text
stringlengths 8
1.01M
|
|---|
WHAT DOES A MATHEMATICS MAJOR STUDY?
Mathematics is one of the fundamental areas of human knowledge. It has held an established position among the humanities for over two thousand years, and in recent centuries it has played a vital role in the sciences. The body of mathematical knowledge is growing today faster than ever before, with old questions being answered and new ones asked at an unprecedented rate.
In the paragraphs below, we sketch some of the courses and topics which mathematics majors study at Wright State University. In each case, the illustrations of what students learn are only small samples of the full curriculum.
Mathematics majors typically begin their college mathematics studies with three semesters of Calculus. Calculus is both the cornerstone of "analysis," one of the major branches of pure mathematics, and an indispensable tool for most of the sciences and engineering. Students learn to discover for themselves familiar formulas from high school geometry -- e.g., area of circle = πr2, volume of sphere = (4/3) πr3 -- and they learn how to investigate similar issues in more complex settings, as illustrated to the right. Students study mathematics describing such diverse phenomena as conservation of energy laws from physics, and continuously compounded interest from banking.
All mathematics majors also take one semester of Linear Algebra, usually by the beginning of the junior year. This basic topic resides within another major branch of pure mathematics, "algebra" and is also of critical importance in an extensive range of applications of mathematics, from management to structural engineering to telecommunications. In linear algebra, the study of linear equations, like those in beginning high school algebra -- but with the possibility of thousands of equations in thousands of unknowns -- leads to abstract mathematical spaces.
The Department of Mathematics and Statistics offers a variety of more advanced courses. Students decide which of these to take according to their interests and the particular degree which they seek.
For example, students apply multivariable calculus in probability and the mathematical foundations of statistical inference in Theory of Statistics. Related choices include Statistical Methods and Introduction to Experimental Designs, in which one learns how to collect and analyze data, for instance in scientific and industrial experiments.
Similarly, students can choose mathematics courses with special significance in computer science. A prime example is Applied Graph Theory, featuring mathematical models and algorithms applicable to such problems as traffic systems, activity scheduling, and design layout. Others are Cryptography (how to encrypt data securely) and Coding Theory (how to send messages that self-correct transmission errors).
Other courses in applied mathematics emphasize mathematical theory and problem solving methods directed toward the physical sciences and engineering. An illustrative example at the junior level is Partial Differential Equations involving several variables (e.g., both space and time). They describe such phenomena as the propagation of electromagnetic waves through space, and the flow of heat in solids.
Courses in pure mathematics concentrate upon the theoretical foundations of algebra, calculus, and other elementary courses, and at the same time point the way toward more advanced topics in modern mathematics. To illustrate, students in Real Variables (or theoretical advanced calculus) learn why the facts and computational methods learned in ordinary calculus are correct; in Modern Algebra, students study various abstract systems which include the familiar objects of school mathematics as special cases (whole numbers, rational numbers, real numbers, etc.)
In addition to taking many courses in the major field of study, every undergraduate at Wright State University is required to take a comprehensive program of General Education studies, including courses in history, English, economics, the sciences, and several other disciplines. Moreover, some degree programs allow students to take "free electives," or courses chosen by students from essentially any area. Wright State's degree programs in mathematics have a generous allotment of these free electives.
|
Calculators
ARTICLES ABOUT CALCULATORS
LIKE hundreds of thousands of other high school students, Greg Myers, 16, began using a graphing calculator in freshman algebra. Graphing calculators, which bear little resemblance to their 1970's ancestors, are sophisticated devices that can run small computer programs and draw the graph represented by complex equations in an instant. In the last few years, they have become mandatory in many high school math classes and can be used on the SAT and advanced placement exams and other standardized...
|
Intermediate Algebra with Applications & Visualization, 4th Edition
Description
The Rockswold/Krieger algebra series uses relevant applications and visualization to show students why math matters and gives them a conceptual understanding. It answers the common question "When will I ever use this?" It covers the traditional topics, but rather than present them as concepts to memorize, with applications tacked on at the end, it teaches students the math in context. By seamlessly integrating meaningful applications that include real data, along with visuals—graphs, tables, charts, colors, and diagrams—students are able to see how math impacts their lives as they learn the concepts. This conceptual understanding makes them better prepared for future math courses and life.
|
Mathematics as a discipline is an important and beautiful human endeavor. It is an intellectual discipline worthy of study for its artistic and logical form as well as for its ability to help describe and interpret the world around us. To the extent of their abilities, all students at Portage High School should have the opportunity to learn to view mathematics as a way of thinking and to appreciate the cultural heritage of mathematics. In addition to grasping ideas and understanding their interrelations, it is important for students to communicate these ideas effectively to others.
Goals
The goals for students relect those stated in the Indiana Mathematics Standards as established by the Indiana State Department of Education. These goals assist students to value mathematics, become confident in one's ability to do mathematics, become a mathematical problem solver, communicate mathematically, reason mathematically, and use technology appropriately.
|
Appendix A Electronic Learning Objects to Support MCV4U
Appendix A: Electronic Learning Objects to Support MCV4U
E-Learning Ontario Web Site: MGA4U Unit 3 Vectors
Activity 2: Vector Laws
The last applet on Vector Laws allows the user to investigate the commutative, associative, distributive
properties of 2-space vectors in geometric form.
At the bottom of activity 2 is a link to the University of Guelph's Physics department where a tutorial for
vectors is provided.
Activity 3: Applications of Geometric Vectors
The second applet in the Velocity Java Applets allows the user to investigate the resultant vector for a
boat crossing a river. The user controls 2-space vectors in geometric form for the boat's velocity and the
current.
Activity 5: Algebraic Vectors
The first applet allows users to interactively explore the connections between geometric and algebraic
forms of vectors in 2-space.
At the end of this activity is a link to a 3-space Graphing Tool that allows students to graph points, lines,
and planes in various forms.
Activity 6: Operations with Algebraic Vectors
There are four applets on addition of vectors, scalar multiplication, unit vectors, and position vectors.
They allow the user to interactively manipulate 2-space vectors.
E-Learning Ontario Web Site: MGA4U Unit 5 Vector Methods with Planes and Lines
Activity 1: Equations of Lines in 2-space
There are five guided and three interactive applets on forms of vector equations, how to convert between
forms, distance from a point to a line.
Activity 3: Intersection of Lines
There are two guided applets on intersection of lines in 2-space and 3-space.
Activity 5: Equations of Planes
There are four guided applets on the forms of equations of planes and how to convert between forms.
Activity 6: Intersection of a Line and a Plane
There is one guided applet.
MCV4U – Overview 15
Appendix A: Electronic Learning Objects (continued)
to Support MCV4U
Activity 7: Intersection of Planes
There is one guided applet on solving systems of planes algebraically.
Activity 8: Task: X, Y, and Z Factor
An open ended task using the 3-space Graphing Tool allows students to consolidate vector concepts.
Vector Applets on the Web
NCTM
This site has two applets. The first illustrates the components of a vector to control a car. The
user interactively controls the speed and direction. The second illustrates vector addition for
an aircraft flying that is acted upon by wind. The user controls the speed and direction of both
the aircraft and wind.
Syracuse University
This applet demonstrates cross product of two vectors in 3-space. It allows users to
interactively change the vectors and see the resulting cross-product. The two vectors are
limited to one plane but the plane can be moved to different viewing angles.
International Education Software
This Japanese site has a collection of applets that cover a wide variety of 2-space and 3-space
vector topics. The controls are not very user-friendly but there are topics covered here like
vector forms of lines in 2-space and 3-space that are not covered on other sites.
Professor Bob's Physics Lab (Rob Scott)
This interactive site has flash applets on various Physics topics. Some topics such as Milliken
and Momentum labs allow students to apply vector concepts.
B.Surendranath Reddy (Physics Teacher in India)
This site has several applets that can be used in MCV4U. For vectors there are applets for
addition, cross product of vectors, converting between Cartesian and directed line segment
forms and several kinematics applets. For calculus there are applets for instantaneous speed
and velocity.
MCV4U – Overview 16
|
Syllabus
Calculators and computers
You are free to use any scientific, non-graphing calculator for
quizzes and exams. You may not use any graphing calculator.
Many computer programs (such as Mathematica and MATLAB) can perform a
wide variety of calculations. You may wish to use a program to check
your homework, but you will still be expected to write out the steps
involved in solving each problem.
|
books.google.co.uk - The Fundamental Theorem of Algebrastates that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The... fundamental theorem of algebra
|
Commercial site with one free access per day. Students can vary a, h and k to explore the effects on the graph. An exploration guide is available to help guide the students through an acitivty. By cli... More: lessons, discussions, ratings, reviews,...
Commercial site with one free access per day. Students are given a set of points and are asked to "zap" as many points as possible. They can use either polynomial form or vertex form. Students can ge... More: lessons, discussions, ratings, reviews,...
This file describes an experimental setup (using a motion detector) in which a ball's distance from start is recorded as the ball rolls up then down an inclined plane. It includes a simulation applet... More: lessons, discussions, ratings, reviews,...
Guided activities with the Graph Explorer applet, designed to let students learning about quadratic functions explore: the parabolic shape of the graphs of quadratic functions; how coefficients affect
|
Students find data on sunrise and sunset times for some locale over the course of a year, then graph the length of the day as a function of the day of the year. They are asked to find a trigonometric function that models the data and to comment on such characteristics as amplitude and period. Data from locales at other latitudes are also investigated. In the solution section, the graphs of the sine waves generated through these experiments are analyzed in detail. This problem is an open-ended assessment task from the Balanced Assessment in Mathematics Program at the Harvard Graduate School of Education. (author/th)
Ohio Mathematics Academic Content Standards (2001)
Patterns, Functions and Algebra Standard
Benchmarks (11–12)
A.
Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
Grade Level Indicators (Grade 11)
4.
Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.
Grade Level Indicators (Grade 12)
3.
Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior.
Principles and Standards for School Mathematics
Algebra Standard
Understand patterns, relations, and functions
Expectations (9–12)
understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
|
undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The capstone of the book is a brief presentation of the Riemann zeta function and of the significance of the Riemann Hypothesis.
|
EDUC 6520: Advanced Pedagogical Content Knowledge:Math 4-5
The purpose of this course is to introduce integral components of the intermediate (4-5 grade) mathematics curriculum. While the focus is on mathematical content, teaching methods including the use of multiple representations and technology will be underscored throughout the semester. The major thrust of the course will be development of the real number system and arithmetic operations, measurement, probability, data analysis, and geometry. Prerequisites: Early Childhood License or Permission of Instructor... more »
Credits:2
Overall Rating:0 Stars
N/A
Thanks, enjoy the course! Come back and let us know how you like it by writing a review.
|
Welcome to Coordinate Algebra and Coordinate Algebra with Support. 1st and 4th period are year long classes that will meet every day, and will have a state end of course test in May that will count as 20% of each student's grade for the course. 3rd period is a semester long class that will meet every day for 1 semester and take the state end of course test in December, it will also count for 20% of those student's grades. This year we are teaching a brand new curriculum that is aligned with the national Common Core standards. We will start of this week right away. I would highly suggest getting signed up for the Cornerstones for Success tutoring program. That way at if any point in the year the student needs help, they will have the appropriate paperwork filled out to participate.
Recommended Materials:
I would recommend that each student gets their own Texas instruments, TI-30xs calculator to use throughout the semester or year. The school has calculators that can be issued to students like a text book, but it is better if the student has their own. Students will also need to have some sort of notebook/folder/binder to take notes in and keep their work organized. They will also need to be prepared with a pencil every day in class.
|
Course
2, Unit 1 - Matrix Models
Summary
In the "Matrix Models" unit, students learn to use matrices to organize
and display data, to operate on matrices in a variety of ways (including
summing rows and columns, comparing two rows, finding the mean of rows
and columns, scalar multiplication, and addition, subtraction, and multiplication
of two matrices), and to interpret the results in context. These ideas
are developed in a variety of contexts, giving students an appreciation
of the widespread use of matrices. In addition, this unit is rich in
connections to other Discrete Mathematics units, such as "Graph Models"
in Course 1, and Algebra units such as "Linear Models" in Course 1.
Key
Ideas from Course 2, Unit 1
Matrix: A rectangular array of rows and columns used to organize
information.
Dimension: A matrix has dimension 2 by 3 if it has 2 rows and 3
columns. A square matrix has the same number of rows as columns.
Matrix operations: Two matrices are added by adding corresponding
cells; thus the entries in row i column j are added to get
the (i, j) cell in the resulting matrix. (Likewise for subtraction.)
Two matrices are multiplied by multiplying each entry of each row of
the first matrix by the corresponding entry of each column in the second
matrix. Thus:
Inverse: A matrix A has an inverse A-1 if A(A-1) = (A-1)A = I,
the identity matrix. For example, if A = ,
then A-1 = .
Also, A(A-1) = I = .
Square matrices: The only matrices to possibly have inverses. Students
can find inverses, if they exist, by using a calculator. For a 2 by
2 matrix, students have a formula. If A = ,
then A-1 = .
Matrix equation of the form Ax = B: This
equation can be solved by multiplying each side of the equation by
the inverse of matrix A, giving x = A-1(B).
This can be used to solve systems of equations in more than one variable.
For example, can
be rewritten as a matrix equation, = .
Which can be solved by multiplying both sides on the left by the inverse
of A, = .
So, = .
|
Department of Mathematics
Student Handbook: Learning Outcomes
All students will be appraised of the departmental learning outcomes objectives which are as follows:
Upon the completion of the core curriculum in Mathematics, the student should be able to:
1) Analyze polynomial and transcendental functions of one or more variables with respect to:
a) operations of functions, graphs, existence of inverse functions b) existence of limits c) continuity, differentiability (both explicit and implicit) and partial differentiation. d) integrability with techniques of integration e) representation of functions through infinite series f) interpretation and summary of information from graphs of functions g) be able to prove limit theorems in simple cases by using Epsilon - Delta methods
2) Demonstrate an understanding for the applications of the derivative and the integral in:
Upon the completion of the Mathematics Program, the student should be able to:
6) Demonstrate quantitative literacy:
a) be able to analyze, interpret, and present data in a logical and scientific manner. b) know basic counting methods, and basic knowledge of statistics and probability
7) Demonstrate an understanding of the principles and techniques of applying mathematics to real world problems:
a) use techniques of linear algebra and differential equations to solve various applied problems b) understand the importance and widespread existence of nonlinear problems and the role of the linear theory in developing insight into these problems c) grasp the concept of "dynamical" systems and their importance in comparison to "static" problems
8) Understand the role of the computer in mathematics by implementing and understanding the importance and limitations of algorithms for:
a) numerical methods for approximating integrals, series and numbers b) different methods for graphing continuous and discontinuous functions in two and three dimensions c) numerical methods for approximating solutions of linear systems and differential equations
9) Communicate clearly and effectively in an organized fashion the basic concepts and principles of mathematics, from calculus to modern applications and theory:
a) communicate, in both oral and written fashion, mathematical concepts and methods in a precise manner b) present historical perspectives and implications of mathematical ideas c) understand research in mathematics by actively doing research in a specific area d) analyze some application problems using modeling techniques to observe patterns, interconnections, and underlying structures
|
Sample Math Problems
Practice is essential. There is no substitute for solving problems. Our sample math problems serve both as practice and reinforcement of the math lessons. Our focus is not just to present the solution, but instead to guide students to find the solution along with Mr. X. Sample math problems are available for Arithmetic, Basic Algebra, Geometry, Advanced Algebra, Trigonometry, and Calculus. Access to the sample problems is one of the benefits of subscribing with Mr. X. Our subscription service also gives access to our library of math lessons. Let Mr. X help with your math homework by subscribing today!
|
Titles in Barron's extensive "Painless Series " cover a wide range of subjects as they are taught on middle school and high school levels. These books are written for students who find the subjects unusually difficult and confusing--or in many cases, just plain boring. Barron's "Painless Series " authors' main goal is to clear up students' confusion and perk up their interest by emphasizing the intriguing and often exciting ways in which they can put each subject to practical use. Most of these books take a light-hearted approach to their subjects, often employing humor, and always presenting fun-learning exercises that include puzzles, games, and challenging "Brain Tickler" problems to solve. This title defines algebraic terms, shows how to avoid pitfalls in calculation, presents painless methods for understanding and graphing equations, and makes problem-solving fun. ATTENTION STUDENTS: " You get a special FREE bonus when you purchase your copy of Barron's "Painless Algebra Barron's is taking " Painless " to the next level: FUN! Sealed inside your copy of " Painless Algebra, " you'll find a code that gives you access to a FREE app. Simply key in that code on your iPhone, iTouch, iPad or Android device, and you download a fun-to-play Algebra arcade game challenge that will reinforce your skill in mastering Algebra!
You can earn a 5% commission by selling Painless Algebra
|
References
This webpage from SERC features GEOLogic questions, which are puzzles that were developed to support students understanding of geoscience concepts while challenging them to develop better logic and ...
