate, I had read about the theorem and had even seen a sketch of the proof in Garrity's book...This will be one of the books that I keep with me as I continue as a graduate student. It has certainly helped me understand concepts that I have missed." Elizabeth D. Russell, Math Horizons
"Point set topology, complex analysis, differential forms, the curvature of surfaces, the axiom of choice, Lebesgue integration, Fourier analysis, algorithms, and differential equations.... I found these sections to be the high points of the book. They were a sound introduction to material that some but not all graduate students will need." Charles Ashbacher, School Science and Mathematics
Book Description
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. This book will help readers to fill in the gaps in their preparation by presenting the basic points and a few key results of the most important undergraduate topics in mathematics: linear algebra, vector calculus, geometry, real analysis, algorithms, probability, set theory, and more. By emphasizing the intuitions behind the subject, the book makes it easy for students to quickly get a feel for the topics that they have missed and to prepare for more advanced courses.
This book has a very particular purpose: to recap some basic concepts from undergraduate mathematics so that you get the "big picture". In other words, for every math course you took as an undergrad, this book provides a good outline of the major ideas and how they fit together. But, it is only an outline; nothing more. If you actually missed out on some topic, or your knowledge of a subject is shaky, then this book won't help much. It will only help by providing a bibliography of some other references for that subject.
This book is meant to organize your undergraduate math knowledge, not to supplement it.
With that said, I'll mention a few words about the content of the book. It is quite well written and definitely extracts the essential ideas for your quick consumption. There are a few topics that I personally feel are missing, such as Gram-Schmidt and Jordan Canonical Forms for Linear Algebra, and UFDs and PIDs from Algebra. In general, it seemed like the book leaned a little more towards analysis than algebra, but the vast majority of important topics were indeed encapsulated in their synopsis.
Good for a very specific audience, but otherwise not wonderfully useful.
There's no doubt about it -- this book designed for people who want to learn some real math. It doesn't take, as the title and description might lead you to believe, a "Math for Engineers" approach.
Each chapter covers, in the span of 10 or 15 pages, what would normally be an entire semester's worth of material, and as a result, is quite dense -- there are alot of ideas crammed onto each page. But unlike traditional advanced math books (which are notoriously dense) the focus is more on developing intuitions than on long strings of equations.
An important strength is that every chapter ends with suggestions on textbooks in that chapter's subject. This turns out to be quite helpful, since one can't reasonably expect to learn everything important about any of these subjects from a brief chapter in any book.
I can envision three main ways in which this book might be useful: First, in combination with one or more of the books in listed in the bibliography for learning a new subject. Second, on its own for review of topics you've seen before. Third, as a reference for "basic" definitions and theorems, as in: "What's a Hilbert space again?"
Overall, this will be a good book to have around, but not a substitute for real study.
I used this book for an opposite purpose to the one the author intended. For me it served to review all the math I *had* learned long ago in school (both undergraduate and graduate), but was starting to forget. The author's informal style and rapid-fire delivery were just right for these topics. The subjects I had truly missed, mainly the more abstract parts of algebra and geometry, I found difficult to follow, though I did come away with some feeling for them. This is not a perfect book. The informal style extends to numerous typos in equations, and modern computer-oriented approaches get short shrift. Nevertheless, I found it a unique resource and a pleasure to read.
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Starting at $41.45 7/3 cornerstone of Elementary Linear Algebra is the authors' clear, careful, and concise presentation of material--written so that students can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system.The Sixth Edition incorporates up-to-date coverage of Computer Algebra Systems (Maple/MATLAB/Mathematica); additional support is provided in a corresponding technology guide. Data and applications also reflect current statistics and examples to engage students and demonstrate the link between theory and practice.
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Keith Devlin
Consortium for Mathematics and Its Applications
Topic: math Age Level: advanced
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Maple
Maple is a computer algebra system that helps you analyse, explore, visualize, and solve mathematical problems.
More information
Maple is a computer algebra system, designed to solve mathematical problems and produce high-quality technical graphics. Maple incorporates a high-level programming language that allows you to define your own procedures; it also has packages of specialized functions you can load to do work in group theory, linear algebra, and statistics, as well as in other fields. You can use it interactively or in batch mode, for teaching or research.
Availability
Install using the Program installer for University PCs
Available for home use. Check the Terms and Conditions before you download
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Elementary student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses.Students who approach math with trepidation will find that Elementary Algebra, Second Edition, builds competence and confidence. The study system, introduced at the outset and used consistently throughout the text, transforms the student experience by applying time-tested s... MOREtrategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the 'big picture' of algebra. While using Elementary Algebra, Second Edition, you will find that the text focuses on building competence and confidence. The authors present the concepts, show how to do the math, and explain the reasoning behind it in a language you can understand. The text ties concepts together using the Algebra Pyramid, which will help you see the big picture of algebra. The skills Carson presents through both the Learning Strategy boxes and the Study System, introduced in the Preface and incorporated throughout the text, will not only enhance your elementary algebra experience but will also help you succeed in future college courses. Book jacket.
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Most students find it difficult to determine exactly how much time, they should spend in activities in studying. This is particularly important right at the beginning of the study so as to make an effective study plan.
This is complicated even more by the fact that every chapter requires different amount of time. Even different subjects require different amounts of time in theory and problems.
The different questions that come to the mind of the student are:
How much time should be spent in reading from the textbook?
Are the notes from tuition/coaching enough? Can I manage without reading the textbook?
Is there anything to read from the textbook in Mathematics?
How much time should I spend in solved examples given in the IITJEE course material>
How much time should be allocated to a certain topic?
How much time should be spent on a problem that is not getting solved before looking at the solution or asking for help?
How much time should I spend in testing at home?
How should I calculate the total time required?
We have tried to answer these questions by giving an indicative time plan.
List of chapters (with recommended time slotted)
Topic
Total
Reading (textbook)
Solved examples
Conceptual problems
Exercises (problems)
Chapter test
Mathematics
1
Complex numbers
21
2
2
1
16
2
2
Quadratic equations
19
1
1
1
16
1
3
Logarithms
6
1
5
1
4
Progressions
10
1
2
1
6
2
5
Permutations and combinations
22
1
2
1
18
2
6
Trigonometry
35
2
2
1
30
2
7
Straight lines
18
2
3
1
12
3
8
Circles
21
2
4
1
14
4
9
Conic sections
34
4
4
1
25
4
10
Binomial theorem
33
2
3
1
27
3
11
Functions, Limits and Continuity
49
4
4
1
40
4
12
Differentiability and differentiation
19
1
4
1
13
4
13
Application of derivatives
33
1
3
1
28
3
14
Indefinite integration
10
1
3
1
5
3
15
Definite integration
10
1
3
1
5
3
16
Area under the curve
19
1
5
1
12
5
17
Differential equations
14
1
4
1
8
4
18
Determinants
23
1
5
1
16
5
19
Matrices
11
1
2
1
7
2
20
Probability
16
1
4
1
10
4
21
Vectors
13
1
3
1
8
3
22
Three dimensional geometry
10
2
2
1
5
2
Total
446
34
65
21
326
66
Physics
1
Units, dimensions, vectors and calculus
15
2
2.5
0.5
10
2
2
Kinematics
13
3
2.5
0.5
7
3
3
Laws of motion
18
2
2.5
0.5
13
2
4
Work, Power and Energy
17
2
2.5
0.5
12
2
5
Center of mass, linear momentum, collision
28
4
3
1
20
4
6
Rotational dynamics
33
4
3
1
25
4
7
Elasticity, fluid dynamics and properties of matter
35
4
3
1
27
4
8
Gravitation
16
2
1.5
0.5
12
2
9
Simple Harmonic Motion
21
3
2.5
0.5
15
3
10
Wave motion
23
4
2.5
0.5
16
4
11
Heat and Thermodynamics
48
5
5.5
2.5
35
5
12
Electrostatics
45
5
3.5
1.5
35
7
13
Electric current and resistance
28
4
3
1
20
4
14
Magnetism
27
4
2
1
20
4
15
Electromagnetic Induction and AC
18
3
2
1
12
3
16
Geometrical Optics
21
4
2
1
14
4
17
Wave Optics
18
4
2
1
11
4
18
Modern Physics
18
5
2
1
10
5
Total
442
64
47.5
16.5
314
66
Chemistry
1
Basic concepts of chemistry
18
3
2.5
0.5
12
2
2
Structure of atom
15.5
3
2.5
0
10
2
3
Periodic properties
10
3
1
6
2
4
Gas laws
21
4
2.5
0.5
14
3
5
Chemical bonding
15
3
2
10
2
6
Chemical energetics
18
3
2.5
0.5
12
2
7
Chemical equilibrium
20
4
1.5
0.5
14
3
8
Ionic equilibrium
23
4
1.5
0.5
17
3
9
Redox reactions
16
3
2.5
0.5
10
2
10
General organic chemistry
29
5
1.5
0.5
22
4
11
Hydrocarbons
16
4
12
3
12
Alcohols and ethers
13
3
10
2
13
Alkyl and aryl halides
13
5
8
4
14
Solutions
26
3
2.5
0.5
20
3
15
Solid state
21
3
2.5
0.5
15
2
16
Chemical kinetics
20
3
2.5
0.5
14
2
17
Electrochemistry
25.5
3
2.5
20
2
18
Nuclear chemistry
14.5
3
1.5
10
2
19
Functional groups containing nitrogen
14
4
10
3
20
Aldehydes and ketones
14
4
10
3
21
Carboxylic acids and their derivatives
19
4
15
3
22
s-Block elements
17
5
1.5
0.5
10
4
23
p-Block elements
24
5
1.5
0.5
17
4
24
d-Block elements
19
5
1.5
0.5
12
4
25
Metallurgy
19
5
1.5
0.5
12
4
26
Qualitative salt analysis
19
5
1.5
0.5
12
4
27
Coordination compounds
15
5
1.5
0.5
8
4
Total
494.5
104
40.5
8
342
78
Chapter tests
210
Full length tests
120
at least 20 tests of various formats, of 6 hrs each
Self assessment
30
Revision / other material
60
Total Time (Required)
1802.5
This is just a recommendation. Students can make changes to the study plan based on their proficiency in the subjects. The actual time spent by the student can vary by 10% – 15% depending on the student's personal style of study. Please consider that the time given here is the minimum that a student needs to spend. The total time spent in studying for IITJEE across 2 years should not be less than 10% of the given.
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My Learning Philosophy: Everybody learns. Your struggles, your efforts determine how well or how much. I expect students to have a desire to learn the mathematics in this course. Students attend class, ask questions, work hard and take responsibility for learning. I am here to help all students learn.
COURSE DESCRIPTION: This course is designed as a foundational course for those students who must take additional mathematics in their chosen majors and do not yet have an appropriate background. The emphasis is the study of mathematics from a functional perspective—including linear, quadratic, rational, absolute value, radical, exponential and logarithmic functions. Systems of equations and inequalities and applications such as curve fitting, mathematical modeling, optimization and exponential growth and decay are included.
Course-related Institutional Learning Outcomes & Assessment Methods
College Algebra MAC1105
Institutional Learning Outcomes
Chapter Test
Quantitative and Analytical Reasoning: The student will understand and apply mathematical and scientific principles and methods.
1. Perform accurate computations using order of operations with and without technology.
X
2. Identify and organize relevant information and complete the solution of an applied problem.
X
3. Interpret and communicate understanding of visual representations of data.
X
4. Demonstrate mathematical number sense and unit sense.
X
Attendance is Mandatory:Every student is required to attend all class meetings. Students are responsible for all information/material/assignments covered in class. Usually something important is covered everyday. It may be something you do not realize yet. Come to class. Listen. Take notes. Ask questions. Engage in your learning.
Contact the instructor if any situation occurs that will affect class attendance.
N0 Make-up Tests will be given unless the Student has written verification of an emergency situation and contacts the instructor in advance.
This class Meets Monday and Wednesday
The Final Exam will be given on Monday, Dec. 10 during class time.
HINTS FOR SUCCESS: Take responsibility for learning this subject, attend class, listen, ask questions, answer questions, participate, take notes, do all the homework you can possibly do, do at least some homework daily, dedicate yourself, when things go bad or when things go good rededicate yourself to learning this subject, use my office hours, find someone to study with, enjoy yourself, congratulate yourself on your progress. Mathematics is man's or perhaps God's greatest invention. Good Luck. Peace.
Materials Needed:
Graphing Calculator (Preferably a TI-Inspire)
3 Ring Binder with 10 dividers
Sharpened Pencil
College Ruled Paper
Graph Paper
Grading/Evaluation:
GRADING: Your grade will be determined by your performance on quizzes, tests, homework assignments and the final exam. The final grade will be determined by a simple quotient; Total Points Attained ¸Total Points Possible.
Mostof your points will be determined by Tests.Some of your points will be determined by homework assignments. Some of your points will be determined by in class homework assignments and quizzes and projects. Some of your points will be determined by the Final Exam. Good Luck. Get all the points you can. At the end of the semester you will receive one of the following grades based on your quotient.
100% - 90% A 79.99% - 77% C + 59.99% - F
89.99% - 87% B+ 76.99% - 70% C
86.99% -80% B 69.99% - 60% D
College Policies – Fall 2012
Academic Integrity – Cheating and/or plagiarism will not be tolerated and may result in an "F" for the course as well as disciplinary action under the Code of Student Conduct. A student may be referred to an Academic Integrity Seminar. This two-hour seminar costs $40 and attendance is required (see Student Planner).
Access Services – It is your responsibility to register with the Access Services Office should you have a verifiable and documented disability which may require reasonable accommodation(s). Further, it is your responsibility to provide your instructor with the Faculty Notification Sheet, which sets forth the reasonable accommodation(s) determined by the Access Services Office. Registration with Access Services should be done at the beginning of the Term. For information see
Classroom Decorum – Disruptive behavior will not be tolerated. Disruptive students will be asked to leave the classroom. Continuous disruptive behavior will result in withdrawal from the course and disciplinary action under the Code of Student Conduct (see Student Planner).
Withdrawal – If you want to withdraw from this class, you must fill out the necessary forms and have them signed by the appropriate parties. If you just "stop coming to class" after the posted drop date, you may receive the grade of "F."
? Add/Drop period: Aug. 20-24
? Drop only: Aug. 20-21
? Last date for a refund: Aug. 24
? First mini-semester withdrawal deadline: Sept. 22
? Last day to withdraw with a "W": Oct. 25
? Second mini-semester withdrawal deadline: Nov. 16
The college reserves the right to evaluate individual cases of non-attendance.
Students should be alerted to the fact that
(1) withdrawals do not count in the CF G.P.A, but may not be viewed favorably at the university level or for financial aid;
(2) a withdrawal counts as an attempt under the forgiveness/withdrawal policy and the course repeat policy;
(3) there are increased costs to take the course on the third attempt.
(4) there may be a reason a withdrawal request may be denied. Please see the College's withdrawal procedures.
College Preparatory Courses–State law requires no more than three attempts TOTAL to complete all college preparatory (English, mathematics and reading) courses. Students registered in college prep courses who receive N grade must repeat the same course and complete it with a grade of C of better before they can register for other courses that require the successful completion of the prep as a requirement.
TOTAL POINTS—
Your grade is based on a compilation of the tatal averagesHomework
Home Learning 15%
Quizzes 25%
Tests 40%
Final Exam 20%
TOTAL POSSIBLE PERCENTAGE: 100%
Grading Scale:
A 90%-100% B+ 87%-89% B 80%-86% C+ 77%-79%
C 70%-76% D 60%-69% F<60%
7 Tests are scheduled for this class. Only 6 will count towards final grade. The lowest test score will be dropped. The lowest test and the lowest quiz score will be dropped. If you miss a test or quiz, then thatscore will be your lowest and will be dropped. If you miss more than one test or quiz then your scores will be averaged into your final grade. (Only 1 Test and 1 Quiz will be dropped). The renaining scores will count towards your final average.
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Professional Commentary: Astronomers now know that the planets in our solar system travel around the Sun in roughly circular orbits. People used to believe that the Sun and other planets orbited the Earth....
Professional Commentary: Several interrelated problems are posed to students. All involve the minute and hour hands of a clock and the path traced by the midpoint of the segment connecting the ends of the hands. Students begin by assuming that both hands are the same length and that the clock runs properly and shows the correct time....
Professional Commentary: This online tutorial discusses the standard and polar forms for complex numbers and the basic operations in each system. Euler's Equation presents the link between the two representations....
Professional Commentary: The short course covers the history as well as the mathematics of complex numbers. The development of the imaginary unit i and the proof of the Fundamental Theorem of Algebra are followed by explorations of the complex plane....
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Microsoft Math is a tool designed to cooperate with the development of mathematical formulae and function graphics.
The world of mathematical calculations is huge and, in order to take the first steps, it is necessary to have not only the support of the professors but also the tools which offer didactic methodologies for learning.
This is the case with Microsoft Math, because it has tools to aid work in Algebra, Trigonometry, Chemistry, Physics and Mathematical Calculus. Each of these areas can be approached from a basic level up to a very advanced level.
Some of the Features which enhance Microsoft Math are: the edition of 2 and 3 dimensional graphics, starting from the structure of the function you wan to study, tools to treat equations at different levels, a numerical constants conversion tool and a wide formulae and equations consultation library.
Microsoft Math Related tags
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Search MAA Reviews:
MAA Reviews
Basic Algebra
Anthony W. Knapp
Table of Contents
Contents.- Preface.- Guide for the Reader.- Preliminaries about the Integers, Polynomials, and Matrices.- Vector Spaces over Q, R, and C.- Inner-Product Spaces.- Groups and Group Actions.- Theory of a Single Linear Transformation.- Multilinear Algebra.- Advanced Group Theory.- Commutative Rings and Their Modules.- Fields and Galois Theory.- Modules over Noncommutative Rings.- Appendix.- Index.
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Given a photo, students will learn the process of linearizing data by taking an nth root, use the data to predict outcomes that are not in the data set, form a better understanding of correlation of determination and the correlation coefficient, and create multiple representations of a data set.
Lesson Objectives:
- Create and use representations to organize, record, and communicate mathematical ideas
- Use Mathematical models to represent and understand quantitative relationships
- Interpret representations of functions of two variables
- Approximate and interpret rates of change from graphical and numerical data
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Visual and interactive way to thorough understanding and mastering Trigonometry without getting wearied on the very first chapter! Java- and web-based math course includes theoretical concepts, hands-on examples featuring animated graphics and live formulas, problem-solving lessons, and customizable real time tests with solutions and evaluations.
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Key Curriculum Releases IMP Year 4, 2nd Edition
IMP is four-year core mathematics curriculum and is aligned with Common Core State Standards. Adoption of the IMP curriculum includes implementation strategies, supplemental materials, blackline masters, calculator guides, and assessment tools.
Year 4 covers topics such as statistical sampling, computer graphics and animation, an introduction to accumulation and integrals, and an introduction to sophisticated algebra, including transformations and composition.
The second edition of Year 4 includes a new student textbook, 2 new unit books, and three updated unit books
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu.
PROFESSOR STRANG: Finally we get to positive definite matrices. I've used the word and now it's time to pin it down. And so this would be my thank you for staying with it while we do this important preliminary stuff about linear algebra. So starting the next lecture we'll really make a big start on engineering applications. But these matrices are going to be the key to everything. And I'll call these matrices K and positive definite, I will only use that word about a symmetric matrix. So the matrix is already symmetric and that means it has real eigenvalues and many other important properties, orthogonal eigenvectors. And now we're asking for more. And it's that extra bit that is terrific in all kinds of applications. So if I can do this bit of linear algebra.
So what's coming then, my review session this afternoon at four, I'm very happy that we've got, I think, the best MATLAB problem ever invented in 18.085 anyway. So that should get onto the website probably by tomorrow. And Peter Buchak is like the MATLAB person. So his review sessions are Friday at noon. And I just saw him and suggested Friday at noon he might as well just stay in here. And knowing that that isn't maybe a good hour for everybody. So you could see him also outside of that hour. But that's the hour he will be ready for all kinds of questions about MATLAB or about the homeworks. Actually you'll be probably thinking more also about the homework questions on this topic.
Ready for positive definite? You said yes, right? And you have a hint about these things. So we have a symmetric matrix and the beauty is that it brings together all of linear algebra. Including elimination, that's when we see pivots. Including determinants which are closely related to the pivots. And what do I mean by upper left? I mean that if I have a three by three symmetric matrix and I want to test it for positive definite, and I guess actually this would be the easiest test if I had a tiny matrix, three by three, and I had numbers then this would be a good test. The determinants, by upper left determinants I mean that one by one determinant. So just that first number has to be positive. Then the two by two determinant, that times that minus that times that has to be positive. Oh I've already been saying that. Let me just put in some letters. So a has to be positive. This is symmetric, so a times c has to be bigger than b squared. So that will tell us. And then for two by two we finish. For three by three we would also require the three by three determinant to be positive.
But already here you're seeing one point about a positive definite matrix. Its diagonal will have to be positive. And somehow its diagonal has to be not just above zero, but somehow it has to defeat b squared. So the diagonal has to be somehow more positive than whatever negative stuff might come from off the diagonal. That's why I would need a*c > b squared. So both of those will be positive and their product has to be bigger than the other guy.
And then finally, a third test is that all the eigenvalues are positive. And of course if I give you a three by three matrix, that's probably not the easiest test since you'd have to find the eigenvalues. Much easier to find the determinants or the pivots. Actually, just while I'm at it, so the first pivot of course is a itself. No difficulty there. The second pivot turns out to be the ratio of a*c - b squared to a. So the connection between pivots and determinants is just really close. Pivots are ratios of determinants if you work it out. The second pivot, maybe I would call that d_2, is the ratio of a*c - b squared over a. In other words it's (c - b squared)/a. Determinants are positive and vice versa. Then it's fantastic that the eigenvalues come into the picture.
So those are three ways, three important properties of a positive definite matrix. But I'd like to make the definition something different. Now I'm coming to the meaning. If I think of those as the tests, that's done. Now the meaning of positive definite. The meaning of positive definite and the applications are closely related to looking for a minimum. And so what I'm going to bring in here, so it's symmetric. Now for a symmetric matrix I want to introduce the energy. So this is the reason why it has so many applications and such important physical meaning is that what I'm about to introduce, which is a function of x, and here it is, it's x transpose times A, not A, I'm sticking with K for my matrix, times x. I think of that as some f(x).
And let's just see what it would be if the matrix was two by two, [a, b; b, c]. Suppose that's my matrix. We want to get a handle on what, this is the first time I've ever written something that has x's times x's. So it's going to be quadratic. They're going to be x's times x's. And x is a general vector of the right size so it's got components x_1, x_2. And there it's transpose, so it's a row. And now I put in the [a, b; b, c]. And then I put in x again. This is going to give me a very nice, simple, important expression. Depending on x_1 and x_2. Now what is, can we do that multiplication? Maybe above I'll do the multiplication of this pair and then I have the other guy to bring in. So here, that would be ax_1+bx_2. And this would be bx_1+cx_2. So that's the first, that's this times this. What am I going to get? What shape, what size is this result going to be? This K is n by n. x is a column vector. n by one. x transpose, what's the shape of x transpose? One by n? So what's the total result? One by one. Just a number. Just a function. It's a number. But it depends on the x's and the matrix inside.
Can we do it now? So I've got this to multiply by this. Do you see an x_1 squared showing up? Yes, from there times there. And what's it multiplied by? The a. The first term is this times the ax_1 is a(x_1 squared). So that's our first quadratic. Now there'd be an x_1, x_2. Let me leave that for a minute and find the x_2 squared because it's easy. So where am I going to get x_2 squared? I'm going to get that from x_2, second guy here times second guy here. There's a c(x_2 squared).
So you're seeing already where the diagonal shows up. The diagonal a, c, whatever is multiplying the perfect squares. And it'll be the off-diagonal that multiplies the x_1, x_2. We might call those the crossterms. And what do we get for that then? We have x_1 times this guy. So that's a crossterm. bx_1*x_2, right? And here's another one coming from x_2 times this guy. And what's that one? It's also bx_1*x_2. So x_1, multiply that, x_2 multiply that, and so what do we have for this crossterm here? Two of them. 2bx_1*x_2. In other words, that b and that b came together in the 2bx_1*x_2. So here's my energy. Can I just loosely call it energy? And then as we get to applications we'll see why.
So I'm interested in that because it has important meaning. Well, so now I'm ready to define positive definite matrices. So I'll call that number four. But I'm going to give it a big star. Even more because it's the sort of key. So the test will be, you can probably guess it, I look at this expression, that x transpose Ax. And if it's a positive definite matrix and this represents energy, the key will be that this should be positive. This one should be positive for all x's. Well, with one exception, of course. All x's except, which vector is it? x=0 would just give me-- See, I put K. My default for a matrix, but should be, it's K. Except x=0, except the zero vector. Of course. If x_1 and x_2 are both zero.
Now that looks a little maybe less straightforward, I would say, because it's a statement about this is true for all x_1 and x_2. And we better do some examples and draw a picture. Let me draw a picture right away. So here's x_1 direction. Here's x_2 direction. And here is the x transpose Ax, my function. So this depends on two variables. So it's going to be a sort of a surface if I draw it. Now, what point do we absolutely know? And I put A again. I am so sorry. Well, we know one point. It's there whatever that matrix might be. It's there. Zero, right? You just told me that if both x's are zero then we automatically get zero.