The Mathematical Association of America (MAA) has sought to improve education in collegiate mathematics. This report outlines standards set forth by the MAA to improve college mathematics education. ...
The Supporting Assessment in Undergraduate Mathematics (SAUM) project provides information on assessment and assessment techniques. The purpose of SAUM is to support faculty members and departments ...
The Visual Display of Quantitative Informationpart of SERC Print Resource Collection This book is a classic outline of how complex information can be presented through graphics, charts, and diagrams. Not only does the book show how to display numeric data graphically, it shows how to ...
|
-12 Students & Teachers
We offer several programs designed for middle and high school students and for teachers. From Math Camps to A Taste of Pi, our faculty and researchers engage the university community and the general public in a range of talks and activities related to mathematics.
SFU and UBC take turn in offering theCalculus Challenge Exam for high school students who may wish to obtain credit for calculus courses they have taken prior to attending university.
In Summer, we hold the annual regional SFU Math Camps for high school students supported by CMS and PIMS. These mathematical enrichment camps are by invitation only and provide an opportunity for students to see and experience hands-on exciting, engaging, and challenging mathematics presented by mathematicians from SFU's Department of Mathematics and by visiting faculty. And, during school year, students and teachers are invited to our A Taste of Pi program.
|
Have additional students using Teaching Textbooks Math 4? This student workbook and answer booklet will allow extra students to complete the course in their own book. Perfect for co-ops or siblings! Workbook is 669 pages, softcover, spiral-bound and answer booklet is included. Math 4 CD-ROMs are NOT included; this book is not designed to be used without the CDs. Teaching Textbooks Grade 4.
|
Numbers deals with the development of numbers from fractions to algebraic numbers to transcendental numbers to complex numbers and their uses. The book also examines in detail the number pi, the evolution of the idea of infinity, and the representation of numbers in computers. The metric and American systems of measurement as well as the applications... more...
How to Pass Numeracy Tests will help you practice for timed tests, revise your maths and numeracy knowledge and improve your test technique. Providing over 350 practice questions it also gives vital advice on how the tests are marked and what you can do to optimise results.
The wide variety of practice includes 20 timed tests on data interpretation,... more...
|
Algebraic expressions. - Mathematics
objective:On completion of the lesson the student will understand some of the short cuts used in writing algebraic expressions, and the student will be able to write algebraic expressions down in a way that is easier to understand.
getting started:
Step 1 PRACTICE
Assess your current knowledge of the chosen topic!
Step 2 TUTORIAL
Consolidate your current knowledge of the chosen topic with a Teacher presented tutorial.
Step 3 EXAM
Test your retention of the mathematics material with the exam.
ASSIGNMENT SHEETS
Print out and complete the assignment sheet to further your knowledge on the material, it's easy
All early levels
of mathematics contain printable worksheets which are
an addition to the interactive computer lesson. Printable worksheets
offers your child continuing development of their written
skills as well as formulating answers for exercises related to
the selected topic. Parents have the option to print
out or review the answers for the selected worksheets.
|
Description
Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases students' mathematical literacy so that they better understand the mathematics used in their daily lives, and can use math effectively to make better decisions every day. Contents are organized with that in mind, with engaging coverage in sections like Taking Control of Your Finances, Dividing the Political Pie, and a full chapter about Mathematics and the Arts.
This Fifth Edition offers new hands-on Activities for use with students in class, new ways for students to check their understanding through Quick Quizzes, and a new question type in MyMathLab that applies math to excerpts from recent news articles. In addition, the authors increase their coverage of consumer math, and provide a stronger emphasis on technology through new Using Technology features and exercises. The new Insider's Guideprovides instructors with tips and ideas for effective use of the text in teaching the course.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
Table of Contents
Preface
Prologue: Literacy for the Modern World
Part 1 Logic and Problem Solving
Chapter 1 Thinking Critically
1A Recognizing Fallacies
1B Propositions and Truth Values
1C Sets and Venn Diagrams
1D Analyzing Arguments
1E Critical Thinking in Everyday Life
Chapter 2 Approaches to Problem Solving
2A The Problem-Solving Power of Units
2B Standardized Units: More Problem-Solving Power
2C Problem-Solving Guidelines and Hints
Part 2 Quantitative Information in Everyday Life
Chapter 3 Numbers in the Real World
3A Uses and Abuses of Percentages
3B Putting Numbers in Perspective
3C Dealing with Uncertainty
3D Index Numbers: The CPI and Beyond
3E How Numbers Deceive: Polygraphs, Mammograms, and More
Chapter 4 Managing Money
4A Taking Control of Your Finances
4B The Power of Compounding
4C Savings Plans and Investments
4D Loan Payments, Credit Cards, and Mortgages
4E Income Taxes
4F Understanding the Federal Budget
Part 3 Probability and Statistics
Chapter 5 Statistical Reasoning
5A Fundamentals of Statistics
5B Should You Believe a Statistical Study?
5C Statistical Tables and Graphs
5D Graphics in the Media
5E Correlation and Causality
Chapter 6 Putting Statistics to Work
6A Characterizing Data
6B Measures of Variation
6C The Normal Distribution
6D Statistical Inference
Chapter 7 Probability: Living with the Odds
7A Fundamentals of Probability
7B Combining Probabilities
7C The Law of Large Numbers
7D Assessing Risk
7E Counting and Probability
Part 4 Modeling
Chapter 8 Exponential Astonishment
8A Growth: Linear versus Exponential
8B Doubling Time and Half-Life
8C Real Population Growth
8D Logarithmic Scales: Earthquakes, Sounds, and Acids
Chapter 9 Modeling Our World
9A Functions: The Building Blocks of Mathematical Models
9B Linear Modeling
9C Exponential Modeling
Chapter 10 Modeling with Geometry
10A Fundamentals of Geometry
10B Problem Solving with Geometry
10C Fractal Geometry
Part 5 Further Applications
Chapter 11 Mathematics and the Arts
11A Mathematics and Music
11B Perspective and Symmetry
11C Proportion and the Golden Ratio
Chapter 12 Mathematics and Politics
12A Voting: Does the Majority Always Rule?
12B Theory of Voting
12C Apportionment: The House of Representatives and Beyond
12D Dividing the Political Pie
Credits
Answers
|
MyMathLab access: All new textbooks purchased at an ACC bookstore include MyMathLab access. It is not included with the purchase of a used book, and may not be included with a new book purchased at a different bookstore. Refer to the handout Information about MyMathLabStudents will feel a sense of accomplishment in their increasing ability to use mathematics to solve problems of interest to them or useful in their chosen fields. Students will attain more positive attitudes based on increasing confidence in their abilities to learn mathematics.
Students will learn to understand material using standard mathematical terminology and notation when presented either verbally or in writing.
Students will improve their skills in describing what they are doing as they solve problems using standard mathematical terminology and notation.
1. Description and classification of whole numbers, integers, and rational numbers using sets and the operations among them
identify and use properties of real numbers
simplify expressions involving real numbers
evaluate numerical expressions with integral exponents
2. Polynomials
distinguish between expressions that are polynomials and expressions that are not
classify polynomials in one variable by degree and number of terms
simplify polynomials
add, subtract, multiply, and divide polynomials (including the use of long division techniques and the distributive law)
factorunderstand and use the exponent laws involving integer exponents
convert numbers into and out of scientific notation and perform multiplication and division with numbers written in scientific notation
solve application problems which lead to one of the following types of equations: linear equations in one variable, systems of two linear equations in two variables, quadratic equations
solve literal equations for a specified variable using addition and multiplication principles
use given data to estimate values and to evaluate geometric and other formulas
solve problems involving the Pythagorean theorem
6. Linear equations in two variables
identify the relationship between the solution of a linear equation in two variables and its graph on the Cartesian plane
understand and use the concepts of slope and intercept
determine slope when two data points are given
graph a line given either two points on the line or one point on the line and the slope of the line
write an equation of a line given one point on the line and the slope of the line, or two points on the line
identify lines given in standard, point-slope, or slope-intercept forms and sketch their graphs
solve systems of linear equations
7. Quadratic equations
find solutions to quadratic equations using the technique of factoringand using the principle of square roots
recognize a need to use the quadratic formula to solve quadratic equations and solve quadratic equations by using the quadratic formula when simplification of square roots other than perfect squares is not needed
8. Description and classification of irrational numbers
simplify perfect square radical expressions
use decimal approximations for radical expressions
9. Rational expressions
determine for which value(s) of the variable a rational expression is undefined
|
All of our updated Lesson Materials have 4 documents (Student
Exploration Sheet, Answer Key, Teacher Guide, and Vocabulary Sheet),
each available as a .doc (for easy editing) or a .pdf (for smaller
filesize). You will need to be logged in to see all four documents.
Functions - and in particular linear functions - are big concepts in math. We hope these Gizmos and these new Lesson Materials will help students really understand what they are, how they work, and why they're important.
|
Classroom Materials Please check with your student that they have their proper materials for class. Their required materials include, on a daily basis, pencils, paper, textbook, and of course homework. Being prepared is the first step toward a successful school year!
Standard and Challenge asssignments Several times per week, the students will have the opportunity to choose between a Standard assignment and a Challenge assignment. The purpose of presenting two assignments is to accomodate those students who wish to have more challenge in the homework. The students will get the same amount of credit for either assignment. Typically, both of the assignments will be from the same page or section of the textbook. When checking assignments with Online Classroom the Standard assignment is indicated with a capital s (S) and the Challenge assignment with a capital c (C).
Chapter 7 Test
On Tuesday 5/21/2013 Pre Algebra students will have the Chapter 7 Test in class. The test is of the same format as the practice distributed to students in class on Friday 5/17/2013. The test will be 16 problems, each worth 3 points for a total value of 48 points.
|
If your work involves math that can't easily be done in a spreadsheet, MathCad 6.0 from MathSoft (617 577-1017) may be the tool for you. Both the Standard ($129) and programmable Plus ($349) versions let you do complicated numerical or symbolic calculations on a sort of computerized scratch pad. Results can then be pasted into other Windows applications. While less sophisticated than computer algebra systems such as Wolfram Research's Mathematica, MathCad is much easier to learn and use.BY STEPHEN H. WILDSTROM
|
Mathematics is a subject that entails counting, computing and calculating of numbers and at times even variables. Earlier, abacus was used by man for the purpose of mastering the skill of counting but with the passage of time more sophisticated calculators were developed. Thanks to technological advancement now there are various types of electronic calculators which are available for purchase, a percentage calculator being one of them. These calculators can be extremely handy in many situations. For instance, if you had to calculate some percentages then it is advisable to use a percent calculator.
Using the technology of a percentage calculator or any other kind of calculator for that matter has its own advantages and disadvantages. Calculators are considered as a normal tool these days, which can prove to be indispensable at times. There are two kinds of calculators: handheld calculators and online versions and an example of the latter would be the percent calculator. Online calculators are different from handheld calculators in the sense that they are far more superior because they provide a lot more functions. Some of these net calculators can even plot an equation into a graphical form.
Popular math calculators such as the percent calculator or other types of calculators are used by people from different walks of life such as technicians, students, engineers and teachers. Online calculators, including the percentage calculator, equip the user with a superior understanding of mathematical operations. These calculators assist them in the process of verifying their knowledge of mathematical formulae and theory. With the help of such a tool, they will be able to visualize a possible value of an unknown answer. Technicians and engineers rely on online calculators heavily because their line of work calls for the use of such devices.
A lot of people have prejudices against mathematics and they are just scared of what the subject entails. On the contrary, mathematics is a subject that is very logical and unless the individual understands the logic behind it, he/she would always find it hard to figure things out. Online calculators like percentage calculator can remove some of the prejudices against mathematics to a certain extent. If you are wondering how a percent calculator or any calculator can help one understand mathematics, then the answer lies in the tendency of such calculators to provide explanations to its workings.
In order to understand how such calculators can help you understand math, make use of a high quality and ultra efficient percentage calculator. This can easily be located online in various websites and you just have to ensure that the option which you have chosen provides explanation of how the answer or solution was obtained. Now use the percent calculator to solve a sum that you do not understand. Once you verify the accuracy of the answer, you can then access the explanation part and see the step-by-step instructions on how the answer was calculated.
If you combine online calculators with online self-tutor resources then you will get the ultimate "dream team" to help you combat all your math problems. Using an online percent calculator is not at all difficult – you just have to enter some information from the sum that you are looking to solve. After this, you just have to click on a mouse button and the percentage calculator would do all the hard work for you and display the answer on your computer screen. So the next time when you are having difficulty with your mathematics homework or anything related to mathematics then make use of free online calculators as these magnify the beauty of mathematics.
Given the rising popularity of the Internet it is but natural for people to shift to an online percentage calculator to assist them in their work. Amongst the many advantages of a percent calculator one of the foremost is its convenience which adds to the fun of solving
|
College Algebra (3rd Edition)
9780321466075
ISBN:
0321466071
Edition: 3 Pub Date: 2007 Publisher: Addison Wesley
Summary: These authors have created a book to really help students visualize mathematics for better comprehension. By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution. In addition, the authors have added a variety of new tools to help students better use the book for maximum effecti...veness to not only pass the course, but truly understand the materialDowningtown, PAShipping:Standard, ExpeditedComments:This book has some WRINKLED pages but can still be used without any problem or hassle. Ships with... [more]This book has some WRINKLED pages but can still be used without any problem or hassle. Ships within 24 hrs of your order. Open Mon - Fri. May have some notes/highlighting, slightly worn covers, general wear/tear. [
|
Book Description: This best-selling text balances solid mathematical coverage with a comprehensive overview of mathematical concepts as they relate to varied disciplines. The text provides an appreciation of mathematics, highlighting mathematical history, and applications of math to the arts and sciences. It is an ideal book for students who require a general overview of mathematics, especially those majoring in liberal arts, the social sciences, business, nursing and allied health fields. Let us introduce you to the practical, interesting, accessible, and powerful world of mathematics today—the world of A Survey of Mathematics with Applications, 8e
|
Synopsis
Mathematica Cookbook helps you master the application's core principles by walking you through real-world problems. Ideal for browsing, this book includes recipes for working with numerics, data structures, algebraic equations, calculus, and statistics. You'll also venture into exotic territory with recipes for data visualization using 2D and 3D graphic tools, image processing, and music.
Although Mathematica 7 is a highly advanced computational platform, the recipes in this book make it accessible to everyone -- whether you're working on high school algebra, simple graphs, PhD-level computation, financial analysis, or advanced engineering models.
Learn how to use Mathematica at a higher level with functional programming and pattern matching
Delve into the rich library of functions for string and structured text manipulation
Learn how to apply the tools to physics and engineering problems
Draw on Mathematica's access to physics, chemistry, and biology data
Get techniques for solving equations in computational finance
Learn how to use Mathematica for sophisticated image processing
Process music and audio as musical notes, analog waveforms, or digital sound samples
|
Credits: 15.
Grading scale: TH.
Cycle: G1
(First Cycle).
Main field: Technology.
Language of instruction: The course will be given in Swedish.
FMAA01 overlaps following cours/es: FMA410, FMA415, FMA645 and FMAA05.
Compulsory for: B1, BME1, C1, D1, K1, M1, MD1, N1.
Course coordinator: Director of Studies Anders Holst, Anders.Holst@math.lth.se, Mathematics.
Assessment: Written test in all subcourses, comprising theory and problem solving. The final grade is the integer part of a weighted mean (weights 1,1,2) of the three grades of the subcourses (at most 5). Computational ability tests (see subcourse A1 below). Some oral and written assignments.
Parts: 3.
Further information: The course Calculus in one variable is taught and examined in two versions, A and B respectively, depending on the student's program. The goals are the same. The present course description is version A.
Home page:
Aim The aim of the course is to give a basic introduction to calculus in one variable. Particular emphasis is put on the role that the subject plays in applications in different areas of technology, in order to give the future engineer a good foundation for further studies in mathematics as well as in other subjects. The aim as also to develop the student's ability to solve problems, to assimilate mathematical text and to communicate mathematics.
Knowledge and understanding For a passing grade the student must
within the framework of the course with confidence be able to handle elementary functions of one variable, including limits, derivatives and integrals.
be able to set up and solve some types of linear and separable differential equations that are important in the applications.
be familiar with the logical structure of mathematics, in the way it appears e.g. in plane geometry.
be able to give a general account of and illustrate the meaning of mathematical concepts in calculus in one variable that are used to construct and study mathematical models in the applications.
be able to account for the contents of definitions, theorems and proofs.
Skills and abilities For a passing grade the student must
be able to demonstrate a good algebraic computational ability and without difficulties be able to calculate with complex numbers.
in the context of problem solving be able to demonstrate an ability to independently choose and use mathematical concepts and methods in one-dimensional analysis, and to construct and analyse simple mathematical models.
in the context of problem solving be able to integrate knowledge from different parts of the course.
be able to demonstrate an ability to explain mathematical reasoning in a structured and logically clear way.
Contents Part 1. The number concept. Calculation with fractions. Inequalities. Square roots. Curves and equations of degree 2. Geometry in the plane. Analytic geometry. The circle, ellipse, hyperbola. Arithmetic and geometric sums. The binomial theorem. The modulus of a number. Trigonometry. Powers and logarithms. The concept of a function. The properties of the elementary functions: curves, formulas. Sequences of numbers.
Part 2. Limits with applications: asymptotes, the number e, series. Continuous functions. Derivatives: definition and properties, applications. Derivatives of the elementary functions. Properties of differentiable functions: the mean value theorem with applications. Curve sketching. Local extrema. Optimization. Some simple mathematical models. Complex numbers and polynomials. The Taylor and Maclaurin formulae. Expansions of the elementary functions. The importance of the remainder term. Applications of Maclaurin expansions. Problem solving within the above areas.
Code: 0108.
Name: Part A1. Higher education credits: 5.
Grading scale: UG.
Assessment: Written test comprising theory and problem solving. Computational ability tests must be passed before the examination. One assignment (oral and in writing) must be passed before the examination.
Contents: See above, part 1.
Code: 0208.
Name: Part A2. Higher education credits: 5.
Grading scale: UG.
Assessment: Written test comprising theory and problem solving. One assignment (oral and in writing) must be passed before the examination.
Contents: See above, part 2.
|
Student placement in a mathematics course is subject to ACT-MATH scores or the COMPASS placement test scores or Academic Services Center approval. Students with ACT-MATH scores of 20 or above may enroll in any math course with numbers up to and including MATH 146 or MATH 165. Students with an ACT-MATH score below 20, or no ACT-MATH score, are required to take the COMPASS placement test. Placement of students is based on the level of achievement on the test.
ASC 090 Math Prep (2 credits)
This course improves basic math computational skills: addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals. Includes a study of percents and application of percents. This course may be required due to Compass test results and the course placement policy. (F, S, Su)
ASC 091 Algebra Prep I (2)
This course is designed for students with little or no algebra background who need to prepare for further study in mathematics or who need to review basic algebra concepts. It includes topics such as real numbers, fundamental operations, variables, equations, inequalities and applications. (F, S, Su, O)
MATH 120 Basic Mathematics I (2)
A review of whole numbers, fractions and decimal numbers in conjunction with the fundamental application of ratios, rates, unit rates, proportions and percents in solving everyday problems. The application of business and consumer mathematics such as simple and compound interest, purchasing and checkbook reconciliation. (F, S, Su)
MATH 135 Applied Mathematics (2)
A review of mathematics including fractions, decimals, percentages and basic algebra which incorporates algebraic fractions and equations with variables. Emphasis is placed on the strategies of problem-solving using agricultural applications. (F)
MATH 136 Technical Trigonometry (2)
A study of the fundamentals of trigonometry. Right triangle trigonometry, the Law of Sines, the Law of Cosines and Vectors. Emphasis is placed on problem-solving for the technology fields. Prerequisite: MATH 132. (F, S, O)
MATH 138 Applied Trigonometry (3)
A theory/lab course studying the fundamentals and applications of trigonometry, including right and oblique triangles, the Law of Sines, the Law of Cosines, vectors, angular velocity, graphs and complex numbers.
MATH 146 Applied Calculus I (4)
Review of algebra, including linear, quadratic, exponential and logarithmic functions. Calculus topics for this course will be limits, continuity, rates of change, derivatives, extrema, anti-derivatives and integrals. Emphasis is placed on real-data application. Course is intended for those majoring in business, management, economics, or the life or social sciences. Prerequisite: MATH 103 or MATH 104 or placement exam. (F) ND:MATH
MATH 147 Applied Calculus II (4)
Integrals, multivariable calculus, introduction to differential equations, probability and calculus, sequences and series, introduction to trigonometric functions, derivatives and integrals of trigonometric functions. Emphasis is placed on real-data application. Course is intended for those majoring in business, management, economics or the life or social sciences. Prerequisite: MATH 146. (S) ND:MATH
MATH X92 Experimental Course (1-9)
A course designed to meet special departmental needs during new course development. It is used for one year after which time the course is assigned a different number.
MATH 299 Special Topics (1-5)
A special purpose class or activity to be used for a mathematics course in process of development, for classes occasionally scheduled to meet student needs or interests, or offered to utilize particular faculty resources. (F, S, Su)
|
Contact
Math
Mathematics Curriculum
Goals:
Students will acquire mathematical skills, including the ability to perform routine computations and symbolic manipulation.
Students will develop an understanding of fundamental mathematical concepts.
Students will become mathematical problem solvers.
Students will learn to value mathematics and the quantitative nature of our world.
High School
The high school curriculum is built around a core of topics that all students will have an opportunity to study regardless of the courses they take. These core topics come from the areas of algebra, geometry, measurement, data analysis and probability. In order to graduate from high school, students must take two-years of high school mathematics, including first-year algebra.
|
See What's Inside
Product Description
This book examines the study of geometry in the middle grades as a pivotal point in the mathematical learning of students and emphasizes the geometric thinking that can develop in grades 6–8 as a result of hands-on exploration. An essay on the accompanying CD-ROM describes the van Hiele framework and how it can help improve teaching strategies and assessment. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
|
MATH 549: Introduction to Number Theory
Course ID
Mathematics 549
Course Title
MATH 549: Introduction to Number Theory
Credits
3
Course Description
Number theory is a branch of mathematics that involves the study of integer properties. Topics covered include factorization, prime numbers, continued fractions and congruences as well as more sophisticated tools, such as quadratic reciprocity, Diophantine equations and number theoretic functions. However, many results and open questions in number theory can be understood by those without an extensive background in mathematics. Additional topics might include Fermat's Last Theorem, twin primes, Fibonacci numbers and perfect numbers. 349/549
|
The math theory is developed in slow, simple stages and is directly applied to the solution of real problems. This method is backed up with "CHECKUPS" which act as a motivator, and "BRUSHUPS" which review the mathematical concepts immediately necessary for the continuance of the electrical development and applications.
Chapter 1 Introduction to Electricity
Chapter 2 Simple Electric Circuits
Chapter 3 Formulas
Chapter 4 Series Circuits
Chapter 5 Parallel Circuits
Chapter 6 Combination Circuits
Chapter 7 Electric Power
Chapter 8 Algebra for Complex Electric Circuits
Chapter 9 Kirchoff's Laws
Chapter 10 Applications for Series and Parallel Circuits
Chapter 11 Efficiency
Chapter 12 Resistance of Wire
Chapter 13 Size of Wiring
Chapter 14 Trigonometry for Alternating-Current Electricity
Chapter 15 Introduction to AC Electricity
Chapter 16 Inductance and Transformers
Chapter 17 Capacitance
Chapter 18 Series AC Circuits
Chapter 19 Parallel AC Circuits
Chapter 20 Alternating-Current Power
Chapter 21 Three-Phase Systems
Chapter 22 Three-Phase Transformer Connections
Chapter 23 Mathematics for Logic Controls
Chapter 24 Signal Distribution
|
Based on lectures given at a summer school on computer algebra, the book provides a didactic description of the facilities available in three computor algebra systems - MAPLE, REDUCE and SHEEP - for performing calculations in the algebra-intensive field of general relativity. With MAPLE and REDUCE, two widespread great-purpose systems, the reader is shown how to use currently available packages to perform calculations with respect to tetrads, co-ordinate systems, and Poincare` gauge theory. The section on SHEEP and Stensor, being the first published book on these systems, explains how to use these systems to tackle a wide range of calculations with respect to tackle a wide range of
calculations in general relativity, including the manipulation of indicial formulae. For the researcher in general relativity, the book therefore promises a wide overview of the facilities available in computer algebra to lessen the burden of the lengthy, error-prone calculations involved in their research
|
MA 371
MA 371 - Real Analysis
Fundamental Concepts a) Set Theory i) Basic notions ii) Relations and functions b) The Real Number System i) Ordered fields ii) The rational numbers as an ordered field iii) The real numbers as a complete ordered field iv) Sequences of real numbers and their properties v) The Cauchy criterion
|
This book consists of four units of study (Counting and Listing -- CL; Functions -- Fn; Decision Trees and Recursion -- DT; and Basic Concepts of Graph Theory -- GT), each divided into four sections. Each section contains a representative selection of problems. These vary from basic to more difficult, including proofs for study by mathematics students or honors students. The first three sections in units CL and Fn are primarily a review of material in A Short Course in Discrete Mathematics needed for this course. saved under Free Mathematics Books/Discrete Mathematics
by
freebooksandarticles
|
Math: Solving Word Problems, 2nd Edition
ISBN10: 1-59863-983-8
ISBN13: 978-1-59863-983-4
AUTHORS: Immergut
Get ready to master the unknown number! Master Math: Solving Word Problems is a comprehensive reference guide that explains and clarifies the difficulties people often face with word problems, in a simple, easy-to-follow style and format. Beginning with the most basic types of word problems and progressing through to the more advanced, Solving Word Problems shows you how to focus first on the words in the problem, and then on the numbers, breaking down the problem into smaller segments to help you work through. Using familiar situations
from everyday life such as percents and discounts, interest, motion and speed, and probability, each type of word problem is taught using step-by-step procedures, solutions, and examples. And end-of-chapter problems will help you practice what you learned. A complete table of contents and a comprehensive index enable you to quickly find specific topics, and the approachable style and format facilitate an understanding of what can be intimidating and tricky skills. Perfect for both students who need some extra help or rusty professionals who want to brush up, Solving Word Problems will help you master everything from simple equations and percents to statistics and probability
|
This weekly feature will provide a peek into the learning that is happening at West Lutheran High School.
Week of May 11, 2012
5/14/2012
Pre-Algebra with Mr. Jensen
The class has been working hard on making sure they are ready for Algebra I next year. In our final chapter, we focus on simplifying expressions. Mainly, trying to make sure to know how the exponents of variables react when being added together, multiplied together, and/or taking a power of a power.
Try the following similar problems....if you know the answers, you are probably ready for Alg I:
|
Search Journal of Online Mathematics and its Applications:
Journal of Online Mathematics and its Applications
Tool Building: Web-based Linear Algebra Modules
by David E. Meel and Thomas A. Hern
Student Responses, Part 2
The Work of Jack
Our purpose for the Transformer activity is to get students to explore, make conjectures, and adjust their conjectures. To give a sense of how students use Transformer2D, we provide the work of a randomly selected student named Jack.
The worksheet link on the preceding page downloads a one-page MS Word document with the full statement of the project. However, students were working with a multi-page version with space to fill in their answers. We provide that version here as a PDF file.
Here are the parts of Jack's work as scanned images -- each opens in a separate window, and you may want to have more than one open at a time.
It is evident from Jack's work that he has a growing need for language to describe the phenomena he is encountering. Terms such as "shear," "reflected," and "rotated" are being drawn from the text to help explain the changes he observes in the geometric figures. As a consequence of this type of encounter, students come to appreciate the discussion of the terms and their meanings because they have a need for precise definitions of what they are experiencing.
Jack's described his reactions to the transformer project in his weekly journal entry:
"In class this week we started looking at transformation matrices, still in the form Ax = b. Now this is the same equation we have been using, but now we are looking at it in a different way. Going into the computer lab we started a project using the transformer on Dr. Meel's personal website. This was an interesting tool, because it allowed us to see the effects of different matrices on the range of values in R^2. The different kinds of transformations include reflections, contractions, expansions, vertical and horizontal shear, and projection onto the x-axis and y-axis.
"The project at first seemed a little overwhelming to be honest. I really had no clue how to tell what each matrix was doing by looking at them. But as I tried more and more matrices, and experimented to see what changes I made would create, it all started to make sense. After messing around for a while with the transformer and reading the book, I was able to go through the packet and do my best to answer the questions. Some of them I am not entirely sure I approached the correct way, but I am sure that I understand transformations a lot better in spite of that. So even if I am slightly off from the answers, I at least have an understanding of what transformations are and how to create some myself."
We are immersing students in a learning situation in which making sense of the environment is one major component as they grapple with their own limited perspectives and enhance their understandings of one or another concept piece by piece. In such an environment, they keep exploring, keep conjecturing, keep trying to organize thinking in new ways to accommodate the new bits of information being displayed, while at times experiencing considerable frustration because they lack the big picture. The tools and the corresponding projects are designed to eliminate the frustration and also to permit the teacher to assist, probe for understanding, point out significant hurdles, suggest alternative lines of thinking, and help equip students to manage their frustration and continue to pursue knowledge.
|
...Within the discipline there are many areas of study including Paleobotany (the study of plant history through fossils), Physiology (the study of plant cells and tissues), Pteridology (the study of ferns), and Plant Pathology (the study of diseases in plants).
If you need any further information,...
...While it is a powerful tool, a strong understanding of the assumptions behind the equations is imperative to prevent misuse. Linear algebra is the study of all forms of math that can be expressed in vectors, from simple vector algebra all the way to advanced applications in differential equation...
|
Rent Book
Buy Used Book
Buy New Book
In Stock Usually Ships in 24 Hours
$17.00
eBook
$15.31
More New and Used from Private Sellers
Starting at $6Your solution to MATH word PROBLEMS! Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success. Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning. It's a no-brainer! You'll learn to solve: Decimal, fraction, and percent problems Proportion and formula problems Number and digit problems Distance and mixture problems Finance, lever, and work problems Geometry, probability, and statistics problems Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill.
|
Rent Textbook
Buy Used Textbook
Buy New Textbook
Currently Available, Usually Ships in 24-48 Hours
$166.40Normal 0 false false false MicrosoftInternetExplorer4 Intended for a 2-semester sequence of ElementaryandIntermediate Algebrawhere students get a solid foundation in algebra, including exposure to functions, which prepares them for success in College Algebra or their next math course. Operations on Real Numbers and Algebraic Expressions; Equations and Inequalities in One Variable; Introduction to Graphing and Equations of Lines; Systems of Linear Equations and Inequalities; Exponents and Polynomials; Factoring Polynomials; Rational Expressions and Equations; Graphs, Relations, and Functions; Radicals and Rational Exponents; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conics; Sequences, Series, and The Binomial Theorem; Review of Fractions, Decimals, and Percents; Division of Polynomials; Synthetic Division; The Library of Functions; Geometry; More on Systems of Equations For all readers interested in elementary and intermediate algebra.
|
Grade 11 Mathematics
MBF3C1:Foundations for College Mathematics, Grade 11, College Preparation
This course enables students to broaden their understanding of mathematics as a problem-solving tool in the real world. Students will extend their understanding of quadratic relations, as well as of measurement and geometry; investigate situations involving exponential growth; solve problems involving compound interest; solve financial problems connected with vehicle ownership; and develop their ability to reason by collecting, analysing, and evaluating data involving one and two variables. Students will consolidate their mathematical skills as they solve problems and communicate their thinking. Prerequisite: Foundations of Mathematics, Grade 10, Applied
MCR3U1:Functions, Grade 11, University Preparation
This course introduces the mathematical concept of the function by extending students' experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; and develop facility in simplifying polynomial and rational expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Principles of Mathematics, Grade 10, Academic
MCF3M1:Functions and Applications, Grade 11, University/College Preparation
This course introduces basic features of the function by extending students' experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modelling real-world situations. Students will represent functions numerically, graphically, and algebraically; simplify expressions; solve equations; and solve problems relating to financial and trigonometric applications. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Principles of Mathematics, Grade 10, Academic, or Foundations of Mathematics, Grade 10, Applied
|
Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question
|
Key To Algebra® and Key to Geometry® Workbooks and Answer Keys
If you want a course that satisfies the algebra or geometry requirements for high school, or need a preliminary course for college bound students, then I can recommend the
Key to Algebra and the Key to Geometry series.
From the publisher: "Each student workbook is a self-contained learning unit--everything you need to build and exercise math skills. Concepts are presented one per page in small, easily digestible steps. The Key to Algebra and the Key to Geometry workbooks are easy to read and understand, and allow students to proceed at their own pace and to study at their own level."
The Answer Books have the pages reproduced with filled-in answers for easy grading.
There are four reduced student pages with answers on each page.
Why We Use Key to Algebra
In our homeschool we give the Key to Algebra workbooks to our children
before they start Algebra 1. I want to make sure the student thoroughly
understand the basic algebra concepts. It also helps them to work faster
because the problems on a page are of one type. Often homeschooled students
understand the algebra concept, but their computation skills are slower
because there's no bell that rings at the end of a class--unless the
homeschool teacher rings one. I usually assign a lot of problems to be done
in a short time.
What's Best About Key to Geometry
It is one of the best courses to teach your student how to do geometry
constructions with a pencil, straightedge, and a compass. The instructions
are easy for the student to follow independently. Grading them is easy
because of the format of the Answer Books. We assign Key to Geometry to each
of our students before they start either
Harold
Jacobs Geometry or
BJU Press Geometry.
|
In school before you stop studying mathematics, please do the
following.