Now what do you think the shape of this curve, the shape of this graph is going to look like? The point is, if we're positive definite now. So I'm drawing the picture for positive definite. So my definition is that the energy goes up. It's positive, right? When I leave, when I move away from that point I go upwards. That point will be a minimum. Let me just draw it roughly. So it sort of goes up like this. These cheap 2-D boards and I've got a three-dimensional picture here. But you see it somehow? What word or what's your visualization? It has a minimum there. That's why minimization, which was like, the core problem in calculus, is here now. But for functions of two x's or n x's. We're up the dimension over the basic minimum problem of calculus. It's sort of like a parabola It's cross-sections cutting down through the thing would be just parabolas because of the x squared.
I'm going to call this a bowl. It's a short word. Do you see it? It opens up. That's the key point, that it opens upward. And let's do some examples. Tell me some positive definite. So positive definite and then let me here put some not positive definite cases. Tell me a matrix. Well, what's the easiest, first matrix that occurs to you as a positive definite matrix? The identity. That passes all our tests, its eigenvalues are one, its pivots are one, the determinants are one. And the function is x_1 squared plus x_2 squared with no b in it. It's just a perfect bowl, perfectly symmetric, the way it would come off a potter's wheel.
Now let me take one that's maybe not so, let me put a nine there. So I'm off to a reasonable start. I have an x_1 squared and a nine x_2 squared. And now I want to ask you, what could I put in there that would leave it positive definite? Well, give me a couple of possibilities. What's a nice, not too big now, that's the thing. Two. Two would be fine. So if I had a two there and a two there I would have a 4x_1*x_2 and it would, like, this, instead of being a circle, which it was for the identity, the plane there would cut out a ellipse instead. But it would be a good ellipse. Because we're doing squares, we're really, the Greeks understood these second degree things and they would have known this would have been an ellipse.
How high can I go with that two or where do I have to stop? Where would I have to, if I wanted to change the two, let me just focus on that one, suppose I wanted to change it. First of all, give me one that's, how about the borderline. Three would be the borderline. Why's that? Because at three we have nine minus nine for the determinant. So the determinant is zero. Of course it passed the first test. One by one was ok. But two by two was not, was at the borderline. What else should I think? Oh, that's a very interesting case. The borderline. You know, it almost makes it. But can you tell me the eigenvalues of that matrix? Don't do any quadratic equations.
How do I know, what's one eigenvalue of a matrix? You made it singular, right? You made that matrix singular. Determinant zero. So one of its eigenvalues is zero. And the other one is visible by looking at the trace. I just quickly mentioned that if I add the diagonal, I get the same answer as if I add the two eigenvalues. So that other eigenvalue must be ten. And this is entirely typical, that ten and zero, the extreme eigenvalues, lambda_max and lambda_min, are bigger than, these diagonal guys are inside. They're inside, between zero and ten and it's these terms that enter somehow and gave us an eigenvalue of ten and an eigenvalue of zero.
I guess I'm tempted to try to draw that figure. Just to get a feeling of what's with that one. It always helps to get the borderline case. So what's with this one? Let me see what my quadratic would be. Can I just change it up here? Rather than rewriting it. So I'm going to, I'll put it up here. So I have to change that four to what? Now that I'm looking at this matrix. That four is now a six. Six. This is my guy for this one. Which is not positive definite.
Let me tell you right away the word that I would use for this one. I would call it positive semi-definite because it's almost there, but not quite. So semi-definite allows the matrix to be singular. So semi-definite, maybe I'll do it in green what semi-definite would be. Semi-def would be eigenvalues greater than or equal zero. Determinants greater than or equal zero. Pivots greater than zero if they're there or then we run out of pivots. You could say greater than or equal to zero then. And energy, greater than or equal to zero for semi-definite. And when would the energy, what x's, what would be the like, you could say the ground states or something, what x's, so greater than or equal to zero, emphasize that possibility of equal in the semi-definite case.
Suppose I have a semi-definite matrix, yeah, I've got one. But it's singular. So that means a singular matrix takes some vector x to zero. Right? If my matrix is actually singular, then there'll be an x where Kx is zero. And then, of course, multiplying by x transpose, I'm still at zero. So the x's, the zero energy guys, this is straightforward, the zero energy guys, the ones where x transpose Kx is zero, will happen when Kx is zero. If Kx is zero, and we'll see it in that example.
Let's see it in that example. What's the x for which, I could say in the null space, what's the vector x that that matrix kills? , right? The vector . gives me . That's the vector that, so I get 3-3, the zero, 9-9, the zero. So I believe that this thing will be-- Is it zero at three, minus one? I think that it has to be, right? If I take x_1 to be three and x_2 to be minus one, I think I've got zero energy here. Do I? x_1 squared will be at the nine and nine x_2 squared will be nine more. And what will be this 6x_1*x_2? What will that come out for this x_1 and x_2? Minus 18. Had to, right? So I'd get nine from there, nine from there, minus 18, zero.
So the graph for this positive semi-definite will look a bit like this. There'll be a direction in which it doesn't climb. It doesn't go below the base, right? It's never negative. This is now the semi-definite picture. But it can run along the base. And it will for the vector x_1=3, x_2=-1, I don't know where that is, one, two, three, and then maybe minus one. Along some line here the graph doesn't go up. It's sitting, can you imagine that sitting in the base? I'm not Rembrandt here, but in the other direction it goes up. Oh, the hell with that one. Do you see, sort of? It's like a trough, would you say? I mean, it's like a, you know, a bit of a drainpipe or something. It's running along the ground, along this direction and in the other directions it does go up. So it's shaped like this with the base not climbing. Whereas here, there's no bad direction. Climbs every way you go. So that's positive definite and that's positive semi-definite.
Well suppose I push it a little further. Let me make a place here for a matrix that isn't even positive semi-definite. Now it's just going to go down somewhere. I'll start again with one and nine and tell me what to put in now. So this is going to be a case where the off-diagonal is too big, it wins. And prevents positive definite. So what number would you like here? Five? Five is certainly plenty. So now I have [1, 5; 5, 9]. Let me take a little space on a board just to show you. Sorry about that. So I'm going to do the [1, 5; 5, 9] just because they're all important, but then we're coming back to positive definite. So if it's [1, 5; 5, 9] and I do that usual x, x transpose Kx and I do the multiplication out, I see the one x_1 squared and I see the nine x_2 squareds. And how many x_1*x_2's do I see? Five from there, five from there, ten. And I believe that can be negative. The fact of having all nice plus signs is not going to help it because we can choose, as we already did, x_1 to be like a negative number and x_2 to be a positive. And we can get this guy to be negative and make it, in this case we can make it defeat these positive parts.
What choice would do it? Let me take x_1 to be minus one and tell me an x_2 that's good enough to show that this thing is not positive definite or even semi-definite, it goes downhill. Take x_2 equal? What do you say? 1/2? Yeah, I don't want too big an x_2 because if I have too big an x_2, then this'll be important. Does 1/2 do it? So I've got 1/4, that's positive, but not very. 9/4, so I'm up to 10/4, but this guy is what? Ten and the minus is minus five. Yeah. So that absolutely goes, at this one I come out less than zero. And I might as well complete.
So this is the case where I would call it indefinite. Indefinite. It goes up like if x_2 is zero, then it's just got x_1 squared, that's up. If x_1 is zero, it's only got x_2 squared, that's up. But there are other directions where it goes downhill. So it goes either up, it goes both up in some ways and down in others. And what kind of a graph, what kind of a surface would I now have for x transpose for this x transpose, this indefinite guy? So up in some ways and down in others. This gets really hard to draw. I believe that if you ride horses you have an edge on visualizing this. So it's called, what kind of a point's it called? Saddle point, it's called a saddle point. So what's a saddle point? That's not bad, right? So this is a direction where it went up. This is a direction where it went down. And so it sort of fills in somehow.
Or maybe, if you don't, I mean, who rides horses now? Actually maybe something we do do is drive over mountains. So the path, if the road is sort of well-chosen, the road will go, it'll look for the, this would be-- Yeah, here's our road. We would do as little climbing as possible. The mountain would go like this, sort of. So this would be like, the bottom part looking along the peaks of the mountains. But it's the top part looking along the driving direction. So driving, it's a maximum, but in the mountain range direction it's a minimum. So it's a saddle point. So that's what you get from a typical symmetric matrix. And if it was minus five it would still be the same saddle point, would still be 9-25, it would still be negative and a saddle.
Positive guys are our thing. Alright. So now back to positive definite. With these four tests and then the discussion of semi-definite. Very key, that energy. Let me just look ahead a moment. Most physical problems, many, many physical problems, you have an option. Either you solve some equations, either you find the solution from our equations, Ku=f, typically. Matrix equation or differential equation. Or there's another option of minimizing some function. Some energy. And it gives the same equations. So this minimizing energy will be a second way to describe the applications.
Now can I get a number five? There's an important number five and then you know really all you need to know about symmetric matrices. This gives me, about positive definite matrices, this gives me a chance to recap. So I'm going to put down a number five. Because this is where the matrices come from. Really important. And it's where they'll come from in all these applications that chapter two is going to be all about, that we're going to start. So they come, these positive definite matrices, so this is another way to, it's a test for positive definite matrices and it's, actually, it's where they come from. So here's a positive definite matrix. They come from A transpose A. A fundamental message is that if I have just an average matrix, possibly rectangular, could be a square but not symmetric, then sooner or later, in fact usually sooner, you end up looking at A transpose A. We've seen that already. And we already know that A transpose A is square, we already know it's symmetric and now we're going to know that it's positive definite. So matrices like A transpose A are positive definite or possibly semi-definite. There's that possibility. If A was the zero matrix, of course, we would just get the zero matrix which would be only semi-definite, or other ways to get a semi-definite.
So I'm saying that if K, if I have a matrix, any matrix, and I form A transpose A, I get a positive definite matrix or maybe just semi-definite, but not indefinite. Can we see why? Why is this positive definite or semi-? So that's my question. And the answer is really worth, it's just neat and worth seeing. So do I want to look at the pivots of A transpose A? No. They're something, but whatever they are, I can't really follow those well. Or the eigenvalues very well, or the determinants. None of those come out nicely. But the real guy works perfectly. So look at x transpose Kx. So I'm just doing, following my instinct here.
So if K is A transpose A, my claim is, what am I saying then about this energy? What is it that I want to discover and understand? Why it's positive. Why does taking any matrix, multiplying by its transpose produce something that's positive? Can you see any reason why that quantity, which looks kind of messy, I just want to look at it the right way to see why that should be positive, that should come out positive. So I'm not going to get into numbers, I'm not going to get into diagonals and off-diagonals. I'm just going to do one thing to understand that particular combination, x transpose A transpose Ax. What shall I do? Anybody see what I might do? Yeah, you're seeing here if you look at it again, what are you seeing here? Tell me again. If I take Ax together, then what's the other half? It's the transpose of Ax. So I just want to write that as, I just want to think of it that way, as Ax. And here's the transpose of Ax. Right? Because transposes of Ax, so transpose guys in the opposite order, and the multiplication--
This is the great. I call these proof by parenthesis because I'm just putting parentheses in the right place, but the key law of matrix multiplication is that, that I can put (AB)C is the same as A(BC). That rule, which is just multiply it out and you see that parentheses are not needed because if you keep them in the right order you can do this first, or you can do this first. Same answer. What do I learn from that? What was the point? This is some vector, I don't know especially what it is times its transpose. So that's the length squared. What's the key fact about that? That it is never negative. It's always greater than zero or possibly equal.
When does that quantity equal zero? When Ax is zero. When Ax is zero. Because this is a vector. That's the same vector transposed. And everybody's got that picture. When I take any y transpose y, I get y_1 squared plus y_2 squared through y_n squared. And I get a positive answer except if the vector is zero. So it's zero when Ax is zero. So that's going to be the key. If I pick any matrix A, and I can just take an example, but chapter, the applications are just going to be full of examples. Where the problem begins with a matrix A and then A transpose shows up and it's the combination A transpose A that we work with. And we're just learning that it's positive definite.
Unless, shall I just hang on since I've got here, I have to say when is it, have to get these two possibilities. Positive definite or only semi-definite. So what's the key to that borderline question? This thing will be only semi-definite if there's a solution to Ax=0. If there is an x, well, there's always the zero vector. Zero vector I can't expect to be positive. So I'm looking for if there's an x so that Ax is zero but x is not zero, then I'll only be semi-definite. That's the test. If there is a solution to Ax=0.
When we see applications that'll mean there's a displacement with no stretching. We might have a line of springs and when could the line of springs displace with no stretching? When it's free-free, right? If I have a line of springs and no supports at the ends, then that would be the case where it could shift over by the vector. So that would be the case where the matrix is only singular. We know that. The matrix is now positive semi-definite. We just learned that. So the free-free matrix, like B, both ends free, or C. So our answer is going to be that K and T are positive definite. And our other two guys, the singular ones, of course, just don't make it. B at both ends, the free-free line of springs, it can shift without stretching. Since Ax will measure the stretching when it just shifts rigid motion, the Ax is zero and we see only positive definite. And also C, the circular one. There it can displace with no stretching because it can just turn in the circle. So these guys will be only positive semi-definite.
Maybe I better say this another way. When is this positive definite? Can I use just a different sentence to describe this possibility? This is positive definite provided, so what I'm going to write now is to remove this possibility and get positive definite. This is positive definite provided, now, I could say it this way. The A has independent columns. So I just needed to give you another way of looking at this Ax=0 question. If A has independent columns, what does that mean? That means that the only solution to Ax=0 is the zero solution. In other words, it means that this thing works perfectly and gives me positive. When A has independent columns.
Let's just remember our K, T, B, C. So here's a matrix, so let me take the T matrix, that's this one, this guy. And then the third column is . Those three columns are independent. They point off. They don't lie in a plane. They point off in three different directions. And then there are no solutions to, no x's that's go Kx=0. So that would be a case of independent columns. Let me make a case of dependent columns. So and I'm going to make it B now. Now the columns of that matrix are dependent. There's a combination of them that give zero. They all lie in the same plane. There's a solution to that matrix times x equal zero. What combination of those columns shows me that they are dependent? That some combination of those three columns, some amount of this plus some amount of this plus some amount of that column gives me the zero vector. You see the combination. What should I take? again. No surprise. That's the vector that we know is in the everything shifting the same amount, nothing stretching.
Talking fast here about positive definite matrices. This is the key. Let's just ask a few questions about positive definite matrices as a way to practice. Suppose I had one. Positive definite. What about its inverse? Is that positive definite or not? So I've got a positive definite one, it's not singular, it's got positive eigenvalues, everything else. It's inverse will be symmetric, so I'm allowed to think about it. Will it be positive definite? What do you think? Well, you've got a whole bunch of tests to sort of mentally run through. Pivots of the inverse, you don't want to touch that stuff. Determinants, no. What about eigenvalues? What would be the eigenvalues if I have this positive definite symmetric matrix, its eigenvalues are one, four, five. What can you tell me about the eigenvalues of the inverse matrix? They're the inverses. So those three eigenvalues are? 1, 1/4, 1/5, what's the conclusion here? It is positive definite. Those are all positive, it is positive definite. So if I invert a positive definite matrix, I'm still positive definite.
All the tests would have to pass. It's just I'm looking each time for the easiest test. Let me look now, for the easiest test on K_1+K_2. Suppose that's positive definite and that's positive definite. What if I add them? What do you think? Well, we hope so. But we have to say which of my one, two, three, four, five would be a good way to see it. Would be a good way to see it. Good question. Four? We certainly don't want to touch pivots and now we don't want to touch eigenvalues either. Of course, if number four works, others will also work. The eigenvalues will come out positive. But not too easy to say what they are. Let's try test number four. So K_1. What's the test? So test number four tells us that this part, x transpose K_1*x, that that part is positive, right? That that part is positive. If we know that's positive definite. Now, about K_2 we also know that for every x, you see it's for every x, that helps, don't let me put x_2 there, for every x this will be positive.
And now what's the step I want to take? To get some information on the matrix K_1+K_2. I should add. If I add these guys, you see that it just, then I can write that as, I can write that this way. And what have I learned? I've learned that that's positive, even greater than, except for the zero vector. Because this was greater than, this is greater than. If I add two positive numbers, the energies are positive and the energies just add. The energies just add. So that definition four was the good way, just nice, easy way to see that if I have a couple of positive definite matrices, a couple of positive energies, I'm really coupling the two systems. This is associated somehow. I've got two systems, I'm putting them together and the energy is just even more positive. It's more positive either of these guys because I'm adding.
As I'm speaking here, will you allow me to try test number five, this A transpose A business? Suppose K_1 was A transpose A. If it's positive definite, it will. Be And suppose K_2 is B transpose B. If it's positive definite, it will be. Now I would like to write the sum somehow as, in this something transpose something. And I just do it now because I think it's like, you won't perhaps have thought of this way to do it. Watch. Suppose I create the matrix [A; B]. That'll be my new matrix. Say, call it C. Am I allowed to do that? I mean, that creates a matrix? These A and B, they had the same number of columns, n. So I can put one over the other and I still have something with n columns. So that's my new matrix C. And now I want C transpose. By the way, I'd call that a block matrix. You know, instead of numbers, it's got two blocks in there. Block matrices are really handy.
Now what's the transpose of that block matrix? You just have faith, just have faith with blocks. It's just like numbers. If I had a matrix [1; 5] then I'd get a row one, five. But what do you think? This is worth thinking about even after class. What would be, if this C matrix is this block A above B, what do you think for C transpose? A transpose, B transpose side by side. Just put in numbers and you'd see it. And now I'm going to take C transpose times C. I'm calling it C now instead of A because I've used the A in the first guy and I've used B in the second one and now I'm ready for C. How do you multiply block matrices? Again, you just have faith. What do you think? Tell me the answer. A transpose, I multiply that by that just as if they were numbers. And I add that times that just as if they were numbers. And what do I have? I've got K_1+K_2. So I've written K_1, this is K_1+K_2 and this is in my form C transpose C that I was looking for, that number five was looking for. So it's done it. It's done it. The fact of getting A, K_1 in this form, K_2 in this form. And I just made a block matrix and I got K_1+K_2. That's not a big deal in itself, but block matrices are really handy. It's good to take that step with matrices. Think of, possibly, the entries as coming in blocks and not just one at a time.
Well, thank you, okay. I swear Friday we'll start applications in all kinds of engineering problems and you'll have new applications
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Review for AP Calculus
To be successful in AP Calculus you need to be able to identify the
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Overview
Main description
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Advanced Algebra: Linear Algebra
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Advanced Algebra: Linear Algebra Word Problems
A company`s employees are working
A company's employees are working to create a new energy bar. They would like the two key ingredients to be peanut butter and oats, and they want to make sure they have enough carbohydrates and protein in the bar to supply the athlete. They want a total of 31 carbohydrates and 23 grams of protein to make the bar sufficient. Using the following table, create a system of two equations and two unknowns to find how many tablespoons of each ingredient the bar will need. Solve the system of equations using matrices. Show all work to receive full credit.
Carbohydrates
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Peanut Butter
2
6
Oats
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1
A football stadium has 60,000 seats
A football stadium has 60,000 seats. The manager divides the stadium into two sections for the exhibition games, the games the home team plays outside its division, and the games it plays within its division. The tickets are worth $19 in section A and $14 in section B. Assuming that all tickets can be sold, how many seats must he assign to each section on says of an exhibition game, a game outside the division, and a game within the division to bring in the following revenues: $970,000 for each exhibition game, $990,000 for each game outside the division, and $1,120...
Advanced Algebra: Linear Algebra Practice Questions
Solve the following simultaneous equations using matrices:
2X-Y=3
X+5Y=7
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Schaum's Outlines present all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.A monograph is concerned with exchange rings in various conditions related to stable range. It discusses diagonal reduction of regular matrices and cleanness of square matrices. It includes topics such...Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate...
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Publisher Comments* Assumes virtually no prior knowledge
* Numerous worked examples, exercises and challenge questions
Modular Mathematics is a new series of introductory texts for undergraduates. Builiding on both the skills and knowledge acquired at A level, each book provides a lively and accessible account of the subject. Examples and exercises are used as teaching aids throughout and ideas for investigative and project work help to place the subject in context.
Synopsis:
provides a lively and accessible account of the subject. Examples and exercises are used as teaching aids throughout and ideas for investigative and project work help to place the subject in context.
Synopsis"Synopsis"
by Elsevier,
provides a lively and accessible account of the subject. Examples and exercises are used as teaching aids throughout and ideas for investigative and project work help to place the subject in context.
"Synopsis"
by Elsevier,
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Calculus is widely recognized as a difficult course that requires extra study and practice. The Calculus Workbook for Dummies will continue the light-hearted, practical approach taken in the original book, while providing practice opportunities and detailed solutions to hundreds of problems that will help students master the maths that is critical for future success in engineering, scince, and other complex disciplines.
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Written by a professional physicist, Calculus in Focus teaches you everything you need to know about first semester calculus. Topics covered include computation of limits and derivatives, continuity, one-sided limits, finding maxima and minima, related rates problems, implicit differentiation, integration and more. Each chapter is packed with sample problems that guide the reader through the procedures used to solve calculus problems. End of chapter exercises, based on the solved problems in the...
First published by Silvanus P. Thompson in 1910, this text aims to make the topic of calculus accessible to students of mathematics. In the first major revision of the text since 1946, Martin Gardner has thoroughly updated the text to reflect developments in method and terminology, written an extensive preface and three new chapters, and added more than 20 recreational problems for practice and enjoyment.
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Based on a series of lectures given by the author this text is designed for undergraduate students with an understanding of vector calculus, solution techniques of ordinary and partial differential equations and elementary knowledge of integral transforms. It will also be an invaluable reference to scientists and engineers who need to know the basic mathematical development of the theory of complex variables in order to solve field problems. The theorems given are well illustrated with examples.
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How does cooperation emerge among selfish individuals? When do people share resources, punish those they consider unfair, and engage in joint enterprises? These questions fascinate philosophers, biologists, and economists alike, for the "invisible hand" that should turn selfish efforts into public benefit is not always at work. The Calculus of Selfishness looks at social dilemmas where cooperative motivations are subverted and self-interest becomes self-defeating. Karl Sigmund, a pioneer in evolutionary...
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This complete guide to numerical methods in chemical engineering is the first to take full advantage of MATLAB's powerful calculation environment. Every chapter contains several examples using general MATLAB functions that implement the method and can also be applied to many other problems in the same category.
The authors begin by introducing the solution of nonlinear equations using several standard approaches, including methods of successive substitution and linear interpolation; the Wegstein method, the Newton-Raphson method; the Eigenvalue method; and synthetic division algorithms. With these fundamentals in hand, they move on to simultaneous linear algebraic equations, covering matrix and vector operations; Cramer's rule; Gauss methods; the Jacobi method; and the characteristic-value problem. Additional coverage includes:
The numerical methods covered here represent virtually all of those commonly used by practicing chemical engineers. The focus on MATLAB enables readers to accomplish more, with less complexity, than was possible with traditional FORTRAN. For those unfamiliar with MATLAB, a brief introduction is provided as an Appendix.
The accompanying website contains MATLAB 5.0 (and higher) source code for more than 60 examples, methods, and function scripts covered in the book. These programs are compatible with all three operating systems: Windows(r), MacOS(r), and UNIX(r).
Description:
A wide variety of problems are associated with the flow
of shallow water, such as atmospheric flows, tides, storm surges, river and coastal flows, lake flows, tsunamis. Numerical simulation is an effective tool in solving them and a great ...
Description:
High resolution upwind and centred methods are today a mature
generation of computational techniques applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a ...
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Third edition of popular undergraduate-level text offers overview of historical roots and evolution of several areas of mathematics. Topics include mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, and more. Emphasis on axiomatic proce...
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Algebra 1 Explorer software will turn struggling students into successful math learners and average students into accelerated math learners!
Spark your students' interest in algebra with state-of-the-art graphing tools and interactive lessons that encourage students to explore and learn algebraic concepts.
The Algebra 1 Explorer features a comprehensive algebra 1 curriculum that is packed with resources for your class, including student self-paced tutorials, interactive class presentation material, as well as student learning activities and worksheets.
The Algebra 1 Explorer has many exploratory graphing tools that will help you emphasize key points during class presentations. And the interactive lessons with compelling visual representations will add excitement and energy to the learning process, keeping students focused on learning.
MathRealm's teaching methodology, employed in all of our curriculum programs, focuses on allowing the student to interact with a concept before the concept is developed. The journey begins with the student discovering the concept through a virtual manipulative. After this interaction the program methodically builds the concept in a logical manner, allowing the student to see why and how the mathematical process works. Problem-solving strategies and real-life occurrences are woven into the concept-building part of the program. Research demonstrates that this approach leads to a deeper understanding.
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As mentioned, all kinds of Math are important as a general point, I'm of the opinion problem solving is likely going to make you better than branching into learning a bunch of math for the sake of becoming a better programmer.
However, Discrete Math is probably a good starting point. – mjsabbyJan 16 '10 at 11Boolean Algebra. Of all the math
courses I ever did understanding
boolean logic stands out as by far
the most useful. Just formally
knowing the basics, such as NOT A
AND NOT B == NOT(A OR B) will get
you far. This is the one math subject that justifies the question
Linear Algebra is surprisingly
useful too. I've done a
considerable amount of 3D work so
that is essential there, but it pops
up all over the place to some
degree.
Statistics. Everyone should have
some grasp of basic stats
And finally one off the wall - Fractal algebra. Partly because it's just fun, but also because understanding what turbulence is and knowing to reach for a Perlin noise function or similar when otherwise you might just use white noise is a major win.
Algebra is a must for anyone aiming to be good at mathematics but for a CS student, discrete mathematics is a must. For the sake of algorithm analysis, you should study statistics along with differential equations and calculus. Also if you intend to study DSP, Fourier analysis is a must.