Look at the chapters Implication Rules
and Chains of Reason. These two
chapters may help you to use and understand precisely rules,
instructions, patterns, definitions and recipes in every
subject and every area of skill or specialization including
mathematics.
Read the first chapters on algebra. The description in them of
the three key skills for algebra and the algebraic examples
will, I hope, help you step from arithmetic to algebraic way of
writing and thinking plus a little beyond. These chapters try
to explain and describe with everyday words, how (algebraic)
shorthand notation is used to describe and do mathematics after
arithmetic. There is more to mathematics than just doing
arithmetic well.
Mathematics courses are preparation for business calculations, for
handling your money sensibly and for courses in sciences,
engineering and technology. You should view mathematics as an
opportunity to strengthen your thinking skills.
In mathematics courses you should not only meet calculations to do
but also the chains and threads of reason and persuasion which
justify them and links them together. Understanding and following
the rules and patterns of mathematics, practices and nurtures an
ability to think and reason well. Mathematics provides a neutral
territory for the practice of rule and pattern-based reason and
logic. The opinions and views you meet in daily life say and care
little about what mathematical conclusions should be.
If you find yourself in a course which gives formulas and numbers
to use in them but does not expect you to use algebra, you are
wasting your time. Your time would be better spent studying
algebra, and then taking a more advanced course that respects your
intelligence. Similarly in college, if you find a course which
gives you formulas and numbers to use in them and also talks at
length about rates of change without expecting you to understand
calculus,6 then a calculus course would be of
better use of your time.
6 Calculus in the first instance
provides formulas for the slopes of (nonlinear) curves and for
the rates of changes of numbers or quantities.
Look at the description of courses you will take in and outside
mathematics. From their description, see what mathematics course(s)
you are expected to take with them (co-requisites) or before them
(prerequisites). Methods taught in a co-requisite mathematics
course are too often covered after they are required in another
course. So take a mathematics course before there is any
possibility the methods in it will be needed in another course.
Then you may master the methods before they are required and not
after.
If you follow the advice and the cautions below, you should have a
mathematical foundation for any subject requiring calculation. In
mathematics do the following.
Master arithmetic. Also master weights and measures. After you
have mastered the rules of arithmetic, learn to use a
calculator.
Master the algebraic way of writing and thinking. Also master
the use of rules and patterns to arrive at conclusions.
Mathematics after arithmetic builds on our abilities to talk
about numbers and quantities, to describe calculations and to
change the way calculations are done. Mathematics after
arithmetic also depends on our ability to follow and understand
rules and patterns. See the first chapters on reason and
algebra in this book.
After algebra, take trigonometry and geometry.
Learn about money matters. Take a course on money calculations,
preferably after a course in basic algebra. Most of us handle
money for credit or investment. In your last year of studies
before starting work, take a course on the mathematics and
arithmetic of personal finances. This course should include
balancing of budgets, description of typical household expenses
for individuals or families in rented or mortgaged properties,
problems involving simple and compound interest, and
mortgage/pension calculations. Traces of such calculations
appear in elementary and high school mathematics, but they are
forgotten years before you need them. A course like the one
described should be offered in schools and colleges for
students in any art or science. Ask for one to be given, if it
is not already offered. All this is practical mathematics.
It should be more widely known.
If you go to college, take a year or two of the mathematics
subject called calculus, a year of probability and statistics,
and a year of matrix computations (or linear algebra). Calculus
courses usually have trigonometry and algebra as prerequisites.
Calculations in many trades, including business, engineering,
computer technology, physics and health science, require
calculus.
Mastering the rules and patterns of mathematics and reason
(there is a connection) is good practice for mastering the rule and
patterns of all disciplines. To master mathematics you need to read
your course notes or course textbook carefully. Examples, solutions
and proofs show you patterns to follow or imitate. Here every step
not understood hides an idea from you.
The problems you find easy to solve should be done to restore and
build your confidence and to reassure yourself that you have
understood what they require. But after you have done a few such
problems, you should look at the ones which appear harder. The
problems which appear to be too hard should be noted and
remembered. You can return to them later by yourself or with help
from another. What is hard for you to solve may be easy for
another, and vice-versa.
When I taught a remedial algebra course, one of my students was a
high school gym teacher. One of his past assignments was to teach
algebra.7
7This should not occur, but in many
school systems it does. When it does, it shows a lack of respect
to students and a lack of purpose for education. It also suggests
circumstances beyond the control of students and teachers.
In some schools due to circumstance beyond their immediate control,
some instructors are required to explain ideas outside their own
specialties. When or if you meet such an instructor, be polite and
do not become a troublemaker. If a teacher sees you as a threat or
troublemaker, you may suffer. When you meet a misplaced instructor,
politely and diplomatically try to transfer to another class in the
same subject or read the course textbook yourself and get a tutor.
Here are several more comments on learning mathematics or
another subject.
How you find a solution to a problem is not important provided
you understand fully the solution. (Some teachers may
disagree.)
If you have to copy solutions blindly then you will not
understand ideas well enough to pass tests and the final
examination.
You should ask another to check that your written responses or
solutions are both understandable and well-written. Mistakes
brought to your attention in any manner improve your
understanding. If such checking improves your ability to avoid
mistakes in the future, then such checking should I believe be
encouraged. Again, some teachers may disagree.
Students who know and identify in their solution those steps
which are doubtful deserve more respect than students who
don't. Knowing exactly where one is sure and where one is not
is the sign of an alert mind.
Correct answers obtained accidentally, for instance by
canceling errors in a solution should not be given full marks.
Errors in a solution show that the subject is not carefully
mastered.
Learning is better done in a cooperative atmosphere where
students help each other to understand instead of a competitive
one, where the success of one student is at the expense of
others. (But you can not always choose your environment.)
Seeing two or more approaches to a subject can be better than
one. What appears hard in one approach may appear easier in
another.
Calculators lessen the need for us to do arithmetic but, in using
them, mistakes can be made. Here you need to know in advance what
kind of answer a computation will yield. If you think you have made
a mistake in entering numbers or instructions, you need to reenter
them again. If a different result appears from before, at least one
of your efforts, the original or the check, will be in error.
(Logic Question: What can you say for sure if the results agree?)
Suggestion: remember or learn how to do arithmetic by hand and how
to estimate the expected size of results for addition, subtraction,
multiplication and division.
Computer programs can perform arithmetic and algebraic or symbolic
operations. They can also draw graphs and solve some equations
rapidly. These programs do not provide solutions to all possible
problems. For the solutions they can provide, you have to
understand the statement of the initial problem. Beyond this, a
computer (or another student) cannot understand the chains of
reasoning for you. Understanding is an personal affair. No computer
and no other person can do this for you. But if you know what to
expect from a calculation, calculators and computer programs can
help you check your expectations and explore mathematical ideas.
Here you can learn from your mistakes. In some cases, computer
software can tutor you. They can tell what to expect in various
circumstances. Today there are computer programs and on-line
computer books which may help you master mathematics and other
subjects. More are appearing everyday. I know of them, but I have
no experience with them.
August 2011 Postscript - Context
In mathematics programs for ages 4 to adult, the main objective
appears to be the preparation of students for college level studies
and careers based on calculus or statistics. But many or most
students do not complete secondary school. While mathematics
programs for ages 5 to 12 may develop numerical and geometric
skills and practices of actual or potential value for daily and
adult life, programs for ages 13 to 17 focus on a preparation for
calculus- or statistic-based college studies, and they do so
imperfectly.
Most secondary school mathematics - in North America at
least - is taught by instructors who have not seen how or why
calculus nor statistics is used in college studies and beyond. Most
instructors started to teach mathematics not because they were
trained in it, but because no one else was available to teach it.
Besides that, year after year, the only immediate reason for
understanding or explaining a skill or practice is its likely
appearance on the next final examination. In a discipline whose
mastery is taken to be a sign of good figuring and reasoning
abilities, how or why the skill or practice will be needed in
college will not be clear to learners and their teachers. Learners
taking advanced courses covering biology, chemistry, physics and
money computations will see high school mathematics in use - better
late than never. Students in trade oriented courses may also see
some applications. But almost everyone else will not. While
mathematical preparation for college studies may be wanted and
respected for its long term value, for most, the short-term value
of doing well on test and finals is not inclusive and not
appealing.
Present day students and teachers will have to do their best with
local course design and materials. In site material, online books
and further site material will help. See the advice in the other
column. Here the description of alternative ends, values and paths
for mathematics and logic-language skill and practice development
may provide some immediate context and motivation.
Doing Your Best with local course design and materials
If you find studying hard or not, then you should try to master
logic - see site logic chapters - and you should be aware of and
try to avoid the domino effects of errors in multi-methods at home
in cooking, at school in mathematics etc, and at work in following
instructions. Logic mastery with the greater precision in reading
and writing it provides is one way to ease or avoid troubles in
following instruction at school today and in work tomorrow.
For reading and writing, all the letters of your alphabet have to
be met and remembered. Anything less will lead to difficulties in
spelling and understanding words. The child who complains there are
too many letters too remember will soon be corrected. For
mathematics mastery, skill in counting, measuring and calculating
with decimals, fractions, percentages and signs is a must. But too
many students are not shown fully how to do so in the last years of
primary school and the first years of secondary school. But such
skill is a must for all further mathematics in secondary school and
college. Site arithmetic steps will help. Use them to check for
gaps in your command of fractions, primes and arithmetic with units
of measure.
In my school days, the introduction and use of algebra included
steps too large for most to follow. Site algebra chapters and steps
provide starter and advanced lessons to give a remedy. In it,
smaller and extra steps provide a more gradual, less steep, paths
to try. See what works. Looking for a remedy began in the author's
high school days. The first break-through came with the fall 1983
invention of three lessons: Three skills for algebra, why slopes
and two logic puzzles. Talking about the three skills adds words to
the exposition of mathematics. The lesson on why slopes appear in
secondary mathematics lesson was an algebraically light calculus
preview. Calculus is the subject of slope related computations
forwards and backward for lines or linear functions y = ax+b and
more generally for curves or nonlinear functions y = f(x).
Preparation for calculus is the main reason for earlier mathematics
in college or secondary school. Calculus and many other subjects
require precision reading and writing. That was one reason for the
two logic puzzles. While lower level mathematics may be taught by
rote in a practice first, theory second or not all at all manner,
that is easier, students heading for calculus-based college
programs should meet and be able to master some of the theory - too
much would be overwhelming. Site logic chapters provide a worky
introduction, almost math-free, to the kind of logic used in
advanced mathematics, logic whose mastery in all or part has take
home value in sharpening skills and in easing or avoiding learning
difficulties at home, work and school.
In the bottom margin of each page there are links to key lessons.
Explore them. If one or a few are not to your liking, try the
others. Site content indicated above should fill a gap or two in
your education. That should be reason to explore more site
material. Good luck.
Alternative Paths for Mathematics Education
The question of how and why learn and teach mathematical and
logical skills and practices has bothered me for decades. As a
student, I wanted to understand why I should study mathematics -
its topics and in general. My fathers story that he was handicapped
in college level chemistry and physics since he had not studied
mathematics extensively provided general motivation. That general
aim did not compensate for the lack of immediate motivation for
some topics - the teacher reply that they were required, or the
expectation that I bring my own motivation, were not helpful. And
when I taught mathematics, I wanted to be able to tell my students
why they should learn this or that - to provide a context. I
remember teaching a final year high school courses to students,
they needed for graduation, but I did not see how its topics would
help them with their academic and career aims. Teaching calculus or
upper level high school mathematics is more rewarding for the
instructor and students when there is a chance they are aiming for
college programs in commerce, science, technology, engineering or
mathematical subjects. At least then, there is a context for
teaching and learning. But four-fifths of secondary students are
mathematics education orphans. Apart from meeting graduation
requirements, mathematics courses cover skills, practices and
concepts with little or no take-home value for them, short- or
long-term, while other skills and practices that could serve their
daily or adult lives go untaught. That is annoying. Ideas follow to
suggest how mathematics education could be more helpful in
developing skills and practices with take-home value for daily and
adult life, and also for college programs which require advanced
mathematics. The ideas may not help your secondary school
education, but they might help the secondary education of your
children and if you go to college, the mathematics education there
of yourself and others - Colleges have more freedom than secondary
schools to change course design and materials.
Imagine mathematic and logic education may come in overlapping
layers or levels. The first level, usually for ages 3 to 14, would
cover all the counting, measuring and figuring skills and practices
that employ numbers and/or maps-plans-diagrams drawn to scale with
actual or potential take-home value for daily- and adult-life. In
that time, date and calendar matters; money matters; logic and
decision making matters; and elementary knowledge of chance,
probability and odds may be useful. The first level would describe
some calculations with words - adding by adding subtotals would be
one example; and other calculations with formulas. Calculation
methods would be given for routine or common situations in daily or
adult life, all in a scout-like, be prepared for what is likely,
context. Instructors would show how to do and record work in steps
that can be seen and checked as done or later. Avoiding the domino
effect of mistakes in counting, measuring, figuring and in general
would be emphasized as an end, value and tool for skill development
and mastery. This first level might cover methods that are easily
understood and verified in class until just before doing so becomes
too repetitive. And if mathematics education was to stop or not be
appreciated beyond this first level, at least the first level would
provide skills and concepts that would or could be useful sooner or
later.
The second level of mathematics would introduce the use of algebra
for solving for unknowns or getting formulas for them, and the use
of algebra to say when different computations give the same result.
In the past, the algebraic shorthand role of letters and symbols
has not been clearly introduced. I saw that in my school days. In
my school days, I was able to rationalize the shorthand roles of
letters in finding formulas for solutions, in solving equations and
in describing when different calculations would give the same
result. But many other students, more gifted than I in their
reading and writing abilities could not. And one of my physic
teachers did not understand algebra. The foregoing combination of
students and teachers in my physic course made my instruction
slower than need-be. Site algebra steps, smaller and extra, provide
a more gradual path, less steep, for developing algebra skills and
practices. A natural context for commutative, associated and
distributive laws in arithmetic is provided by the new concept of
equivalent computation rules. The latter has little or no take-home
value for daily or adult life, but it makes preparation for
mathematics-based college programs more accessible. Learning how to
use formulas forwards and backward allows several related formulas
with take-home value given earlier to be replaced by one. The
foregoing may lead to a fuller and stronger mastery of money
related calculations, there be take-home value in that for daily or
adult life. Stopping here might leave a favourable impression of
mathematics,
Three further topics, study in any order, in mathematics have some
actual or potential take-home or intellectual value for daily and
adult life.
The Euclidean model for reason is introduced by site logic
chapters in a math-free way. Those chapters may improve reading
and writing skills, and help people see the difference between
one- and two-way implications. Not seeing this easy difference
is a source of confusion in following and giving instructions,
in digesting information in and outside of mathematics; and in
agreements. The latter and their small-print need to be read
and fully understood to avoid surprises. Remember in making an
agreement, all parties need an acceptable exit clause - an
option to use if things do not turn out as wanted. Good luck.
The site simplified coverage of Euclidean Geometry, error-free
we hope, shows how implication rules can be used in sequence to
arrive at conclusions - often further implication rules.
Euclidean geometry provides a neutral territory for
illustrating the deductive use of implication rules, alone or
in sequence. All is simpler than you think.
Not all is certain. Earlier mathematics and life may provide
examples of chance, likelihood, probability and odds.
Probability theory in mathematics provides ways to estimate
what is likely to happend when not is certain. That help people
lessen or avoid risk. Mastery of equivalent computation rules,
how to use formulas forwards and backwards, and simple
operation with sets, should allow students to understand the
first elements of this theory and its notation. Learning about
probability, and above expected value of possible outcomes in
daily life or in playing games will help decision making in
matters of chance, when not all is certain, including the
methods for arriving at conclusions. The elementary study of
probability theory provides one model, more algebraic than
geometric, of mathematical reason.
The natural logarithm appears in the backward use of the
compound interest or growth formula. Secondary mathematics is
not the place to describe the origins of this logarithm
function and the anti-log or inverse exponential function. But
a short, full theory of logarithms, roots and powers may be
develop from the algebraic description of properties of the
natural logarithm and its inverse. The elementary study of the
latter theory provides another model of mathematical reason.
Here it might include the forward and backward use of
exponential growth and decay models.
Site geometry steps cover and employ rectangular and polar
coordinates in the plane. The development of complex numbers
from the properties of these coordinates and their interaction
might provide another easy topic. In the prepartion of students
for geometry-based careers and for calculus-based college
programs, this topic would set the stage for the mastery of
trigonometry. Many geometric problems can be solved by drawing
to scale and then measuring. Some of these problems can be
solved by sketching and using trigonometric calculations in
place of drawing to scale precisely and measuring missing
angles or lengths. Trigonometry in the first instances allows
calculation guided by a sketch to be used in place of drawing
to scale. There-in lies a context for trigonometry.