Even if you plan to never go outside the realm of algorithms and data structures, some math is really useful.
Discrete maths and number theory is a must imho.
An entry level (linear) algebra class is always useful. As noted above, algebra pops up everywhere.
Statistics is a must no matter what you intend to do(within CS).
Calculus, well, nice to have, but I would prioritize the discrete, statistics, number theory and algebra first(in that order).
All in all, most entry level math classes are probably good to have. Then you can build upon that later if you find you like it and/or need it. It's great that you already are thinking about math. Too many underestimate the usefulness of mathematics.
I think knowledge of Probability and Statistics is a must in every field of computer, including software development. Sooner or later, a developer needs to profile his/her program and analyze the statistical results and then this knowledge comes really essential.
It sounds as though you already know that all programmers, web developers or otherwise, could benefit from giving their brains a bit of a maths work out. Studying maths is unlikely to inform your hands-on-technical knowledge but it will refine your logical thinking skills.
I'd agree with Mark Harrison on studying Discrete Maths, a great subject encompassing a lot of concepts used in computer science. I'd second the book that Mark recommends - its actually the one that I used in my undergraduate studies. It was pretty good back then and such textbooks generally improve through subsequent editions.
Discrete Maths is a big topic however, and its very difficult to work one's way through such a meaty textbook. One way of jumpstarting your studies could be via a book of Discrete Maths puzzles with solutions. (I have this book and its written in an engaging and fun style). Work through the puzzles and if you get stuck then explore the new concepts in greater depth using the Grimaldi book.
All programmers will benefit from studying Algorithms too. I stumbled across some video lectures from an MIT course on Algorithms. I learnt a lot from these!
The ability to collect, analyze and present data to others is a skill that you can always use whether you're a programmer or you move on to other things like management. I currently develop software, and in the past hardware, that is always very speed/performance critical. When debugging and looking for bottlenecks, it is easy to point fingers and make wild assumptions. When you have data to back your argument, it is easy to prove your point.
Just a few of the cases where I have used statistics:
determining data flow volumes to aid in making hardware/IT purchase decisions for new sites
analyzing lab and fields results to determine precision and accuracy of equipment
You don't need to be able to do high-level calculus to utilize statistics. You just need to understand, and then apply the concepts. It helps to know how to use a few tools too, like Excel or R, so that you don't have to roll your own scripts.
I think maths in general is quite important as I think it helps you think in a structured and logical way that would help you if you want to be a developer.
For the time being I am finding anll the maths, calculus and also statistics very useful. I am currently doing a Masters in Biometrics and all the years of learning maths and statistics if truely proving very helpful.
Regards
Shivam
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It's a pilot project, so it remains to be seen if this approach makes a difference in improving the pass rates for students in lower-level math courses like College Algebra, which have been at around 60…
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Summary
4-color hardback text w/complete text-specific instructor and student print/media supplement package AMATYC/NCTM Standards of Content and Pedagogy integrated in Exercise Sets, Sourced-Data Applications (students are also asked to generate and interpret data), Scientific and Graphing Calculator Explorations Boxes, Mental Math exercises, Conceptual and Writing exercises, geometric concepts, Group Activities, Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews 6 step Problem-Solving Approach introduced in Chapter 2 and reinforced throughout the text in applications and exercises helps students tackle a wide range of problems Early and intuitive introduction to the concept of graphing reinforced with bar charts, line graphs, calculator screens, application illustrations and exercise sets. Emphasis on the notion of paired data in Chapters 1 and 2 leads naturally to the concepts of ordered pair and the rectangular coordinate system introduced in Chapter 3. Graphing and concepts of graphing linear equations such as slope and intercepts reinforced through exercise sets in subsequent chapters, preparing students for equations of lines in Chapter 7
Table of Contents
Preface
ix
Review of Real Numbers
1
(72)
Symbols and Sets of Numbers
2
(9)
Fractions
11
(8)
Exponents and Order of Operations
19
(6)
Introduction to Variable Expressions and Equations
25
(5)
Adding Real Numbers
30
(6)
Subtracting Real Numbers
36
(5)
Multiplying and Dividing Real Numbers
41
(7)
Properties of Real Numbers
48
(7)
Reading Graphs
55
(18)
Group Activity: Creating and Interpreting Graphs
62
(1)
Highlights
63
(4)
Review
67
(3)
Test
70
(3)
Equations, Inequalities, And Problem Solving
73
(90)
Simplifying Algebraic Expressions
74
(7)
The Addition Property of Equality
81
(7)
The Multiplication Property of Equality
88
(6)
Solving Linear Equations
94
(10)
An Introduction to Problem Solving
104
(8)
Formulas and Problem Solving
112
(10)
Percent and Problem Solving
122
(9)
Further Problem Solving
131
(8)
Solving Linear Inequalities
139
(24)
Group Activity: Calculating Price Per Unit
149
(1)
Highlights
150
(6)
Review
156
(3)
Test
159
(1)
Cumulative Review
160
(3)
Graphing
163
(64)
The Rectangular Coordinate System
164
(11)
Graphing Linear Equations
175
(10)
Intercepts
185
(10)
Slope
195
(13)
Graphing Linear Inequalities
208
(19)
Group Activity: Financial Analysis
217
(1)
Highlights
218
(4)
Review
222
(2)
Test
224
(1)
Cumulative Review
225
(2)
Exponents And Polynomials
227
(54)
Exponents
228
(10)
Adding and Subtracting Polynomials
238
(8)
Multiplying Polynomials
246
(6)
Special Products
252
(5)
Negative Exponents and Scientific Notation
257
(9)
Division of Polynomials
266
(15)
Group Activity: Making Predictions Based on Historical Data
273
(1)
Highlights
274
(2)
Review
276
(2)
Test
278
(1)
Cumulative Review
279
(2)
Factoring Polynomials
281
(62)
The Greatest Common Factor and Factoring by Grouping
282
(7)
Factoring Trinomials of the Form x2 + bx + c
289
(6)
Factoring Trinomials of the Form ax2 + bx + c
295
(9)
Factoring Binomials
304
(5)
Choosing a Factoring Strategy
309
(5)
Solving Quadratic Equations by Factoring
314
(10)
Quadratic Equations and Problem Solving
324
(19)
Group Activity: Choosing Among Building Options
333
(1)
Highlights
334
(3)
Review
337
(2)
Test
339
(1)
Cumulative Review
340
(3)
Rational Expressions
343
(68)
Simplifying Rational Expressions
344
(7)
Multiplying and Dividing Rational Expressions
351
(5)
Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator
356
(7)
Adding and Subtracting Rational Expressions with Unlike Denominators
363
(6)
Simplifying Complex Fractions
369
(7)
Solving Equations Containing Rational Expressions
376
(7)
Ratio and Proportion
383
(7)
Rational Equations and Problem Solving
390
(21)
Group Activity: Comparing Formulas for Doses of Medication
399
(1)
Highlights
400
(5)
Review
405
(2)
Test
407
(1)
Cumulative Review
408
(3)
Further Graphing
411
(42)
The Slope-Intercept Form
412
(6)
The Point-Slope Form
418
(7)
Graphing Nonlinear Equations
425
(8)
Functions
433
(20)
Group Activity: Matching Descriptions of Linear Data to Their Equations and Graphs
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Generalizing patterns makes up a large part of the Minnesota academic standards for algebra as they develop from Kindergarten to grade 8. Concepts of equality, variable, and function are topics that middle school students struggle with greatly in algebra and are all connected when students generalize patterns.
IN THIS SESSION TEACHERS:
Solve problems that may be used as a way to get students to make connections between arithmetic and algebra..
Learn ways to ask questions that facilitate the development of algebraic habits of mind.
Learn how to emphasize equivalent expressions and introduce algebraic proofs.
Learn ways that children develop patterns over time focusing on recursive and explicit features.
Standards Specifically Addressed
MINNESOTA ACADEMIC STANDARDS
Grade 3: Use single-operation input-output rules to represent patterns and relationships and to solve real-world and
mathematical problems.
Grade 4: Use input-output rules, tables and charts to represent patterns and relationships and to solve real-world and
mathematical problems.
Grade 6: Recognize and represent relationships between varying quantities; translate from one representation to another;
use patterns, tables, graphs and rules to solve real-world and mathematical problems.
Grade 8: Understand the concept of function in real-world and mathematical situations, and distinguish between linear
and non-linear functions.
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From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
Strong algebra skills are crucial to success in applied calculus. This text is designed to bolster these skills while students study applied calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra topics and points out potential problem spots. The table of contents is organized so that the algebra topics are arranged in the order in which they are needed for applied calculus.
The geometry of rectangles, triangles, and circles, expressing one quantity as a function of one or more other quantities.
9. Exponential and Logarithmic Functions.
Working with graphs, expressions and equations of exponential functions, inverse functions, and logarithmic functions.
Appendices.
A. Completing the Square and Deriving the Quadratic Formula.
B. Factorials.
C. The Binomial Theorem.
Answers to Exercises.
Index.
Features & benefits
Organization. The book is organized by topic from most standard calculus courses for management and the life sciences. For instance, Chapter 4 provides the algebra background students will need as they are studying the limit, and Chapter 5 provides the algebra background for coverage of the derivative.
Student-friendly. The book is written in an easy style that can be understood by all students on their own, without input by the instructor.
Flexibility. Just-in-Time Algebra can be used as a student companion in a standard applied calculus course, or as a second text for courses where time is allotted for a review of algebra.
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guide2
Course: MATH 20082, Fall 2009 School: Bethel VA Rating:
Word Count: 134
Document Preview
263B Math Guide for Test 2 Here are some sample questions from sections 5.5, 6.16.3. Some topics that we covered are not represented by these questions, but are still fair game. 1. Evaluate each of the following integrals. A method is suggested, but you may use another method if you prefer.
5
(a) Rules/algebra:
2 4
x2 - 3 dx x
2
(b) Substitution:
3 1
xex dx
(c) dx.
10
(d) Parts:
0
xe2x Partial Fractions:
9 263B Guide for Test 3 Here are some sample questions from sections 6.5, 6.6, 7.1. Some topics that we covered are not represented by these questions, but are still fair game. 1. Consider the integral1cos(3x2 + 1)dx .0(a) Compute an approxi
MAC-layer approaches for security and performance enhancement in IEEE 802.11 by Hao-Li WangA thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHYMajor: Computer Engineering Pro
Incremental impact analysis for object-oriented softwareby Luke BishopA thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCEMajor: Computer Engineering Program of Study Committee
Math 163A A06Fall 2007Your first Good Problem is due in class on Friday September 7. Good Problems are graded half on content and half on presentation. The only specific presentation skills that you are responsible for so far are described in the2 5x + 6 2 x (x 2h)2 (b) lim h0 h x1 (c) lim
Math 163A Guide for Test 3 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Compute the following derivatives: (a) f (x) = 2 + x + f (x) = (b) y = (c) Dx
Math 163A Guide for Test 4 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Sketch the graph of a single function that has all of the following properties:
Math 163A A06 Fall 2007 Guide for the Final Exam The final exam is Tuesday 20 November, at 2:30pm, in our classroom. This is also the final deadline for any late good problems. The exam is cumulative, but there will not be any questions specifically
Algorithms and procedures to analyze physiological signals in psychophysiological research by Joset Amy EtzelA dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHYMajor:
Math 446/546Spring 2005p.1Guide for Test 1 The test is in class on Friday 22 April. Bring a calculator (for arithmetic). We did not fully cover Section 6.6 in the homework. I suggest that you do Section 6.6 problems 2b, 3b, and 4b. Guarante
Math 446/546Spring 2005p.1Guide for Final Exam The final exam is on Tuesday 7 June from 12:202:20pm, in our classroom. Bring a calculator (for arithmetic). The exam is cumulative, so any question that would be fair for the first two tests is
Math 266B Winter 2006 Guide for Test 1 The first test is in class on Friday 20 January. Here are some sample questions, so that you have an idea of what to expect. 1. (a) (b)0ex dx =2x3 dx = x-3 dx = 3x-1 dx =3(c) (d) (e)1sin(3)dx =(f)
Math 266B Winter 2006 Guide for Test 2 The second test is in class on Friday 3 February. Here are some sample questions, so that you have an idea of what to expect. You can use the following table of integrals for any of the questions: 1 dx = ln |
Math 266B Winter 2006 Guide for Test 3 The third test is in class on Friday 17 February. Here are some sample questions, so that you have an idea of what to expect. 1. Solve each differential equation. dy = 2 cos(3t) , where y(0) = 7. dt dy (b) = 2y
Math 266B Winter 2006 Guide for Test 4 The fourth test is in class on Friday 3 March February. Here are some sample questions, so that you have an idea of what to expect. 1. Solve the systems of equations: (a) x+y =1 3x + 3y = 1 3x - 2y - z = -9 -
Math 266B Guide for Test 1 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. (a) (b)0ex dx =2x3 dx = x-3 dx = 3x-1 dx =3(c) (d) (e)1sin(3)dx =
Math 266B Guide for Test 2 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. You can use the following table of integrals for any of the questions: 1 dx = ln
Math 266B Guide for Test 3 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Solve each differential equation. dy = 2 cos(3t) , where y(0) = 7. dt dy (b) = 2
Math 266B Guide for Test 4 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Solve the systems of equations: (a) x+y =1 3x + 3y = 1 3x - 2y - z = -9 -x +
Math 266B Guide for Final Exam Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Suppose f is a differentiable function with the following properties: f (0)
Maestro: A remote execution tool for visualization clusters by Aron Lee BierbaumA thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCEMajor: Computer Engineering Program of Study
Math 163AFall 2003Your first Good Problem is due in class on Friday September 12. Good Problems are graded half on content and half on presentation. The only specific presentation skills that you are responsible for so far are described in the La
Math 163A Fall 2003 Guide for Test 1 The first test is in class on Friday 26 September. Here are some sample questions, so that you have an idea of what to expect. `*' means a number or function that would be filled in. 1. (a) Find the equation for t-2 - 5x + 6 2 x - (x - 2h)2 (b) lim h0 h x-1 (c)
Sampled charge reuse for power reduction in switched capacitor data convertersbySaqib Qayyum MalikA dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHYMajor: Electri
Math 163A Guide for Test 3 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Sketch the graph of a single function that has all of the following properties:
Math 163A Guide for Test 4 Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. A company wishes to manufacture a box with a volume of 6 m3 that is open on top
Investigation of Magneto-optical Properties for Optical Fiber Based Devices.byRashmi BahugunaA dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHYProgram of Study Co
Math 163A Fall 2006 Guide for the Final Exam The nal exam is Saturday 18 November, at 12:20 pm, in our classroom. This is also the nal deadline for any late good problems. The exam is cumulative, but there will not be any questions specically from Ch
Security and social implications of radio frequency identificationbyAdrienne N. HuffmanA thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCEMajor: Electrical Engineering Prog
Math 344Winter 2007Your first Good Problem is due in class on Thursday January 11, as part of homework set 1. Good Problems are graded half on content and half on presentation. The only specific presentation skills that you are responsible for so
Math 344Sample Test Problems from Part II1Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. (a) Using pivoting if needed, find the LU decomposition of
Math 344Sample Test Problems from Part III1Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. What are: Polynomial Interpolations, Splines and Least Squ
Math 344Sample Test Problems from Part IV1Here are some sample questions from old tests. Some topics that we covered are not represented by these questions, but are still fair game. 1. Write a Matlab program to do n steps of the modified Euler
Supervisory control of discrete event systems for bisimulation and simulation equivalence specifications by Changyan ZhouA dissertation submitted to the graduate committee in partial fulfillment of the requirements for the degree of DOCTOR OF PHILO
APPENDIX A ACCURACY, PRECISION, ERRORS, UNCERTAINTY, ETC. In common speech, the words accuracy and precision are often used interchangeably. However, many scientists like to make a distinction between the meanings of the two words. Accuracy refers to
System-level design refinement using SystemCbyRobert Dale WalstromA thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCEMajor: Computer Engineering Program of Study Committee:
Graphing Functions with M AT L AB 1One of the nice features of M AT L AB is that this computer algebra system allows you to easily define functions. For example, consider the linear function u(t) = 0.6t + 1.2. We are now going to define this functio
Compositions of Functions and Their Graphs 1Let f (x) = x3 - 4x2 + x - 1. 1. Use the ezplot function to plot a graph of this function on the interval [-2, 4]. (Instructions for ezplot are in the "Graphing Functions with M AT L AB" Exercise.) 2. Let
APPENDIX B SUGGESTIONS FOR TABULATING AND PLOTTING DATA Much of the data you take can and should be collected into tables for clarity. Any plot you make should have an associated table, and the location of the table (page number in the notebook) shou
Cooperative Spatial Multiplexing System by Andreas DarmawanA thesis submitted to the graduate faculty in partial fulllment of the requirements for the degree of MASTER OF SCIENCEMajor: Electrical Engineering Program of Study Committee: Sang W. Ki
APPENDIX C INSTRUMENTS USED IN THE PHYSICS 321 LABORATORY The instruments we use in the laboratory provide or measure DC and AC voltages and/or currents. The tables below list the specifications for devices actually used in the Physics 321 Laboratory
APPENDIX D DISCUSSION OF ELECTRONIC INSTRUMENTS DC POWER SUPPLIES We will discuss these instruments one at a time, starting with the DC power supply. The simplest DC power supplies are batteries which supply an essentially constant voltage as long as
Curve Fitting, Loglog Plots, and Semilog Plots 1In this M AT L AB exercise, you will learn how to plot data and how to fit lines to your data. Suppose you are measuring the height h of a seedling as it grows. The height (measured in centimeters) wil
DC MEASUREMENTSEXPERIMENT 1:DC INSTRUMENTS AND MEASUREMENTS9/4/03In this experiment we will learn about some of the laboratory instruments which will be used during the semester, and gain experience with measuring DC currents and voltages. We
Math 266A Winter 2005 Guide for Test 1 The first test is in class on Friday 21 January. Here are some sample questions, so that you have an idea of what to expect. `*' means a number or function that would be filled in. 1. (a) Find the equation for t
Finding limits in M AT L AB 1If this M AT L AB excercise is being counted in your grade, then please record your answers to all the questions on a separate sheet for submission. In this exercise, you will learn how to use M AT L AB to find limits of
Spatial signal processing in wireless sensor networks by Benhong ZhangA dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHYMajor: Electrical Engineering Program of Study
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Honors Algebra I is the first of five sequential courses in the academic math track and is designed for the math students who possess a strong background in their math skills. Honors Algebra I is offered to students in grades 8. Major units of study include; solving first degree equations and inequalities; solving second degree equations, using four basic operations; learning monomials, polynomials, and algebraic fractions using factoring - all varieties, graphing linear equations; solving work problems with application of the above skills; and applying systems (3 ways). Math vocabulary and spelling skills are developed. Use of the graphing calculator will be applied to various concepts throughout the course. Students are exposed to related careers and emphasis is placed on the need for math skills in life work. Major assignments may include graphing projects. The depth of course coverage and the complexity of the algebra problems offered in the program serve to differentiate the different program levels of honors and college prep.
Program Purpose:
Students who complete the Honors Algebra One course will be exposed to problem solving, applications of algebra, reasoning skills, making geometric models and using the latest technology to develop a clear understanding of mathematical concepts as outlined in the New Jersey core curriculum standards.
Students will be given extensive opportunities to develop higher level thinking skills necessary for success in future advanced mathematics courses.
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Mathematics
Mathematics Lab
10
11
12
2 Semesters / 1 Credit(s)
This course is required for students who did not meet the State standard on the End of Course Assessment for Algebra I. Curriculum will focus on strengthening basic mathematical skills, conceptualization and communication of mathematical ideas and reinforcement of skills necessary for success in Algebra I.
NOTE: THIS COURSE COUNTS AS AN ELECTIVE TOWARD GRADUATION BUT DOES NOT FULFILL MATHEMATICS GRADUATION REQUIREMENTS.
Pre-Algebra
9
2 Semesters / 2 Credit(s)
This course is designed to help students who have previously struggled in mathematics. It is meant to help freshmen students prepare for taking and succeeding in Algebra I the following year. Topics covered in the course include simplifying expressions, solving equations, working with fractions and decimals, proportions, graphing equations, spatial thinking, and introductory geometry. Students will also be exposed to the basics of many Algebra 1 topics. This course counts as a mathematics course for the General Diploma ONLY.
Algebra I (1 & 2)
10
11
1 Semester / 2 Credit(s)
This course is designed for those students who did not receive a C- average or better in Algebra I or did not pass the ISTEP+ Algebra I Graduation Exam. This course will meet everyday.
Algebra I
9
10
2 Semesters / 2 Credit(s)
Algebra Idevelops traditional principles such as: solving equations and inequalities, performing operations with real numbers and polynomials, working with integer exponents and factoring polynomials, doing exercises with relations and functions, graphing linear equations and inequalities, graphing and algebraically solving linear systems, solving quadratic equations, and introducing topics from probability and statistics. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Honors Algebra I
9
2 Semesters / 2 Credit(s)
The same topics as in Algebra I are covered with more emphasis on problem solving and critical thinking skills in order to challenge the mathematically talented student. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Prerequisite: A High School Placement test Basic Skills Math Grade Equivalent of 10.0 and higher along with a Reading and Language Grade Equivalent above 8.0
Geometry
9
10
2 Semesters / 2 Credit(s)
The purpose of Geometry is to use logical thought processes to develop spatial skills. Students work with figures in one, two- and three-dimensional Euclidean space. The interrelationships of the properties of figures are studied through visualization, using computer drawing programs and constructions, as well as through formal proof and algebraic applications. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
This Geometry course is designed for those students who did not receive a C- average or better in Geometry. This course will meet everyday.
Honors Geometry
9
10
2 Semesters / 2 Credit(s)
This course covers the same topics as Geometry, but with greater emphasis on complex direct deductive proof and indirect proof and on utilization of more advanced algebraic techniques. Content is extended to include topics such as analytic geometry and the interrelationships of inscribed polyhedra. A GRAPHING CALCULATOR IS REQUIRED.
Prerequisite: B- or better in Honors Algebra I; Freshman enrollment is based on a math readiness test given in June.
Algebra II
11
12
2 Semesters / 2 Credit(s)
This course further develops the topics learned in Algebra I with extensive work on learning to graph equations and inequalities in the Cartesian coordinate system. Topics include: relations and functions, systems of equations and inequalities, conic sections, polynomials, algebraic fractions, logarithmic and exponential functions, sequences and series, and counting principles and probability. A GRAPHING CALCULATOR IS REQUIRED.
This course expands and develops the topics learned in Honors Algebra I. Content areas include the topics listed for Algebra II with greater emphasis on preparation for upper level mathematics content. The course is required for students who plan to take AP Calculus, and it is recommended that this course be taken at the same time as Honors Geometry unless Honors Geometry was taken as a freshman. A GRAPHING CALCULATOR IS REQUIRED.
Prerequisite: B or better in Honors Geometry; Freshman enrollment is based on a math readiness test given in June.
Pre-Calculus
11
12
1 Semester / 1 Credit(s)
This course continues the foundation concepts necessary for college level mathematics. Topics studied include: relations and functions, polynomials, rational and algebraic functions, logarithmic and exponential functions, analytic geometry, and data analysis. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
Knowledge of trigonometry is necessary for successful performance in college level mathematics. Topics covered in this course include: trigonometry in triangles, trigonometric functions, trigonometric identities and equations, and polar coordinates. A GRAPHING CALCULATOR IS REQUIRED FOR THIS COURSE.
This course covers the same topics as Pre-Calculus/Trigonometry listed above. Greater emphasis is placed on applications and developing the depth of understanding and skills necessary for success in AP Calculus. This course is required for students who plan to take AP Calculus. A GRAPHING CALCULATOR IS REQUIRED.
This course is intended for students who have a thorough knowledge of college preparatory mathematics. It covers both the theoretical basis for and applications of differentiation and integration. Concepts and problems are approached graphically, numerically, analytically and verbally. All students enrolled in this course will take the AP Calculus (AB) Exam. A GRAPHING CALCULATOR IS REQUIRED.
This course introduces and examines the statistical topics that are applied during the decision-making process. Topics include: descriptive statistics, probability, and statistical inference. Techniques investigated include: data collection through experiments or surveys, data organization, sampling theory and making inferences from samples. Computers are used for data analysis and data presentation. This course should not be taken as a replacement for Pre-Calculus/Trigonometry in a college preparatory course of study. A SCIENTIFIC CALCULATOR IS REQUIRED.
This course expands students' mathematical reasoning and problem-solving skills as they cover topics such as logic, graph theory, matrices, social choice, game theory, sequences, series and patterns. The course will encourage students to make mathematical connections from the classroom to the world after high school, while learning the importance of mathematics in everyday life. This course is offered as an addition to Pre-Calculus/Trigonometry, not a replacement. A SCIENTIFIC CALCULATOR IS REQUIRED
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Math
Sierra Vista's Algebra II is part of the Geometry and Beyond Workshop which meets Tuesdays and Thursdays from 8:30am-10am. The course is offered via the ALEKSsystem.
Text: The ALEKS system is aligned with the district approved McDougal Littel Algebra 2. McDougal Littel offers support including Powerpoint Presentations, animations, games, exercises and an online version of the text at their Classzonewebsite.
How does this work?
Workshop: Students attend the workshop for three hours each week. While there, they complete lessons/topics in the ALEKS system. Students take notes on these lessons and keep a notebook. When they have difficulty with a lesson they can reference the textbook and the McDougal Littel Classzone Website. A credentialed teacher is also available for assistance and consultation. Students are encouraged to spend 2-4 additional hours per week on campus or at home working on the course.