Mathematics education may continue with further topics - see site
steps. The further ones also required by mathematics-based college
programs. But those further topics would not have any actual or
potential take-home value for daily or adult life like the three or
four above. For students not heading for mathematics and statistic
based college studies, the three or four topics described above
might leave a favourable impression. Leaving a favourable
impression, one that includes multiple skills and practices with
actual or potential take-home value, and perhaps a thirst to learn
would be better than covering too much, and leaving a bad
impression or an alienated view of mathematics in the process. To
avoid the latter, less done to perfection would be better.
Mathematics education should continue beyond the first level only
while the underlying topics are easy for students to master, and
have some relevance for future studies or for life in the street.
2011-08 Remark 1: Statistics appears too much in the mathematics
programs in UK, USA and New Zealand. The use of averages has some
practical value. Pie charts may be use to illustrate proper fractions and
percentages. Beyond that, the study of statistic in high school appears
to have little or not take-home value for the daily and adult life. The
year after year placement of statistics in late primary and in secondary
school mathematics is mostly a distraction from the development of skills
and practices with actual or take-home value, and from the preparation of
students for calculus-based college studies. Less would be best.
2011-08 Remark 2: Teachers, if course design was to treat each year
of instruction as if it was the last chance to provide students skills
and practices with take-home value, the direction and priorities of
course design would be different. They would, if I am not mistaken, after
coverage of the easier skills and practices with actual or potential
take-home value for daily- or adult life, identify and put those
skills and practices in or helpful to calculus preparation with the
greatest and than faintest take-home value first or as early as
possible. That would be subject to the inclusiong of further skills and
practices that make skill development less steep and more gradual.
2011-08 Remark 3: Teachers,
In computer programming, subprograms or code that has inputs but not
outputs is redundant - can be removed or archived. In mathematics, skills and practices whose practical or
intellectual value to students and society is unclear or confused - due
to years and years of committee based course design - can
likewise be eliminated or archived. Critical path analysis - careful
thought - is needed in course design to choose goals and objectives,
clear and concrete, and identify what is needed. Read about which way
to go below
|
Maple16
What is Maple?
The result of over 30 years of cutting-edge research and development, Maple helps you analyze, explore, visualize, and solve mathematical problems. With over 5000 functions, Maple offers the breadth, depth, and performance to handle every type of mathematics. Maple's intuitive interface supports multiple styles of interaction, from Clickable Math™ tools to a sophisticated programming language. Using the smart document environment provided by Maple, you can automatically capture all of your technical knowledge in an electronic form that combines calculations, explanatory text and math, graphics, images, sound, and diagrams.
Learn about some of Maple's many features by exploring the information below.
Smart
Popups and Drag-to-Solve™ are part of Maple's collection of Clickable
Math tools, which also include palettes, interactive assistants,
context-sensitive menus, tutors, and more. These tools make it easy to
learn, teach, and do mathematics with Maple. Learn More...
In
Maple, you can embed interactive components in your document. Using
elements such as sliders, buttons, dials, and gauges, your document
becomes both a technical report and an easy-to-use application. Learn More...
The
variable manager provides easy access to all variables in your Maple
session, allowing you to better manage your documents, quickly assess
the state of your computations, and inspect variable values without the
need to navigate through the document. Learn More...
Maple
puts over 30 different palettes at your disposal to help with numerous
tasks, including building and editing mathematical expressions, keeping
track of variables, and sharing documents with other users. Learn More...
To
help you find a symbol that you need from a collection of over 1000
symbols, Maple allows you to use your mouse to sketch the symbol in
the Symbol Recognition palette as you would draw it on paper. Learn More...
Live
Data Plots help with insight, understanding, and publication of your
data, all at the click of a button. These plots make it even easier for
you to present your data in a form that is visually appealing and
conveys meaning. Learn More...
Maple
is the only technical computing system that allows you to take
advantage of multithreading in your own programs. The Maple programming
language offers direct access to launching and controlling threads. Learn More...
Distributed systems offer fantastic gains when it comes to solving
large-scale problems. By sharing the computation load, you can solve
problems too large for a single computer to handle, or solve problems in
a fraction of the time it would take with a single computer.. Learn More...
With
the NAG connectivity feature, you can combine the pre-eminent modelling,
exploration and application development abilities of Maple with the
power and breadth of NAG numeric routines. Learn More...
|
College Algebra
Our most popular course! College Algebra is required for just about any degree, but especially if you're majoring in Math, Science, Computer Science and Businesses.
This online course will give you a working knowledge of college-level algebra and its applications! Learn how to solve linear and quadratic equations and word problems at your own pace. Master the mysteries of polynomial, rational and radical equations and applications in the comfort of your own home.
Choose your online College Algebra
|
Tutors
Maple provides a large collection of built-in, point-and-click learning tools for most key topics in calculus, algebra, and more. Over 50 interactive tutors offer focused learning environments in which students can explore and reinforce fundamental concepts.
Many tutors allow students to work step-by-step through problems. Students can perform steps themselves, ask for hints, or ask Maple to perform the next step. For example, one tutor gives students practice applying the different rules of integration. Another provides help with performing Gaussian elimination on matrices, allowing students to focus on the steps without getting lost in the arithmetic.
Tutors frequently use 2-D and 3-D plots and animations, reinforcing concepts that are sometimes difficult to visualize. Examples include volumes and surfaces of revolution, eigenvector plots, Newton's method, gradients, space curves, conic sections, and DE plots.
Tutors are available for single variable, multivariable, and vector calculus, precalculus, linear algebra, complex variables, numerical analysis, and differential equations.
Watch a Demonstration
The following demonstration illustrates the use of tutors:
|
Algebra 2:Algebra 2 requires students to make an in-depth study of the number systems to solve equations and inequalities in both real and complex numbers. Algebra skills are applied to linear, quadratic, and logarithmic functions. Emphasis will be place on algebraic application and graphing techniques with and without a calculator.
IB Math Studies: This course consists of eight topics:1) Introduction of graphic displaycalculator, 2) Numbers and algebra, 3) Sets, logic, and probability, 4) Functions, 5) Geometry and trigonometry, 6) Statistics, 7) Introductory differential calculus, and 8) Financial mathematics. As part of the International Baccalaureate program, this course, within the curriculum of mathematics, follows the mission of developing intellectual, personal, emotional, and social skills to live, learn, and work in a rapidly globalizing world.
|
Holt Mathematics Course 3 Answer Key
You are here because you browse for Holt Mathematics Course 3 Answer Key. We Try to providing the best Content For pdf, ebooks, Books, Journal or Papers in Chemistry, Physics, mathematics, Programming, Health and more category that you can browse for Free . Below is the result for Holt Mathematics Course 3 Answer Key query . Click On the title to download or to read online pdf & Book Manuals
Holt Mathmatics Course 3, Pre-Algebra, and Algebra
Min8.pdf. Holt Mathmatics Course 3, Pre-Algebra, and Algebra. Apr 14, 2007 Explanation of Correlation. The following document is a correlation of Holt Mathematics Course 3, Pre-Algebra, and Algebra 1 to the April 14,ΗΤΤΡ://GΟ.ΗRW.CΟΜ/RΕSΟURCΕS/GΟ_ΜΤ/ΜΝ/ΜΙΝ8.ΡDF
Problem Set #3 Answer Key 1. Define the relevance of the following
Full ps3 key.pdf. Problem Set #3 Answer Key 1. Define the relevance of the following. Problem Set #3 Answer Key. 1. Answer the following multiple-choice questions . ease of genetic approaches, and two, that you must switch over to a more You're a conceptual artist, inspired by biology, who wants to make a piece titledΗΤΤΡ://ΟCW.ΜΙΤ.ΕDU/CΟURSΕS/ΒRΑΙΝ-ΑΝD-CΟGΝΙΤΙVΕ-SCΙΕΝCΕS/9-09J-CΕLLULΑR-ΝΕURΟΒΙΟLΟGΥ-SΡRΙΝG-2005/ΑSSΙGΝΜΕΝΤS/FULL_ΡS3_ΚΕΥ.ΡDF
|
Click on the Google Preview image above to read some pages of this book!
The Bedside Book of Algebra is a collection of problems that will offer something of interest to everyone, in this easy to follow format, it guides the reader through this fascinating subject.
The principles of algebra were founded by al-Khwarizmi many centuries ago, in a time when mankind had no calculators, computers, or electonic gadgets. There were no telephones and the only means of communication was by messenger on horseback and boat. Yet the usefulness of algebra in almost every walk of life involving numbers has ensured not only its survival but also its continued development right up to the present day.
The Bedside Book of Algebra is a collection of problems, some with a very practical application, others designed as purely theoretical puzzles, that will offer something of interest to everyone. Each section is written in an easy-to-follow format and guides the reader progressively through this fascinating subject. Understand algebra, and all other branches of mathematics and arithmetic will suddenly open up in front of you. Also included are feature spreads on major figures from the history of algebra, giving biographical details and explaining their theories and findings. The Bedside Book of Algebra is ideal for quiet bedtime reading, or something to wake up the grey matter while enjoying your early morning beverage.
About the Author
Michael Willers was born in 1967 in the city of Victoria, British Columbia, Canada. Throughout his school years, the one subject that provided light was mathematics (and, later, science). He graduated high school in 1985 and entered an engineering program at the University of Victoria, paying his way by working as a shop assistant and consulting as a computer programmer for various companies. By the time he graduated with a degree in Electrical Engineering in 1991, he was sick of it, and promptly entered the university's teacher-training program - partly to overcome a fear of public speaking. Michael finished his training in 1992 and has been teaching high-school mathematics ever since. From 2000 to 2005 he worked on a Masters degree, focusing on mathematical connections to other subjects, such as art and philosophy. It was during this time that he started to look in-depth at the likes of Fibonacci, Pascal and Descartes, among others.
|
Unit specification
Aims
This course introduces students to (i) analytical and numerical methods for solving partial differential equations (PDEs), and (ii) concepts and methods of vector calculus. It builds on the first year core applied mathematics courses to develop more advanced ideas in differential and integral calculus.
Brief description
The first half of the course consists of an introduction to the important topics of Fourier series, partial differential equations and analytical methods for solving first and second order PDEs (including Laplace's equation and the heat and wave equations). The second half of the course includes an introduction to the specialist topic of numerical analysis (finite difference methods for solving differential equations) together with an introduction to fundamental results underlying vector calculus (the divergence theorem and Green's and Stokes' theorems). The tools introduced in the course will be essential for understanding subsequent applied mathematics and numerical analysis options in the remaining semesters of the BSc and MMath degree programmes.
solve analytically, using the method of separation of variables, the heat and wave equations (in one space variable) and Laplace's equation (in two space variables) on rectangular and circular domains;
Vector Calculus. Line, surface and volume integrals. Scalar and vector fields: differential and integral calculus. Field lines and field surfaces, the general field surface of a vector field. Grad, div and curl in Cartesian and curvilinear coordinates. Identities. Divergence, Green's and Stokes' theorems. Indicial notation, summation convention. Zero, first, second and higher order tensors. [11]
|
My Advice to a New Math 175 Student:
My advice to a new Math 175 student would be to remain calm.
Having
not
taken even an algebra course for 3 years, I was a little bit worried
about
this class in the beginning. It really helped me though to do the
Review
Chapter in the front of the textbook. So that is where I would start.
I was
always the type of student who hated doing homework and usually waited
until
the day before the test before I began to study. However, I learned in
this
class that my old habits would not get me by this time. So even though
nobody likes to do homework, I advise you to work on ALL the assigned
problems every night before class or at least the night after the
lecture.
It is really important not to fall too far behind in this class. I
would
also not be afraid to ask questions from the homework in class and when
you
need extra help, I suggest going to talk to your professor. Dr. Hoar
helped
me on numerous occasions. I found it easier to understand the material
when
I went in to his office and we worked on them one on one.
So just remember to keep up with the homework, go over as many
different
problems as you can until you feel you are familiar with them, and also
go
and get help from either the professor or the tutor labs. The tutor
labs are
good, although I found it more beneficial to get help from Dr. Hoar. I
know
that these suggestions may sound generic but they are really necessary
if you
want to succeed in this class. Trust me! Good luck people!
|
Textbooks for PDE between "Strauss and Folland"
|
Read carefully the text, consulting the lecture notes,
and write down questions about
everything
that is not clear.
The notation will be a substantial difficulty.
Make a reference list of all symbols that are unfamiliar.
Find out the meaning (you may ask me) and write down the explanation.
Record the subsection (and the page number) which was the source of difficulty.
By the time you finish the reading, you will have the answers
to some of the questions.
Read again the same portion of text, this time with a pen and paper.
Write down the proofs and the solutions of the examples in the text.
This will help you get inside the reasoning thread and become more familiar with the notation.
Hopefully during the second reading/writing your understanding of the material
will improve,
and maybe the number of the questions
will be reduced.
Take the list of questions and read them, consulting the
textbook.
Think of possible answers to each question.
Send me the questions and what you think might be a possible answer.
|
Course Description: Real number system, order of operations. Algebraic problem solving, solving linear equations. Cartesian coordinate system, graphs of equations. Exponents and radicals.Factoring polynomials, solving equations by factoring.Credits not applicable toward graduation.Four Credits.
Note: This course serves as a pre-requisite for MATH 110 (College Algebra), MATH 130 (Introductory Statistics), or MATH 155 (Mathematics, A Way of Thinking). You must earn at least a "C" grade to qualify for the next course in your sequence.
(b) Students will work through pre-algebra ALEKS modules indicated as necessary.
2. Students will improve their mastery of algebraic skills.
(a) Students will take ALEKS assessment of algebra knowledge and skills.
(b) Students will work through the ALEKS modules indicated as necessary.
(c) Students will take indicated exams to demonstrate their learning.
3. Students will develop their ability to apply algebraic thinking and procedures to problem solving.
(a) Students will work through the ALEKS modules that focus on problem solving.
Course Procedures and Policies:
MATH 001: Math 001, "Introductory Algebra", is a not-for-graduation-credit course intended to prepare students for the various courses for which 001 is a pre-requisite, namely MATH 110 (College Algebra), MATH 130 (Introductory Statistics), and MATH 155 (Mathematics, A Way of Thinking). The material is essentially the first year of algebra, which would typically be taken initially in high school, which explains why this course is numbered 001, and why the 4 credits you will earn here do not count toward graduation, even though they do count toward full-time status.
Your placement score indicated that you have not mastered the material in this course, whatever the reason. Your goal here is to finally learn this material and master the necessary skills so that you can be successful in the courses you eventually need to take as part of your college program
ALEKS: ALEKS (Assessment and LEarning in Knowledge Spaces) is a web-based program designed to carefully assess what students know and what they are ready to learn, and then to methodically tutor them in the given material, in this case Introductory Algebra.
Probably the best thing about ALEKS is that it allows each student to take a course specifically designed for their needs – each student in the class will be working at their own pace and working on material they are ready to learn. The implication of this is that I will not be "lecturing" on textbook sections the way you might be used to seeing. My role as instructor here is to monitor your learning and to engage in individual tutoring as the need arises.
Another advantage to using ALEKS is that since it is web-based you can work on your course at your convenience. ALEKS will remember where you left off and will always make sure that you have shown readiness before presenting new material.
By the way, even though you will be expected to do a considerable amount of ALEKS work on your own time, it is very important to understand that it is important to DO YOUR OWN WORK! If you get someone else to do the work you will only be frustrated when ALEKS thinks you know more than you do and starts asking questions you are not ready for. Also the exams must be taken on your own so having someone work through the online material for you will not help your performance on those exams, and hence on your grade for the course.
Textbook: The textbook we will be using is published by McGraw-Hill, who also handles ALEKS for institutions of higher education. Our text has been precisely integrated with ALEKS, so that you can use your book for explanations, worked examples and practice problems as we move our way through the course material. In fact I will be using a feature called "textbook integrations" in which the material will be presented in the same order as the book covers it and quizzes will be given as you finish chapters in the text.
Number of
Absences
Points
0
+25
1
+20
2
+15
3
+10
4
+5
5
0
6 or more
-2 points each
Attendance: A major factor in learning mathematics is a regular and focused schedule of practice. Can you imagine learning to play the piano by only practicing a few minutes a week! You need to practice virtually every day, and for considerable time each day. It takes the same sort of discipline to solidly learn algebra.
My attendance policy is given in the table at the right. Because it is so important that you put in the time, I have a system that rewards regular attendance. I think that a student who has missed as many as 6 classes should seriously consider dropping the course, but as far as my grading system is concerned, I will subtract 2 points for each absence beyond 5, so a person who attends every class will earn 25 points, while a person who misses, say, 8 classes will LOSE 6 points.
In general I will not distinguish between "excused" and "unexcused" absences, although I do consider absences due to participation in a school event, such as an athletic trip or a theatrical production, to NOT be "absences". In this case, however, it is still important that you put in the extra time to catch up.
ALEKS hours
this week
Points
6 or more
+5
5 or more
+4
4 or more
+3
3 or more
+2
less than 3
-1
ALEKS Time: ALEKS keeps track of how much time you have put in
as well as how much progress you have made. I will be using your ALEKS time as part of the grading scheme, as summarized in the table at the right. Each week there will be a grade assigned based on the time you have spent working on ALEKS over the previous week.
The times INCLUDE the 3 hours plus spent in class, so that 6 hours for 5 points means you would need to work at least 3 hours outside of class to earn those points. In general college students are expected to work 2 hours outside of class for each hour in class. I have made this number a little smaller because I am trying to build in some time for studying the text book.
Some people will need more time to learn the material that others – life is not fair and some people learn things more quickly than others. I do expect each of you, however, to put in roughly 12 total hours per week working on learning the material. This does mean that some of you who are farther along than others might end up finishing the course at some point during the semester! ALEKS will tell you how far along you are and some of you will have a starting point farther along than others.
By the way, there are several "short weeks" this fall. Labor Day (9/4) week and Mid-semester break (10/20) week have only three class meetings rather than four, so these two weeks the 6-5-4-3 (hours) will become 5-4-3-2. Thanksgiving (11/23) week is very short, only 1 class meeting, on Monday; this week the numbers are 2 hrs = 5 points, 1 hours = 3 points, 1 hour = 1 point.
Exams and Quizzes: ALEKS has the ability to construct quizzes at points indicated by the instructor. Since I have integrated ALEKS with our text book I will ask it to give you a quiz on the material in each chapter. These quizzes will be 10 questions worth 2 points each, for a total of 20 points per quiz. There will be 11 such quizzes, a Review chapter and then 10 chapters covering the course material. ALEKS will keep you updated on the next deadline – I have set up a schedule which will allow you to work through the course material by the end of the semester. These quizzes may be taken at any point prior to the deadline, and may be taken twice – the higher of the two grades will count. I will also go through and look at your work so that I may give some partial credit if it is appropriate to do so.
You will also take a paper and pencil exam of my design at midterm and during finals week. I imagine that some of you will not be on schedule and this will no doubt affect your performance on these two exams, but part of success in a course is learning the material within a designated amount of time.
In fact, the final exam will be worth 200 points and will be a combination of two things – 100 points will be based on a paper-and-pencil test you will take during finals week and 100 points will be based on the percentage of ALEKS topics you show mastery of in a final assessment to be taken the last couple days of the semester.
Grading System: At present, and I want to reserve the right to make adjustments to this system as the semester wears on, I see your grade being determined by these four factors:
(5) Final Exam: 200 points possible (100 for the percent of ALEKS topics, 100 on exam)
This makes for a total of 620 points. Grades will be assigned according to the scale: A = 90% or higher, B = 80% or higher, C = 70% or higher, D = 60% or higher. You need at least a "C" grade to be allowed to advance to the next course in your sequence.
Help outside the classroom:
If you find yourself having trouble with the material PLEASE get some help! There is regular tutoring provided in the learning center, either on a scheduled or a drop-in basis, and I want to encourage you in the strongest terms to come see me if you have questions. I have regular office hours, listed at the top of the syllabus, and it is my role here to help you master the material. I'd much rather work with you and try to get you over the hurdles than have you fail in the course. So take an active role here in monitoring your learning and do something about it if you are having trouble!
Schedule: Because ALEKS allows students to work at their individual pace you will be at a variety of places in the material throughout the semester. Still, in order to pass the course and move into the subsequent course you will need to finish the material within the semester's time constraints.
It is possible that some of you will actually complete the ALEKS course before the calendar indicates the semester is over, and that's fine. I will still have you take the midterm exam on October 13 and the final exam on December 14 with the rest of the class. If you do finish early your ALEKS time and attendance points will be based on the amount of time you were working on the material.
It is also possible that some of you may reach December without completing the material. ALEKS offers a guarantee that if you put in a reasonable amount of time during the semester and do not pass the course your license to use ALEKS can be extended so that you can continue to work on finishing the course during the following semester – in this case you will be given a grade of "I" (Incomplete) so that you can work on completing the course during the next semester. Of course, this is far from ideal since it means you could not yet enroll in the course you need to take for your major, so it should be your goal to see that that does not occur.
Americans with Disability Act Statement
ALEKS
Your textbook should come with a username and password so that you can log onto ALEKS (Assessment and LEarning in Knowledge Spaces). Then to be enrolled in my specific course you need the course code, which is:
GQGG6-3KTQN
The first day of class you will each log in and we will take a look at the basics of using ALEKS. I will ask you to work your way through the tutorial so that you become familiar with how to enter mathematical expressions. Then on the second day of class I will have you take the initial ALEKS assessment to get a baseline rating of your skills and readiness for the material in this course.
|
CRASH COURSES
MATH CRASH COURSES a.a. 2012 - 2013
Math crash courses are designed to reinforce a few mathematical concepts at a pre-university level. This helps students to start university with greater confidence and comprehension. The main topics covered include: equations, inequality, coordinate geometry, trigonometry, power functions, exponentials and logarithms.
For the class timetable check the Timetable, Calendars and Room section starting from 2 August 2012.
In this section you can view and download the programs of the math crash courses.
|
What is Mu Alpha Theta? Mu Alpha Theta is the national high school and two-year college Mathematics Honor Society. They are dedicated to making kids enjoy math even though it can get really intense.
Mu Alpha Theta. What's the point? How does it help? Well, they're many different reasons on why you should join. If you're a person who really cares about college,Mu Alpha Theta gives you the chance to get scholarships!! Also when you apply for college or job you'll get a better chance of getting in the college/job because Mu Alpha Theta will put it on your application form/resemè!!! Another reason why people join is for the money and grants. Yes, I said it money, the thing that people need and fight for. Mu Alpha Theta gives you the chance to win money and other magnificent prizes. The best part is Mu Alpha Theta isn't your usual boring math class, they do all sorts of fun activities for you to enjoy and learn math at the same time. So come on and try to get in Mu Alpha Theta, there is lots of competition, other people want to join but now you have a reason to get in.
The next important thing is "how to get in". You need a really good backround if you want to change. So if you were really bad but got a change in heart that might not work. You must have at least 3 semesters of algerbra,2 semesters of geometry or one semester of advanced mathematics (these are high school standers). Apart from the paperwork the last thing you need is a GPA of at least 3.0. Getting in is the hard part but are you ready? You know now how to get in.
After all the nonsense is over it's not exactly smooth sailing. You forgot about the competitions and practices. The practices are not as bad because luckily you get to do fun activities but the hard part is when you have to do the intensive training. You might also have to tutor somebody who isn't as high as you. Practices can be fun and intense. Your real challenge is the competitions. It consists of 3 rounds, round one is two topic tests,it's multiple choice, and you get the choice Circle/Pi/Polygons and Equations/Inequalities which is on the second test. Round two is just like the first round. You can choose Number Theory/Applications. The last round is an individual test of general mathematics (on level topics). Most important, all 15 open answers are to be done in thirty minutes with no caculator. This is the competitions,can you handle it??
I hope this can help you understand what Mu Alpha Theta is. If you need more information go to Get ready for Mu Alpha Theta,it's coming!!
|
Linear Algebra II
Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)Mathematics and Statistics2011-11-11T11:22:52Course Related MaterialsFoundations of Development Policy, Spring 2009
" This course explores the foundations of policy making in developing countries. The goal is to spell out various policy options and to quantify the trade-offs between them. We will study the different facets of human development: education, health, gender, the family, land relations, risk, informal and formal norms and institutions. This is an empirical class. For each topic, we will study several concrete examples chosen from around the world. While studying each of these topics, we will ask: What determines the decisions of poor households in developing countries? What constraints are they subject to? Is there a scope for policy (by government, international organizations, or non-governmental organizations (NGOs))? What policies have been tried out? Have they been successful?"Duflo, EstherBusinessSocial Sciences2010-10-07T04:39:16Course Related MaterialsBusiness organisations and their environments: culture
We know that culture guides the way people behave in society as a whole. But culture also plays a key role in organisations, which have their own unique set of values, beliefs and ways of doing business. This unit explores the concepts of national and orgBusinessSocial Sciences2009-08-13T00:25:40Course Related MaterialsTopics in Philosophy of Science: Social Science, Fall 2006
This course offers an advanced survey of current debates about the ontology, methodology, and aims of the social sciences.Haslanger, SallyHumanitiesSocial Sciences2008-01-27T10:00:48Course Related MaterialsNorms
This module will define a norm and give examples and properties of it.Justin RombergMichael HaagMathematics and StatisticsScience and Technology2007-10-30T11:42:00Course Related MaterialsNormas
Este modulo definirá una norma y da unos ejemplos y sus propiedades.Justin RombergMichael HaagScience and Technology2007-08-20T05:12:00Course Related MaterialsManagerial Psychology Laboratory, Fall 2004
Core subject for students majoring in management science. Surveys individual and social psychology and organization theory interpreted in the context of the managerial environment. Laboratory involves projects of an applied nature in behavioral science. Emphasizes use of behavioral science research methods to test hypotheses concerning organizational behavior. Instruction and practice in communication include report writing, team decision-making, and oral and visual presentation. Twelve units may be applied to the General Institute Laboratory Requirement.Ariely, DanBusiness2006-03-20T23:57:00Course Related MaterialsManagerial Psychology Laboratory, Spring 2003
Surve laboratories and its effect on communication patterns in the organization. 15.301 is a core subject for students majoring in management science. A laboratory is a required element of the course for these students. It involves projects of an applied nature in behavioral science. Emphasizes use of behavioral science research methods to test hypotheses concerning organizational behavior. Instruction and practice in communication include report writing, team decision-making, and oral and visual presentation.Allen, Thomas JohnBusiness2006-03-20T23:57:00Course Related MaterialsForms of Political Participation: Old and New, Spring 2005
How, stability and change in political regimes, the capacity of states to carry out their objectives, and international politics.Tsai, LilySocial Sciences2006-03-20T23:56:00Course Related MaterialsFoundations of Development Policy, Spring 2004
ExplDuflo, EstherSocial Sciences2006-03-20T23:47:00Course Related Materials
|
Architect's Handbook of Formulas, Tables, and Mathematical Calculations compliles a vast range of practical, concise formulas, tables, and calculation methods useful to improve the design process. It is a problem-solving and decision-making tool for the practicing architect and interior designer. The material included in this book gives you the answer to the many types of problems you face every day- those dealing with overall site and space planning, sizes of building components, material selection, finishes, construction assemblies, and building systems. In addition, you will find useful ""rules of thumb"" and basic reference data.
The organization of this Handbook is based on how architects actually work through a project and make decisions – from establishing early programming needs, to making preliminary design and building system choices, to evaluating specific material selctions. The tables and calculation methods selected are practical, proven reference information helpful for all phases of a job. To make the tables and formulas even more useful, step-by-step procedures for using them and easy-to-follow examples are included where appropriate.
Recommendations:
Save
7.55%
Save
2.44%
Save
25.73
|
The Academic Foundations Mathematics area is responsible for helping students master basic math skills so that they can take their academic courses with a solid foundation.
Offered in this area are the following three courses:
DEV084 - Basic Mathematics I Provides instruction in basic arithmetic for whole numbers, fractions and decimals with the goal of developing computational skills, number-sense, and problem-solving skills. Prepares students for further study in mathematics by employing effective study strategies and a variety of teaching/learning experiences.
DEV085 - Basic Mathematics II Review of basic arithmetic skills in whole numbers, decimals, and fractions with emphasis on problem solving situations. Instruction into the meaning and use of percentages, ratios, proportions, and measurements. Brief introduction into signed numbers.
DEV108 - Introduction to Algebra Introduction to beginning algebra concepts including operations with rational numbers, identifying and combining like terms, solving one-variable linear equations/inequalities, and laws of exponents. Additional topics include the recognition of simple algebraic patterns and the study and use of some basic geometric formulas.
Faculty use a variety of instructional methods including individualized instruction, self-paced approach, lecture format, and the utilization of technology such as calculators, computers, and instructional DVDs. All of these are well integrated throughout the courses to maximize students' learning ability.
|
Mathematics
More than mere computation, the content and methods of mathematics derive from and reside in the core of human thought itself. The discipline's lifeblood flows from problems present in spatial and numerical processes, in difficulties that arise from logic, science, applications, and social phenomena, and from problems that arise internally from the many sub-disciplines of pure and applied mathematics.
At the foundations of the discipline, mathematical researchers puzzle over problems of consistency, the nature of truth, problems of logic, and determining means for ensuring correct derivation of mathematical theorems. The skills that accrue to students engaged in these kinds of analyses, and the intellectual capacity to transfer mathematical methods to myriad applications, afford expanding life and career opportunities to well-schooled practitioners of mathematics.
The mission of the department of mathematics at Wheaton College is to prepare students to be transforming agents of Christ in a needy world beset by difficult problem. Most, if not all, of these problems require careful analysis and the application of insightful problem solving skills. Located on the main floor of Wheaton's science building, students and faculty in the department of mathematics and computer science interact with physical and biological scientists, geologists and social scientists as well as other disciplines at the intellectual cross-roads of Wheaton's curriculum to develop cross-disciplinary approaches to problem solving.
Department graduates enter graduate schools in mathematics, computer science, or related disciplines. Others undertake careers or advanced training in actuarial science, teaching, economics, business, and statistics. No matter their eventual fields of service, while at Wheaton mathematics and computer science students study and work individually and in small groups with department faculty whose professional interests include differential geometry, dynamical systems, fractal geometry and chaos theory, math modeling, computing, applied mathematics, probability and statistics, knot theory, math analysis, and modern algebra. Multiple opportunities exist for funded summer research, faculty-student mentored publishing efforts, and working in the department as a recitation or teaching assistant. In the end, department majors hone their skills for a lifetime of service for "Christ and His kingdom."
|
Welcome to my page.
Open Algebra Textbooks
Writing these textbooks was quite the effort! It is nice to be done... Elementary and Intermediate Algebra both have been published by Flat World Knowledge
Elementary Algebra is an open textbook designed to be used in the first part of a two part algebra course. It is written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.
Intermediate Algebra offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. In addition to the creative commons license and FWK features, this textbook offers a true pedagogical improvement over the current very expensive options.
Latest Activity
I spent the summer converting my old algebra study guide to ePub format and was able to have it listed in the Google Play store.[ Click here to read my blog post about it ]It was not that hard to do and the Google Play eBook reader is quite good! It looks ok on all of my devices and transfers between them seamlessly. Downside: You have to buy this FREE ebook. Not a problem for Google Play…See MoreElementary Algebra Videos - a visually searchable playlist.Feel free to copy-and-paste these links into your LMS -- it works…See More I will update this post when I am finished.See More
I am trying to build a circle of educators interested in education technology and open educational resources. If you add me to a circle I will circle you back as soon as I can. If you are just starting out with Google+ it may seem like a ghost town at first. I suggest searching for "shared education circles" and adding a shared circle or two for an instant start. Certainly,…See More
John Redden's Blog
|
Book Description: This text uses the language and notation of vectors and matrices to clarify issues in multivariable calculus. Accessible to anyone with a good background in single-variable calculus, it presents more linear algebra than usually found in a multivariable calculus book. Colley balances this with very clear and expansive exposition, many figures, and numerous, wide-ranging exercises. Instructors will appreciate Colley's writing style, mathematical precision, level of rigor, and full selection of topics treated. Vectors: Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Some n-Dimensional Geometry. New Coordinate Systems. Differentiation in Several Variables: Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newton's Method. The Chain Rule. Directional Derivatives and the Gradient. Vector-Valued Functions: Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator. Maxima and Minima in Several Variables: Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema. Multiple Integration: Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration. Line Integrals: Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields. Surface Integrals and Vector Analysis: Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations. Vector Analysis in Higher Dimensions: An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem. For all readers interested in multivariable calculus.
|
Graphing is the useful procedure in mathematics for explaining complex equations, functions and relations
and solving them. Graphing of any equation means its corresponding 2D paper representation of ...
Steps for Graphing Calculator
Step 1 :
Choose different values for "x" zero and solve for y to get different coordinates.
Step 2 :
Mark all the coordinates in the graph and join it.
Examples on ...