On Campus: If students would like to work online on campus, computers with internet access are available in Room 703 and the Sierra Vista Testing Center. Tutors are available in Room 703 from 8:00am-2:30pm.
At Home: Students may do all of the same activities they do in the workshops from any computer with internet access! They may email the teacher for assistance.
Weekly Progress Grades
Every Friday progress reports are sent to Master Teachers.These reports give teachers the amount of time each student has spent on the program, the number of topics practiced and the number of topics mastered.Weekly Progress letter grades are based on the following minimums:
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Discuss examples of the math techniques that you have learned this term that you see implementing in the future in your specific area of study ...
View the answer
You said "these critical thinking skills" but did not list them so I cannot answer the question until you list these skills that the question is ...
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User: Discuss examples of the math techniques that you have learned this term that you see implementing in the future in your specific area of study
Weegy: You said "these critical thinking skills" but did not list them so I cannot answer the question until you list these skills that the question is about. [ [ Also you said "math techniques you have learned" but I am not you and was not in the course so I did not learn them thus cannot answer the question. Please be clearer in your question so I may answer it for you, thank you! ] ] Auto answered|Score .6|rathipearl|Points 207|
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Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
Richard Aufmann is Professor of Mathematics at Palomar College in California. He is the lead author of two best-selling developmental math series, a best-selling college algebra and trigonometry series, as well as several derivative math texts. The Aufmann name is highly recognized and respected among college mathematics faculty.Joanne Lockwood is co-author with Dick Aufmann and Vernon Barker on the hardback developmental series, Business Mathematics, Algebra with Trigonometry for College Students, and numerous software ancillaries that accompany Aufmann titles. She is also the co-author of Mathematical Excursions with Aufmann.
List price:
$74.95
Edition:
3rd
Publisher:
Brooks/Cole
Binding:
Trade Paper
Pages:
432
Size:
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Saxon Math 87 Or Lifepac 7?
Question:I currently have the lifepac 7th grade kit, but he will have used saxon 65, 76, so would it be better to just try and get saxon 87 before going into saxon 1/2?
Answers:
It sounds like you are currently using the Saxon math books, but if you are not, I HIGHLY recommend you use their placement test (see placement test link below).
Saxon used to recommend the following sequence:
Math 6/5
Math 7/6
Alg. 1/2 (OR Math 8/7)
Alg. 1
Math 8/7 was only recommended if a student was weak in basic arithmetic, as it had more arithmetic practice.
With the newer editions, they recommend the following sequence:
Math 6/5
Math 7/6
Math 8/7
(Alg. 1/2)
Alg. 1
With the newer Math 8/7 book, they have included more pre-algebra work, so the Alg. 1/2 is now only used if the student needs more of an intro to algebra before beginning Alg. 1. They do recommend skipping Math 8/7 (and using Alg. 1/2 instead) for an accelerated math student (see recommended sequence link below).
From the Saxon website:
What do you recommend for students after they have completed Math 7/6?
The recommended path after finishing Math 7/6 is to take Math 8/7. If your child finishes Math 8/7 with at least 80% mastery, skip ahead to Algebra 1. In previous editions, many people skipped Math 8/7 because they found it to be a weaker text than Algebra ½. In the newer third edition, pre-algebra has been added to Math 8/7, making it a much stronger book. Saxon 87 covers a lot I was going to skip it and I'm so glad i took it. I t covers review and algebra. It was very helpful. We were tested and the kids who took 87 scored higher then the kids who skipped and took algebra half. Given those two choices I'd have to say Saxon 8/7
However, 8/7 is an optional level, and if you have done the two previous books, you might actually be ready for Saxon Algebra 1/2.
Honestly though, what I'd go into is Teaching Textbooks Pre-Algebra.
have to agree with the other posters. While Saxon 8/7 is theoretically an optional book, it contains a LOT of fabulous review material that will ensure you are completely ready for Algebra 1/2. Good Luck!
This article contents is post by this website user, EduQnA.com doesn't promise its accuracy.
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Linear Algebra With Application Alt. Edition - 8th edition
Summary: Introductory courses in Linear Algebra can be taught in a variety of ways and the order of topics offered may vary based on the needs of the students. Linear Algebra with Applications, Alternate Eighth Edition provides instructors with an additional presentation of course material. In this edition earlier chapters cover systems of linear equations, matrices, and determinants. The more abstract material on vector spaces starts later, in Chapter 4, with the introduction of the vector s...show morepace R(n). This leads directly into general vector spaces and linear transformations. This alternate edition is especially appropriate for students preparing to apply linear equations and matrices in their own fields.Clear, concise, and comprehensive--the Alternate Eighth Edition continues to educate and enlighten students, leading to a mastery of the matehmatics and an understainding of how to apply it. ...show less
2012 Hardcover New Book New and in stock. 10/23/201214496795
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Book details
Book description
The goal of this book is to teach undergraduate students how to use
Scientific Notebook
(SNB
) to solve physics problems. SNB
software combines word processing and mathematics in standard notation
with the power of symbolic computation. As its name implies, SNB
can be used as a notebook in which students set up a math or science
problem, write and solve equations, and analyze and discuss their
results.
Written by a physics teacher with over 20 years experience,
this text includes topics that have educational value, fit within the
typical physics curriculum, and show the benefits of using SNB.
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A First Course in Complex Analysis With Second Edition of A First Course in Complex Analysis with Applications is a truly accessible introduction to the fundamental principles and applications of complex analysis. Designed for the undergraduate student with a calculus background but no prior experience with complex variables, this text discusses theory of the most relevant mathematical topics in a student-friendly manor. With Zill's clear and straightforward writing style, concepts are introduced through numerous examples and clear illustrations. Students are guided and su... MOREpported through numerous proofs providing them with a higher level of mathematical insight and maturity. Each chapter contains a separate section on the applications of complex variables, providing students with the opportunity to develop a practical and clear understanding of complex analysis. Previous Edition 9780763746582
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Saxon Teacher, Algebra 2 is designed to supplement the 3rd edition homeschool kit for Algebra 2. Using this set of CDs without the textbook will lead to an incomplete understanding of the concepts. The program is written with the assumption that the textbook is being used while working on the computer CDs.
This program consists of 5 CDs and works on both Windows and Mac computer systems. The grade level is 8th to Adult and the cost including the textbook is reasonable. This set includes over 110 hours of Algebra 2 content, including instruction for every part of every lesson, as well as complete solutions for every example problem, practice problem, problem set, and test problem. There are two types of CDs included in this program. Lessons which include practice and problems sets, then the instruction CD which includes tutorials of each lesson and the answer key for teachers. Some of the following lessons are taught by a professional teacher and the teacher works through each set of problems. The practice sets are on one CD and are continuous videos; however, at the end of each set there are references given at the end of each lesson for students or teachers needing additional help. I found this to be a very helpful feature in returning to the text for help.
The CDs cover a wide range of problems from Polygons, Trinomials, Negative Exponents, Geometry, Trigonometry, Rounding, Factoring, and Formatting. All the problems and practice sets are equivalent to one full semester of Algebra 2 and the student has enough information and training to be able to do Pre-Calculus work when the CDs have been completed. The mastery aim is for 80% at completion.
The CD format offers students helpful navigation tools that are easy to access and are within a customized player and is compatible with both Windows and Mac.
The CDs are very well planned and have a general understanding of Algebra 2. One can easily follow the step-by-step instructions of the teacher. This program along with the text would take a student through to a successful completion of the Algebra 2 course.
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Academic level (A-D)
Subject area
Grade scale
Learning outcomes
The course will give the basic understanding and knowledge of electrical networks and mathematical methods for analysis of linear models. The course is an essential base for further studies in many different areas where piecewise linear or linear models are used.
Aim
After the completed course the student will have the ability to:
describe properties of passive and active components
explain concepts in the mathematical model used for description of the circuits
identify the most common passive and active circuits and describe their properties
apply the solution methods such as nodal analysis and mesh analysis
use superposition and two-terminal equivalents
solve transient problems in switching circuits
master AC steady state analysis using phasors
be acquainted with graphical solution techniques for nonlinear components
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Math Solver
0.00 (0 votes)
Document Description
To
Math problems made Easy
One
Solving Math problems is not easy! A lot of students have difficulty with Math questions but employing some of these techniques will help you to solve Math problems easily :
Read it carefully - Math problem solving involves reading the problem slowly and carefully in order to understand what is it that you need to solve. At times you miss out important information when you give it a quick reading.
The following steps are genrally followed to solve Math problems:
Break
Change it into an equation - It is important to convert what you read in words into an equation you can solve. So basically you need to change the English into numbers!
Always cross check - Once you get the answer to your Math problem you should always go back and recheck. Sometimes you might miss out on small details and going over the problem and solution again helps.
Ask new topic is introduced you should write it down, review it and in case of any doubts bring it up with your tutor. This process of continuously revising what you learn will gear you up to solve math problems online.
Add New Comment
To solve Math problems quickly and accurately you need an understanding of various
math concepts and solving math problems is not an easy task.
TutorVista has a team of expert online Math tutors to ...
Before talking about linear programming, I would like to tell you the meaning of "linear". Linear
is a Latin word which means pertaining to or resembling a line.
In mathematics, linear equation means ...
Before talking about linear programming, I would like to tell you the meaning of
"linear". Linear is a Latin word which means pertaining to or resembling a line.
In mathematics, linear equation means ...
To Math solve problems quickly and accurately you need an understanding of various math concepts and solving math problems is not an easy task. TutorVista has a team of expert online Math tutors toDifferential Equation is a type of equation which contains derivatives in it. The
derivative may de partial deerivative or a ordinary derivative.The eqution may contain
derivative of any order.
It ...
Content Preview
Math Solver To Math problems made Easy One Solving Math problems is not easy! A lot of students have difficulty with Math questions but employing some of these techniques will help you to solve Math problems easily : Read it carefully - Math problem solving involves reading the problem slowly and carefully in order to understand what is it that you need to solve. At times you miss out important information when you give it a quick reading. The following steps are genrally fol owed to solve Math problems: Break Change it into an equation - It is important to convert what you read in words into an equation you can solve. So basical y you need to change the English into numbers! Always cross check - Once you get the answer to your Math problem you should always go back and recheck. Sometimes you might miss out on small details and going over the problem and solution again helps. Ask
new topic is introduced you should write it down, review it and in case of any doubts bring it up with your tutor. This process of continuously revising what you learn will gear you up to solve math problems online. Help with Math Topics TutorVista's expert tutors will make solving Math Solver very easy. Our expert tutors will work with you in a personalized one-on-one environment to help you understand Math questions better thereby ensuring that you are able to solve the problems. Solve problems in topics like: * Algebra * Geometry * Calculus * Pre-Algebra * Trigonometry * Discrete Mathematics Students frequently need help with fractions, solving algebra expressions, geometry problems, equations, ratios, probability and statistics measurements and calculus. Each of these topics has its own approach for solving problems. TutorVista's online tutoring in math can help students understand the methods for solving Math Solver in each of these categories.
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...Students need a mastery of arithmetic with fractions, decimals, percents, and negative numbers in order to well in this class. This course is integral to success in higher level math. In fact, students who struggle with SAT math, Calculus, and college level courses often lack conceptual understanding of Algebra 1.
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Formats
Book Description
Publication Date: Sep 17 2009 | Series: 50 IdeasProduct Description
About the Author
Tony Crilly is Reader in Mathematical Sciences at Middlesex University, having previously taught at the University of Michigan, the City University in Hong Kong, and the Open University. His principal research interest is the history of mathematics, and he has written and edited many works on fractals, chaos and computing. He is the author of the acclaimed biography of the English mathematician Arthur Cayley.
--This text refers to an alternate
Hardcover
edition.
This book contains short introductions to 50 topics in mathematics that will make readers more numerically literate. It can help you understand a newspaper or magazine article that becomes a little more technical than you expected. It is also a good starting place to learn more about mathematical concepts you just find interesting.
Each chapter is self-contained and delivers a two- to four-page capsule treatment of its topic. Most chapters contain definitions of key ideas, relevant historical quotes, and timelines across the bottom of the first two pages. The book makes particularly effective use of graphs and diagrams to illustrate important concepts. Boxes set off from the text effectively summarize supporting information. Example boxes include "666 - The Number of the Numerologist" (p. 39), "Building with Triangles" (p. 87), and "From Meteorology to Mathematics" about chaos theory (p. 107).
Several chapters are particularly informative for such brief introductions. "Logic" (p. 64) outlines the basic processes of deductive logic and points to more advanced types of formal reasoning. The "Calculus" chapter (p. 76) is a good overview for a high school or college student who considers taking a calculus class and wants to know what to expect. "The Four-Color Problem" (p. 120) introduces a practical topography issue relevant to the work done by cartographers and graphic artists. Finally, the Matrices chapter (p. 156) is an introduction to a type of algebra that many students find difficult. The basics of matrix manipulation are explained using the practical problem of airline flight scheduling.