Content Preview
Best graphing calculator for high school is the TI 84 Plus **************** The FCC's regulations "may not reflect the latest evidence on the effects" of cellphones. US Government Accountability Office (GAO) report .August 7, 2012 ************************ "The new TI-84 Plus is a wonderful calculator. If anybody has had the TI-83 or 83 Plus, they know how easy and reliable it is. The 84 Plus is an all-around imprivement on the older version and even worth the additional $15-$20. I have had it since school started and have noticed than any problem I enter, it is solved immediately upon pressing enter, or solve. The speed is a great improvement over the 83-Plus".Read the rest of this review here The kids of today are not like those of our time. Back in the 1990s the older people had to settle for two dimensional graphs and if one wanted an upper triangular matrix then they had do all the row operations by themselves. Today, kids are using advanced graphing calculators. We recommend the TI84 Plus ( read the Full review here )which is the newer version of the famous TI 83. Although it is not as advanced as the TI89 Titanium it allows you to learn to do calculations yourself. In addition the TI84 Plus is acceptable in standardized tests such as SATs. However the TI84 has all you need in high school.
|
0470171324
9780470171325
Game Theory: A fundamental introduction to modern game theory from a mathematical viewpointGame theory arises in almost every fact of human and inhuman interaction since oftentimes during these communications objectives are opposed or cooperation is viewed as an option. From economics and finance to biology and computer science, researchers and practitioners are often put in complex decision-making scenarios, whether they are interacting with each other or working with evolving technology and artificial intelligence. Acknowledging the role of mathematics in making logical and advantageous decisions, Game Theory: An Introduction uses modern software applications to create, analyze, and implement effective decision-making models.While most books on modern game theory are either too abstract or too applied, this book provides a balanced treatment of the subject that is both conceptual and hands-on. Game Theory introduces readers to the basic theories behind games and presents real-world examples from various fields of study such as economics, political science, military science, finance, biological science as well as general game playing. A unique feature of this book is the use of Maple to find the values and strategies of games, and in addition, it aids in the implementation of algorithms for the solution or visualization of game concepts. Maple is also utilized to facilitate a visual learning environment of game theory and acts as the primary tool for the calculation of complex non-cooperative and cooperative games.Important game theory topics are presented within the following five main areas of coverage:Two-person zero sum matrix gamesNonzero sum games and the reduction to nonlinear programmingCooperative games, including discussion of both the Nucleolus concept and the Shapley valueBargaining, including threat strategiesEvolutionary stable strategies and population gamesAlthough some mathematical competence is assumed, appendices are provided to act as a refresher of the basic concepts of linear algebra, probability, and statistics. Exercises are included at the end of each section along with algorithms for the solution of the games to help readers master the presented information. Also, explicit Maple and Mathematica® commands are included in the book and are available as worksheets via the book's related Web site. The use of this software allows readers to solve many more advanced and interesting games without spending time on the theory of linear and nonlinear programming or performing other complex calculations.With extensive examples illustrating game theory's wide range of relevance, this classroom-tested book is ideal for game theory courses in mathematics, engineering, operations research, computer science, and economics at the upper-undergraduate level. It is also an ideal companion for anyone who is interested in the applications of game theory. «Show less
Game Theory: A fundamental introduction to modern game theory from a mathematical viewpointGame theory arises in almost every fact of human and inhuman interaction since oftentimes during these communications objectives are opposed or cooperation is viewed... Show more»
Rent Game Theory 1st Edition today, or search our site for other Barron Game
|
Investigations in Geometry
CAS MA 150
Credits:
4
An immersion experience in mathematical thinking and mathematical habits of mind. Students investigate topics in Euclidean and non-Euclidean geometry starting from basic elementary material and leading to an overview of current research topics.
Note that this information may change at any time. Please visit the Student Link for the most up-to-date course information.
|
Geometry Seeing, Doing, Understanding
9780716743613
ISBN:
0716743612
Edition: 3 Pub Date: 2003 Publisher: W H Freeman & Co
Summary: Jacobs innovative discussions, anecdotes, examples, and exercises to capture and hold students' interest. Although predominantly proof-based, more discovery based and informal material has been added to the text to help develop geometric intuition.
|
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
TC 9-524 APPENDIX CFORMULAS SINE BAR OR SINE PLATE SETTINGSine bars or sine plates usually have a length of 5 inches or 10 inches. These standard lengths are commonly used by the tool maker or inspector. The sine bar or sine plate is used for accurately
Mech 221: Computer Lab 1Hand in the solutions to the three questions in the lab at the end of the lab. Success in many kinds of engineering requires skill at numerical approximation. The computer labs this term build this skill. This lab in particular wi
Mech 221: Computer Pre-Lab AssignmentsTo help you succeed in this course, there will be pre-lab assignments for every computer lab. The goals of the pre-labs are to reinforce concepts taught in class that will be used in the labs and to introduce any new
Mech 2 Computer Lab GuidelinesThe following guidelines apply to all computer labs in Mech 2. For most of MECH 221, 222, and 223, you will complete one computer lab per week.SoftwareThe computer labs use MATLAB (MECH 221 and MECH 222), Unigraphics NX2 (
Mech 221 Math Problems from Old Tests, Week #1Brian Wetton September 20, 2010Notes: This contains all questions on Mech 221 tests and exams from 200609 on the material on numerical methods from the rst four lectures. Some of this material may appear on
Mech 221 Math Problems from Old Tests, Week #2Brian Wetton September 28, 2010Notes: This contains all questions on Mech 221 tests and exams from 200609 on the material on direction elds and rst order linear dierential equations. Some of this material maMath 152, Spring 2010 Assignment #11Notes: Each question is worth 5 marks. Due in class: Wednesday, April 7 for MWF sections; Tuesday, April 6 for TTh sections. Solutions will be posted Wednesday, April 7 in the afternoon. No late assignments will be acc
HOMEWORK ASSIGNMENT #7 MATH101.209 - INTEGRAL CALCULUSAll the assignments are from the textbook, unless otherwise specied. Write down the initials of your last name on the top right corner of your papers. Section 7.3 Problems 4, 16, 18, 28, 29, 30, 34, 3
HOMEWORK ASSIGNMENT #8 MATH101.209 - INTEGRAL CALCULUSAll the assignments are from the textbook, unless otherwise specied. Write down the initials of your last name on the top right corner of your papers. Section 7.7 Problems 16, 20, 22, 28, 34, 46 Sect
HOMEWORK ASSIGNMENT #9 MATH101.209 - INTEGRAL CALCULUSAll the assignments are from the textbook, unless otherwise specied. Write down the initials of your last name on the top right corner of your papers. Section 8.1 Problems 16, 18, 32 Section 8.3 Prob
HOMEWORK ASSIGNMENT #10 MATH101.209 - INTEGRAL CALCULUSAll the assignments are from the textbook, unless otherwise specied. Write down the initials of your last name on the top right corner of your papers. Section 9.1 Problems 2, 3, 12 Section 9.3 Probl
Name:April 2006 Marks [33] 1. Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for a correct answer placed in the
Name:April 2007 Marks [33Name:April 2008 Marks [21Marks [3]1. Short-Answer Questions. Put your answers in the boxes provided but show your work also. Each question is worth 3 marks, but not all questions are of equal diculty. At most one mark will be given for an incorrect answer. Unless otherwise state
Be sure that this examination has 11 pages including this coverThe University of British Columbia Sessional Examinations - April 2007 Mathematics 101 Integral Calculus with Applications to Physical Sciences and Engineering Closed book examination Time: 2
April 2008 Marks [21] 1.Mathematics 101Page 2 of 11 pagesShort-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal diculty. Full marks will be given for corre
The University of British Columbia Final Examination - April 24, 2009 Mathematics 101 All Sections Closed book examination Last Name First Signature Section : Student Number Instructor : Special Instructions: No books, notes, or calculators are allowed. U
Be sure that this examination has 12 pages including this coverThe University of British Columbia Sessional Examinations - April 2005 Mathematics 101 Integral Calculus Closed book examination Time: 2.5 hoursPrint NameStudent NumberSignatureInstructor
|
Academics
CampusCE
Understand how algebra is relevant to almost every aspect of your daily life, and become skilled at solving a variety of algebraic problems. This unique and thought-provoking course integrates algebra with many other areas of study, including history, biology, geography, business, government, and more. As a result, you will acquire a wide variety of basic skills that will help you find solutions to almost any problem.
INTRODUCTION TO ALGEBRA
Item: E735
|
Mapping science onto the world
For mathematicians, mathematics is an essence unto itself. Engineers are a little more pragmatic. For them, mathematics is the language that describes the science they map onto the world.
Engineering undergraduates enter their programs with a strong background in both math and science. While some fly through the very demanding four years of study, others have moments (or entire terms) where their trusty friend mathematics abandons them, suddenly seeming aloof, foreign and unwieldy.
"It's not their fault," said George Vatistas (Mechanical and Industrial Engineering) emphatically. "Certainly a number will struggle and leave the program, but most should succeed." He and the Faculty of Engineering and Computer Science (ENCS) believe that the way mathematics has been taught in engineering (in general, not just at Concordia) has not sufficiently bridged the gap to applied science theory, so they're doing something about it.
EMAT 213 is the first in a series of common math courses taken by undergraduate students in the faculty. It is prerequisite for many of the classes which follow, and thus a key to future success. That success requires not just understanding the math on an abstract level (strangely, not a problem for most engineering students) but understanding what the numbers, equations and operations mean in the real world.
In order to build and solidify that understanding for students, EMAT 213 has been completely dissected and reassembled. It's part of the overall strategic plan in ENCS which Vatistas helped formulate during his tenure as an associate dean. One of the goals, he explained, is to move towards excellence in teaching across the faculty by revamping both the materials and methods used in the classroom.
"The abstract side of math is very beneficial. However, engineers must be able to synthesize and produce products for the benefit of society. So, why would we talk to engineering students about mathematics in a purely theoretical sense?" Vatistas asked. "We expect them to apply it; the course should reflect that."
"These are smart kids," he added. "We (and their textbooks) were making great logical leaps, and we expected them to come along for the ride. Instead, we were undermining their self-esteem." As far as Vatistas is concerned, that's not what undergraduate education is about.
"In engineering education, our first approach in dealing with a concept should be to prove its validity either experimentally or using logic. [We must] convince the students that what we have developed is realistic by measuring it against reality. [And we must] persuade them that it is of value by means of examples from relevant applications."
What is relevant obviously varies from program to program, so the course focuses on examples which are applicable across a broad range of systems. "Systems may not necessarily resemble each other physically," Vatistas explained, "but they are considered analogous if, and only if, they can be described by the same set of equations." So the mass-spring-damper system that is often used in mechanical engineering is analagous to a resistance-inductance-capacitance system in electrical engineering and to models for heat flow through wall systems in building engineering. Each system is an excellent teaching tool for EMAT 213.
"Applying the math to models. which students will see over and over again throughout their studies, helps them understand how the equations describe the systems and the changes which occur within them," said Vatistas. While the changes to EMAT 213 are new, they are producing promising outcomes.
|
RELATED LINKS
Mathematics
(MATH)
001—Introductory Algebra, 3 Cr.
Real
number system, order of operations. Algebraic problem solving, solving linear
equations. Cartesian coordinate system, graphs of equations. Exponents and
radicals. Factoring polynomials, solving equations by factoring. A grade of C
or higher is required to take 111 or 130. Credit is not applicable towards
graduation.
090—Pre-Algebra with Study Skills and
Learning Strategies, 1.5 Cr.
Math attitude, study habits and
preparation for tests. Math timeline and biography. Math learning style. Time management and scheduling. Math anxiety.
Whole numbers, integers and introduction to algebra. Fractions and equations, applications of
fractions and equations. Decimals, percents, ratio, rate and proportion. Order
of operations. Introduction to statistics. This is a half-semester course.
Credits not applicable toward graduation. Graded CR/NC.
091—Introductory Algebra, 1.5 Cr.
Real number system, properties and
order of operations. Area and perimeter of rectangles, areas, and circles.
Algebraic problem solving, solving linear equations and inequalities. Cartesian
coordinate system, graphing linear equations and inequalities in two variables.
Systems of linear equations. Exponents and radicals. Factoring polynomials,
algebra of rational expressions, solving equations by factoring. This is a half-semester
course. Credits not applicable toward graduation.
111—Intermediate Algebra, 3 Cr.
This
course builds on the concepts and skills developed in MATH 091, or an
equivalent first-year algebra course, and prepares students for MATH 112
(College Algebra) or 113 (Trigonometry.) It covers linear equations and inequalities,
graphs and functions, system of equations and inequalities, polynomials and
factoring, rational expressions, radicals and complex number, and quadratic
functions and equations. Prerequisite: acceptable placement score or grade of C
or higher in 091.
112—College Algebra, 3 Cr.
This course builds on the concepts and
skills developed in MATH 111, or an equivalent second-year Algebra course, and
prepares students for MATH 270 (Managerial Mathematics) or serves as a
co-prerequisite for MATH 220 (Calculus I.) Topics include functions and their
graphs, polynomial and rational functions, exponential and logarithmic
functions, matrices and linear systems, sequences and series. Prerequisite:
acceptable placement score or grade of C or higher in 111.
113—Trigonometry, 3 Cr.
This
course focuses on the concepts and applications of trigonometry. The primary
goal is to prepare students for their calculus course. Topics covered include
the basics of the trigonometric functions and their graphs and applications,
trigonometric identities and equations, the Law of Sines and Law of Cosines,
vectors, complex numbers, conic sections, parametric equations and polar
coordinates. Prerequisite: acceptable placement score or grade of C or higher
in 111.
130—Introductory Statistics, 3 Cr.
An introductory course which deals with
the organization and processing of various types of data, normal and binomial
distributions, estimation theory, hypothesis testing based on the normal
distribution, the t-distribution, the Chi-square distribution, and the
F-distribution, and correlation and regression. Prerequisite: acceptable placement
score or grade of C or higher in 091 or 001.
155—Mathematics: A Way of Thinking,
3 Cr.
An investigation of topics such as the
history of mathematics, number systems, the mathematics of voting, graphing
theory, geometry, logic, probability, and statistics. There is an emphasis
throughout on problem-solving. Prerequisite: acceptable placement score or
grade of C or higher in 091 or 001.
220—Calculus I, 4 Cr.
Limits and continuity. Derivatives and
applications. Differentiation of polynomial, rational, trigonometric,
logarithmic and exponential functions. L'Hopital's Rule. Prerequisite:
acceptable placement score, or at least three years of high school algebra and
trigonometry with at least a B average, or a grade of C or higher in 112 and
113.
This course is intended to be a
one-semester survey of calculus topics specifically for biology majors. Topics
include limits, continuity, derivatives, integration, and their applications,
particularly to problems related to the life sciences. The emphasis throughout
is more on practical applications and less on theory. Prerequisite: placement
score into 220 or grade of C or higher in 180.
230—Elements of Statistics, 4 Cr.
Probability,
random variables, mathematical expectation, estimation of parameters, tests of
hypotheses, regression, correlation, and analysis of variance are some topics
covered. Computers are heavily used for problem-solving and data analysis.
Prerequisite: acceptable placement score or grade of C or higher in MATH 112.
255—Mathematics for Elementary and
Middle School Teachers I, 3 Cr.
Principles,
goals, and methods of teaching elementary school and middle school mathematics.
Topics include set theory, number systems, whole numbers, number theory and
integers and the associated binary operations. Emphasis on problem solving.
Offered every semester. Prerequisite: grade of C or higher in 155 or a Math ACT
score of 22 or higher.
260—Introduction to Abstract
Mathematics, 4 Cr.
Sentential and quantifier logic,
axiomatic systems, and set theory. Emphasis is on the development of
mathematical proofs. Prerequisite: grade of C or higher in 112.
270—Managerial Mathematics, 3 Cr.
Several topics applicable to the study
of business are covered. In particular, the course considers systems of linear
equations and linear programming, the mathematics of finance, and an
introduction to probability. Emphasis in the course is on applications. Prerequisite:
acceptable placement score or grade of C or higher in 112.
Theory and application of probability;
discrete and continuous variables; the binomial, Poisson, geometric, normal,
gamma, and chi-square are examples of distributions studied. Offered as needed.
Prerequisite: grade of C or higher in 221; grade of C or higher in 130 or 230.
Study of selected algebraic topics such
as: groups, rings, and fields; ring of integers, polynomials; field of real
numbers, complex numbers; finite fields. Offered every other year.
Prerequisite: grade of C or higher
in 260. W
355—Content and Methods in
Mathematics for Elementary and Middle School Teachers II, 3 Cr.
Principles, goals, and methods for
teaching mathematics in elementary and middle school. Topics include rational
numbers, real numbers, and geometry. Emphasis on problem-solving. Prerequisite:
grade of C or higher in 255; admission to teacher education program.
365—Numerical
Analysis and Modeling, 3 Cr.
The purpose of this course is to
introduce the student to a variety of mathematical models, solution techniques,
and basic programming. A variety of models and solution techniques are covered,
as chosen by the instructor. Basic programming topics include input/output,
if-then statements, loops, and arrays. A variety of numerical techniques are
covered with may include Runge Kutta methods, fixed point iteration, Newton's
method, and Monte Carlo simulation. Requirements include an
application/modeling project with a written report and class presentation.
Prerequisite: acceptable placement score or grade of C or higher in 221.