Tony Crilly's book has a good topic index and an adequate two-page glossary, but lacks references to supporting literature. This is an unfortunate omission in an introductory book. Readers should be encouraged toward further reading when they are most eager for more knowledge. This is a recurring flaw in the "50 ideas" series.
3.0 out of 5 starsNot up to the standard of the other books in this series.Dec 3 2008
By Colorado Metallurgist - Published on Amazon.com
Format:Hardcover
I have read and reviewed 50 Physics Ideas and 50 Philosophy ideas. I enjoyed both books and therefore looked forward to reading this one. Unfortunately, I was deeply disappointed with this book. This is probably mostly due to the fact that the subject does not lend itself to the format of this series (which allots only 4 pages per idea). Mathematical ideas require some development, which is completely missing here. A statement of each idea is given, but in my opinion not sufficiently explained before a few examples are given, and then on to another idea. I found many subjects to have been presented in a somewhat haphazard and incomplete manner. Instead of focusing on the underlying mathematics, too many of the chapters seem focused on "bar bet" problems, such as the "birthday" problem, with the underlying mathematics (in this case probability theory) given only incidental coverage. Very valuable space is taken up by mentioning, but not developing additional ideas in each section, while some ideas, such as the Logarithm are not even included at all. To site just one example, in the section on Chaos Theory there is mention of the Navier-Stokes equation, but it is not developed or even given. It is introduced because it is contains nonlinear terms (but there is no discussion as to what a nonlinear term is), but the fact that it is this nonlinear behavior is the root cause of chaotic solutions is not discussed. The idea of phase space is also mentioned in the same section, but is never defined or even described. If I had never heard of Chaos Theory I doubt that I would have come away from reading this book with a idea of what it is or why the mathematics behave as it does. This was not the case for the Physics and Philosophy Ideas. In the case of these books, the 4 pages allotted to each idea were enough to understand what the idea was and why it was important. If you want an overview of mathematical ideas I recommend Kline's Mathematics for the Non-mathematician. Reading it will be more work, but you will get much more out of it.
I am giving the book 3 stars because there are some reasonably presented sections in the book, but in my opinion not enough to warrant a higher rating.
14 of 14 people found the following review helpful
5.0 out of 5 starsBest Short Survey of Mathematics I Know OfFeb 16 2009
By Irfan A. Alvi - Published on Amazon.com
Format:Hardcover
This is the best short survey of mathematics I know of, and I think the format works very well (4 pages each for 50 key mathematical ideas). I personally read one idea per day for 50 days, and this book delivered a bright spot for every one of those days.
To address another reviewer's comments, I do agree that this book has lower and upper bounds to be aware of.
The lower bound is that readers should come to the book with at least a general familiarity with the subject of mathematics, including having at least heard of concepts like complex numbers, calculus, probability, chaos, abstract algebra, group theory, Fermat's last theorem, the Riemann hypothesis, etc. And certainly readers should come to the book with a genuine interest in mathematics. In other words, this isn't a book for readers with no background or interest in mathematics, nor readers with a fear of mathematics.
The upper bound is that the book doesn't (and can't) develop the mathematical ideas in step-by-step detail. Rather, the book goes into just enough detail to give a meaningful sense of what the ideas are about, and it does this quite well, with nice features like timelines, examples, historical asides, etc. This book isn't a mathematics textbook, nor does it purport to be, so it shouldn't be judged on that basis.
The only thing I really found lacking was that the book doesn't include suggestions for further reading. But this omission isn't enough to lower my rating from 5 stars, and I highly recommend the book to anyone looking for a short survey of mathematics. Tony Crilly provides a wonderful and panoramic guided tour of the subject, spanning from elementary to fairly advanced ideas, and does it in a way that both entertains and reveals the rich beauty of mathematics. For readers who take the tour and feel sorry to see it end, I suggest moving next to the massive and outstanding The Princeton Companion to Mathematics.
9 of 10 people found the following review helpful
5.0 out of 5 starsGreat book for transition from High School Math to University MathSep 17 2008
By Wu Bing - Published on Amazon.com
Format:Hardcover
This is a concise book covering from ancient Greek Math to 21st century Modern Math. The author has smartly picked the most important 50 ideas, from Greek time, to modern Algebra, FLT, Riemann, Fractal, Genetic Math... it is surprisingly fun stuff to read, unlike other boring Math textbooks, yet it opens the reader's eyes to the beauty and wonderful Math world. For high-school math students going to the University, this is a good transition book, laying the foundation for them to grasp the abstract math ideas in the University, where unfortunately the Math professors would hardly tell the students the roots of these math ideas dated since 3,000 years ago.
Some interesting highlights: 1. Hardy-Weinberg Genetic Law: Cambridge Prof Hardy, as the greatest Pure mathematician in 20th century, prided Pure Math as being 'useless', yet he discovered independently the Genetic Law with Dr. Weinberg (Germany). He wrote the math proof at the back of an envelope after a cricket match in 1903. You can see Hardy's Pure math is not 'useless' at all - he proved the Genetic Law without being a Bio-Scientist, just by applying the beautiful Probability theory (the proof can be found in this tiny book).
2. Abstract Algebra: Since 825AD Arab Mathematician Al-khwarizimi introduced 'Al-jabr' dealing with numbers, Viète (France) in 1591 AD introduced symbols for known and unknown variables, Algebra was transformed into dealing with non-numbers 'Modern Algebra' by Emmy Noether (Germany) in 1920, with axiomatic structures and its 'isomorphism', etc. Bourbaki (France) in 1939 re-built the whole math using Set Axioms under rigourous structures.
Together with the other Twin book: "50 Physics Idea You Really Need to Know" (by Joanne Baker) form a great gift for your to-be-university children or friends. They will definitely be enthused by Science & Math and excel in these 2 subjects in the University.
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19 2003 | Series: Made Simple
Brushing up on math has never been easier!
I have tutored a great number of students in math. Most of these students (even ones who have progressed to Pre-Calculus and beyond) are lacking an understanding of basic mathematics (or have long since forgotten). For adult students who have been out of math class for 20 years or more, especially if they consider themselves to be poor math students, I prefer to start at "the beginning".
I particularly like the discussion of "completing the square" as this topic is usually not well discussed in modern math textbooks (leaving students thoroughly confused about a fairly simple technique). There is also a nice discussion of geometric constructions, which I remember vividly from ninth grade geometry, but have yet to meet a student in the past ten years who has even heard of this (although I had a great, tough teacher).
I am currently using this book to prepare an adult student for the Accuplacer exam (math placement exam in college), but have a reference copy for myself as well. I believe it would be useful for homeschool families, adults taking a math class, and anyone who wants a good reference of the most relevant math topics.
If there is a better, all-around beginning math book, I haven't found it. Highly recommended.
13 of 15 people found the following review helpful
1.0 out of 5 starsMistakes abound, not a good bookJun 25 2007
By P. Stearns - Published on Amazon.com
Format:Paperback
This book is ok for a review... if you have already learned this stuff long ago and understand the CONCEPTS very well.
There are several errors in this book. Some are surprisingly simple. For example, page 26 has a list of prime numbers. They list 45 as a prime number. Page 38 has a demonstration that uses wrong numbers. How can an editor allow such blatant errors? If you dont already know math, this book could lead you astray. Of course, I only did the first 2 chapters.
19 of 24 people found the following review helpful
3.0 out of 5 starsdecent supplement - not an introductionJan 1 2004
By A Customer - Published on Amazon.com
Format:Paperback|Amazon Verified Purchase
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Credit: 0.5 units Lessons: 10 lessons, 10 submitted Exams: 2 exams Grading: Computer and Faculty Evaluated Prerequisites: Successful completion of Algebra I. Successful completion of Geometry is strongly suggested. Description: This course begins with a review of the essentials of algebra. Then it presents linear functions; linear equations and inequalities; and linear equations in three variables. It concludes with the factoring of polynomials and the solving of quadratic equations.
The course content will be viewable prior to the start date. Read more about our online semester courses. The semester courses follow a specific calendar, and the normal 9-month completion policy does not apply to these courses. Therefore, students who have not completed all work by the due date for the course final (listed on the course calendar) will automatically be withdrawn from the course.
Lesson assignments need to be created in Microsoft Word or another word processor that saves files as .doc (Word 97–2003 document) or .rtf (Rich Text format).
Materials Note: Students will need access to a scientific calculator (e.g., TI-35) or a graphing calculator (TI-83+ or newer is highly recommended) The same textbooks are also used for the spring semester.
Preview This Course — A preview includes general information about the course and, if available, one lesson and one progress evaluation.
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Questions About This Book?
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc.
Summary
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. New to the Third Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically.
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Dover's impressive collection of popular science books covers technology and invention, space and time, basic machines and computers, forces and fields, chaos, biographies of Einstein and Newton, and much more. We publish books by the famous pioneering scientists of yesterday as well as gifted authors of the 21st century, including George Gamow, Michael Faraday, Martin Davis, Morris Kline, Emilio Segrè, Ian Stewart, and Clifford A. Pickover.
To visit our main Math and Science Shop, please click here. And be sure to join our Math and Science Club for a 20% everyday discount, free newsletter, and other exclusive benefits.
Recommendations...Game Theory and Politics by Steven J. Brams Many illuminating and instructive examples of the applications of game theoretic models to problems in political science appear in this volume, which requires minimal mathematical background. 1975 edition. 24 figuresProducts in General and Popular MathematicsOur Price:$15.95
Advanced Trigonometry by C. V. Durell, A. Robson This volume is a welcome resource for teachers seeking an undergraduate text on advanced trigonometry. Ideal for self-study, this book offers a variety of topics with problems and answers. 1930 edition. Includes 79 figures.
Our Price:$19.95Our Price:$10
Our Price:$23.95
The Analytic Art by Francois Vičte, T. Richard Witmer Originally published in 1591, this work pioneered the notion of using symbols of one kind (vowels) for unknowns and of another kind (consonants) for known quantitiesOur Price:$12.95
Our Price:$9.95Our Price:$14.95
Challenging Mathematical Problems 2 Vol Set by Dover Save Over 11%! This 2-volume set of Challenging Mathematical Problems with Elementary Solutions features over 170 challenging problems ranging from the relatively simple to the extremely difficultOur Price:$8.95Our Price:$14.95
A Concise History of Mathematics: Fourth Revised Edition by Dirk J. Struik Compact, well-written survey ranges from the ancient Near East to 20th-century computer theory, covering Archimedes, Pascal, Gauss, Hilbert, and many others. "A work which is unquestionably one of the best." — Nature.
Our Price:$9.95
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:: 2012 AP* Summer Institutes at UT Austin
Use of a multi-representational approach to examine problems algebraically, graphically, numerically and verbally
Use of technology to reinforce these relationships, to experiment and to develop concepts
Examination of topics covered in both the Algebra II and PreCalculus classroom such as transformations, inverses, and various other aspects of functional analysis
Improving assessments to better prepare students for AP mathematics
Sharing best practices and designing Pre-AP problems and lessons in subject-area breakout sessions
Exploring online resources in a lab setting
Classroom-based activities that tie together multiple concepts
What participants should bring:
* Laptop computer
* Notepad or spiral notebook
* Graphing calculator
* 30 copies of a favorite lesson or activity
* One test given in the past year
* Current textbook(s)
* Post-it notes
Lead Consultant: Jill Bell
Jill Bell currently teaches Pre-AP Algebra II GT, Pre-Calculus and AP Calculus AB at Ronald Reagan High School in San Antonio, Texas. A 20+ year veteran of the classroom, Ms. Bell has been a College Board consultant for seven years and has presented various workshops across the country on Pre-AP Mathematics, AP Calculus, and SAT Math preparation. She is active in Rotary and has been honored by "Who's Who Among America's Teachers." She holds with a BS in Secondary Education from Baylor University and secondary certifications in Mathematics, Spanish and Gifted/Talented.
* Trademark Notice: College Board, AP, Advanced Placement Program, AP Vertical Teams, Pre-AP, and the acorn logo are registered trademarks of the College Board. Used with permission.
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Next: Graphs of Linear Systems
Previous: Theoretical and Experimental Probability
Chapter 7: Systems of Equations and Inequalities
Chapter Outline
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Chapter Summary
Description
This chapter introduces students to linear systems of equations and inequalities as well as probability and combinations. Operations on linear systems are covered, including addition, subtraction, multiplication, and division.
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MAT
310
- Number Theory
This is an introductory course in Number Theory. The course will explore the properties of, and the relationship between, the natural numbers, integers, rational numbers, and irrational numbers. This course will explore and prove theorems related to topics in number theory such as: Pythagorean Triples, Divisibly, The Fundamental Theorem of Arithmetic, Congruences, the Chinese Remainder Theorem, Prime numbers, Modulo arithmetic, Pell?s Equation, Diophantine's Approximation, and the Gaussian Integers.
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