Topics
in Euclidean and other geometries; foundations of geometry; place of Euclidean
geometry among other geometries. Offered every other year. Prerequisite: grade
of C or higher in 260.
499—Mathematics Seminar, 1 Cr.
Selected topics of current interest in
mathematics are researched and presented. Students, faculty, and occasional
guest speakers share in the presentations. Offered as needed. Restricted to
students with junior standing or higher. Permission of instructor required. May
be repeated for credit.
|
Yi, Jung-AYoo, Jae-GeunLee, Kyeong HwaToward students' full understanding of trigonometric ratios.J. Korean Soc. Math. Educ., Ser. D, Res. Math. Educ. 17, No. 1, 63-78 (2013).2013Korean Society of Mathematical Education, SeoulENG60trigonometric ratiosmathematical termsmathematics teachingstudents understandingSummary: Trigonometric ratios are difficult concepts to teach and learn in middle school. One of the reasons is that the mathematical terms (sine, cosine, tangent) don't convey the idea literally. This paper deals with the understanding of a concept from the learner's standpoint, and searches the orientation of teaching that make students to have full understanding of trigonometric ratios. Such full understanding contains at least five constructs as follows: skill-algorithm, property-proof, use-application, representation-metaphor, history-culture understanding [{\it Z. Usiskin} (2012). What does it mean to understand some mathematics? In: Proceedings of ICME12, COEX, Seoul Korea; July 8--15, 2012 (pp. 502--521). Seoul, Korea: ICME-12]. Despite multi-aspects of understanding, especially, the history-culture aspect is not yet a part of the mathematics class on the trigonometric ratios. In this respect this study investigated the effect of history approach on students' understanding when the history approach focused on the mathematical terms is used to teach the concept of trigonometric ratios in grade 9 mathematics class. As results, the experimental group obtained help in more full understanding on the trigonometric ratios through such teaching than the control group. This implies that the historical derivation of mathematical terms as well as the context of mathematical concepts should be dealt in the math class for better understanding of some mathematical concepts.
|
Intermediate Algebra, 4th Edition INTERMEDIATE ALGEBRA, 4e, algebra makes sense!
|
An Interactive Introduction to Complex Numbers
Basic Calculations Applet
The Basic Calculations Applet allows users to input complex numbers in either cartesian or exponential form and display them in vector form. The applet also shows the results of conjugation and basic arithmetic of complex numbers. The applet, constructed using GeoGebra, requires Java 1.4.2 or later.
|
Math 2
Book Description: Reviews basic computation including whole numbers, fractions, decimals, and percent. Emphasis is placed on applications in the areas of consumerism, personal money management, and measurement. Also introduces probability, statistics, basic algebra, and the fundamentals of geometry.
|
"Very few books do justice to material that is suitable for both professional software engineers and graduate students. This book does just that, without losing its focus or stressing one audience over the other." Marlin Thomas, Computing Reviews
Book Description
Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and related topics such as modular arithmetic. The authors present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details.
|
You mean "memorize" as opposed to "learn"? By "learn" I mean "use often enough to understand what situations require that formula and know how that formula is derived". I find that much easier than memorizing formulas and far more valuable.
There's a lot more memorization involved in learning a language than in learning mathematics. How old were you when you learned your native language?
I have a very poor memory, and had always worried about being able to memorize the formulas. I found that very little actually needed to be memorized, since derivations are fairly simple. I agree with mathwonk in that way.
Of Course there are a few results that there needs to have a certain degree of memorization. For example, a result like the Wallis Product. It is not too difficult to follow the proof and understand the concepts, but there are just a few steps that seem like the person who originally did those steps was just toodling around with some maths and stumbled upon it. And its nice that he stumbled upon a nice looking form of the product, when there are at least 5 other equivalent expressions. In my case, for the Wallis Product I had to remember some steps in the proof, when usually the steps are really quite self-explanatory (at least in hindsight).
Memorizing for the low levels
Learning for the high levels
Forgetting for the geniuses
Obviously in math, once you have done enough questions relating to that formula, then it will become second nature and you won't even need to think twice to recall the formula.
You should learn the really basic ones, since the others are just built upon the basic formulas. If you truly understand how a formula works, then you can easily derive it when you needed it, although it may cause some inconvenience.
Say, the trig identities for example. Some of them looks like God's Wrath...but you can derive them from basic ones if you know what you are doing.
Of course, if you take IB Mathematics Higher Level, your Data Booklet has everything you need for you :P
|
More Information
QA-1 Natural Arithmetic
$15.00
Ben Iverson. Volume 1 in set of 3. The basic introduction to Quantum Arithmetic. This should be understandable to a student which is 10 years of age, and is grounded in addition, subtraction and multiplication. A little basic algebra would be helpful. These books are designed for school text books. Although they can be started at a young age, it does not mean that more highly educated persons will find it easy. They may find it quite difficult because they will have certain learnings to forget. 8.5" X 5"
|
Writing Algebraic Story found a Dr. Seuss book that demonstrates the idea of linear equations. I started the lesson by reading the book to the students and creating linear equations to follow the storyline. Then we created our own, following a model I put together.
By downloading this file, you will get the Dr. Seuss book title, the lesson plan I followed, the notes I gave to the students, the examples I showed them, and a few examples that my own students made.
It took me a few months to find this book, but it demonstrates linear equations (using y=mx+b) perfectly! The students might be a bit shocked to find out they are about to read a children's story, but they will enjoy it. Who doesn't like being read to? :)
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
7343.12
|
Education college math courses for teachers are now available in several sites. Graduate and post graduate courses can be taken on the web. Formal education and free programs may be studied.
Coursework
Practically every mathematics branch may be studied. This includes trigonometry, algebra, geometry, calculus and advanced computations. Each of these subjects have different levels (geometry I, algebra I, II and so on). They are also very detailed. There are introductory subjects such as fractions, decimals, whole number reviews and so on. An algebra course will cover topics like bilinear forms, linear transformations, vector spaces and polynomials.
There are also subjects that will teach you about the number system. Teachers will also have to study theorems, graphs and mathematical sciences. Mechanics of deformable solids, fluid mechanics, particle mechanics and computer algebra are also taught in these classes. There are also courses about statistics, probability theory and Stochastic processes.
More advanced concepts include manifolds, algebraic topology and general topology. The latter is also known as point set topology. This area includes topics about dimension theory, open spaces, separation axioms and continuous functions.
Free Courses
These can be taken by teachers too. While they have a lot of information, not all of them earn credit. Online universities offer credit earning courses. The price of each course varies. States have different rules regarding the credit hours needed. You have to check your universality regarding credit transfer policies and rules.This will vary from university to university.
Format
Websites have different set ups. Most of these programs are self-paced, although some of them assign instructors to their students. The majority of these sites divide the lessons into modules. Some courses only take a few hours to finish. But since these courses are self paced, students can decide to study full or part time. This will determine the length of each course.
Other Information
Online classes have lectures, assignments and video guides. Just like other online classes, there are discussion forums, boards and email. Some of them now use Skype and other means to enhance communication.
Some of these courses prepare teachers for the state exams. These subjects will cover a lot of topics.
Different study materials are available. There are several lessons available. Study materials include electronic notes, lectures, question sheets, problem sets and tests.
Online college math courses for teachers have to be accredited. Free programs do not earn credit. By taking one of these classes, you will be better prepared for the job.
Online courses for teachers classroom can get you ready to get a certification or degree. In the past you had to take courses in a traditional class. But the Internet has made it possible to take some of the coursework on the Internet.
Coursework Overview
Teacher education colleges are run by teachers and other qualified experts. Subjects include studying the meaning of school administration, office management and record keeping, Key Issues in educational management and characteristics of good head teachers.
Among the subjects you have to study are school and community, management of co-curricular activities, organizational structure, school discipline and human relations. Aspiring students also have the option of going through decision making, types of administrators and administrative functions in education.
Other Areas of Study
Universities with classes for teachers include subjects such as administrative functions in education, nature and scope of educational management, supervision and inspection, meaning and scope of school organization. Basic concepts that have to be studied are nature, aims, objectives and principles of school Administration. You also have to learn about the difference between administration, supervision & management.
Features and Format
These colleges offer more than just classroom management lessons. Teachers can take up courses on all aspects of educational leadership, physical education and its foundations. You also have to study curriculum and elementary teaching. An aspiring teacher can take up a teacher study program.
They can also take a major in other related courses. You can study for a degree or a certificate. There are graduate and undergraduate programs for students. Some of these classes have live and online classes. You have to undergo fieldwork and Internet based coursework.
Other Information
Issues in education are almost always covered in these courses. Among these subjects are training vs. development, productivity vs. human relations and efficiency and effectiveness. There are also subjects on management of school libraries, using the internet cyber bullying, management of school time-table and service training. You will also study challenges in school administration, common weaknesses of teachers, workload and common problems.
Would-be teachers also have to study guiding principles for schools, how to place emphasis on co-curricular activities and organization of education. Courses on school discipline include old and new concepts of discipline, factors that affect discipline and importance of decision making.
Online courses for teachers classroom is becoming a standard feature in universities and colleges. After completing these web based courses, you will be ready for the state teacher exam.
Online continuing education courses for social workers allow a professional to complete these requirements without going to school. Online options are available in many websites and colleges that have CEU courses.
Coursework Overview
Colleges with social work degrees and their online counterparts arrange their subjects in different sections. Divisions vary but there are several topics which are almost always covered. These include studies in human relations, social psychology basic and advanced concepts.
You will find topics about becoming a helper, ethics, responsibilities and the law. Other subjects are about case management, social work in rural areas and drug abuse and correctional services. Other available subjects are about child welfare services, social work in health care, mental health, administration and research.
License and CEUs
These topics are often reviewed in CEU courses. These programs are available only for social workers who have a valid license. This can be obtained by completing a master's or bachelor's degree in social work. These CEU courses have to be taken every three years to renew the license. After the CEUs are completed, the social worker will submit them to the state.
Requirements
Social work classes are
required to complete a specific number of CE units; the number varies by state. Some do not require you to submit evidence of CEU. However, audits are performed on a random basis. Proof of CEUs have to be submitted. For this reason, continuing education records have to be kept for four years.
Format and Features
A course in ethics is mandatory in most states as is pain management. Other topics that may be needed are common errors in workplace social interactions, stress and frustration, social issues in the workplace and in general. Continuing education classes include learning about the self, schemas, inference and emotion.
Other Information
There are also courses that focus on nonverbal communication, attributing the causes of behavior and research methods. The latter is often studied because methods are always changing. After completing the required courses, you will earn the credits needed to complete the program. If continuing education is not completed, they will be in violation of the state's health code. This can result in probation or license suspension.
Online continuing education courses for social workers work in different ways, depending on the state. To renew licenses, a certain number of CEU hours have to be completed. You need to consult the state licensing board for more information.
Online ESL courses for teachers differ by state. However, they share some common features like videos, audio and other multimedia features. Schedules are also flexible, allowing teachers to study during the day or night.
Coursework Overview
Those who want to instruct ESL must have an education major in college. These courses focus on teaching skills. This is necessary for those who want to operate classes proficiently. Other courses include intercultural communication and applied linguistics. Students also discover methods for teaching conversational English. These courses explain how to speak the language at work.
You will also learn how to assign meaningful homework and plan classes. Specific courses teach you how to work with adults or K-12 students who are studying English. Teachers also learn word meanings, pronunciation and grammar rules.
Formats of these virtual schools vary. There are courses designed for educators who want to work full time. Others are for those who want to teach part time.
Additional Subjects
These universities with education majors also teach about instructional techniques. This is necessary for instructing students with different cultural backgrounds. ESL teachers must have a bachelor's or master's degree.
There are also courses specific for teaching classes. At the same time they are taught how to run computer software tools. More advanced courses focus on the other tasks of a teacher. This includes being a community resource.
Continuing Education
This is required in many states. There are ESL educators with a bachelor's degree who can get a master's degree. This can lead to a better job and pay. Formal continuing education is necessary for their professional development. These courses help ensure their teaching methods are current. Many of these courses are can now be studied on the web.
Other Information
These subjects emphasize teaching methods and cultures. These classes may include a semester of educating. Clinical training is also required in many courses. All ESL educators must also have a license. The requirements differ depending on the state.
Admission requirements will vary depending on whether it is a public or private schools. In other states, you have to take an adult education license. They also study about job placements and the places where you can find work. There are many other courses that you can study on the web.
Online ESL courses for teachers are now being taken by several educators. The convenience that they afford is something that many are now discovering.
Free online courses for teachers aide offer several topics that aspirants and professionals can take. Teachers' aide programs and free courses give you the opportunity to learn the skills necessary to become successful. You will discover the core skills required for the job.
Coursework Overview
Major courses are speaking effectively with parents. Teachers also discover how to develop a good working relationship with coworkers. There are also topics on work team effectiveness and development. As a teacher's assistant, you will find out how to help pupils access the curriculum and develop their numeracy and literacy skills.
These assistants also learn how to help students with their social, emotional and behavioral needs. Aides must also learn about health, hygiene, safety and security. As a student you just take up subjects concerning education and its implications and how to organize the environment.
Other Tasks of Teachers Assistants
A teachers' assistant must also study how to use integrate communication and information technology in the classroom. An assistant must also be aware of how to help students with physical and / or sensory impairments.
Free educational sites for teachers assistants explore different methods for helping students with interaction, learning, cognition, interaction and communication problems. They also have to support students with multilingual and bilingual abilities.
Other Subjects in Free Courses
Their activities are not limited to the classroom. They must also help students with their health and well-being. Teachers assistants must also help when it comes to maintaining the safety of the environment for students.
These free resources also explain what it takes to promote the emotional development of a student. Additional topics include evaluation of learning activities, observing and reporting pupil performance. As an assistant they must also keep stock of pupil records.
Format
These courses use video, audio and forums to help students. Using these resources, you will find it easier to learn how to help during learning activities and maintain good relationships with other students.
Other Information
You will also find out about management of pupil behavior, helping literacy and numeracy activities and working with other people. There are also subjects that focus on supporting your colleagues, planning and evaluating learning activities. Among the other subjects that have to be studied are dealing with behavior problems, communicating with pupils, praise and encouragement.
Free online courses for teachers aide also teach you how to observe changes in students. An integral part of the course is guiding pupil behavior.
If you are a military family always moving from one deployment to the next and your kids need help with their studies; your kids are home-schooled; your young ones are
Image via Wikipedia
being tutored at home or you want to give your kids lessons for advanced studies; your best option is to enroll them in online courses designed for kids of various ages.
There are online courses for kindergarten through sixth grade if your kids are within this age bracket and they are usually assigned their own online instructor.
This will give you the assurance that your kids are being given the best learning tools and assistance even if they are not enrolled in a regular school.
What Courses Can Your Kids Take Online?
For lessons specific to online courses for kindergarten through sixth grade, these are broken down per level to allow the students to take courses appropriate for their age as well as for their level of knowledge.
The core courses however are basically the same for each grade level, it's the extent of the extent and coverage of the courses that vary from one level to the next.
Depending on the online school that you have selected, the curriculum may be categorized differently. For instance, Language Arts may be categorized as Language Arts and Reading while History may be presented as Social Studies.
How are the Courses Presented?
Since the courses are designed for younger students, these have to be presented in a way that they will not get easily bored with the lessons as well as explained in a manner that they can easily comprehend what is being taught to them.
Audio-visual learning aids, interactive practice tests and animated graphics to emphasize certain areas covered by the lessons are the most common methods of how each course is presented online.
You can enroll your kids for a full course or you can also choose specific areas that you want them to focus on. When choosing courses, it is important to check the child's capacity in term of course load so they can work on their courses more effectively.
If you enroll them in more than two courses at a time, they might feel overwhelmed and pressured and thus greatly affect the quality of their online education.
Entrance tests may also be required by the online school prior to admission to check how much the child already knows. This will help the school in determining what grade level they are most suitable forFor licensed and practicing teachers who wish to advance in their careers, a Master's Degree could be the answer. With a Master's Degree, more doors will be opened for you with regards to your career advancement.
Understandably, going back to school may not be your priority for two reasons: time and budget. Graduate school is expensive, maybe even more expensive than taking your undergraduate courses.
For this reason, not a lot of professionals consider getting a higher education. However, with education now also offered online, teachers and other professionals now have a better chance of continuing their education and getting a Master's Degree.
Inexpensive Online Courses for Educators
To find inexpensive online courses for educators, you should first check those that are established specifically to provide online courses alone but whose courses, modules and instructors are affiliated with some of the top universities worldwide.
You may discover that online courses from campuses that have put their presence on the internet are basically priced the same whereas courses offered by websites thru their affiliate schools are much more affordable.
There are also online schools that are actually non-profit organizations which offer more affordable graduate courses with the help of financiers, donors and other supporters.
On the other hand, you can also ask the online university or college if they offer discounts and under what terms and conditions can you avail of these.
Your other options for tuition reduction are applying for a full or partial scholarship and applying for other financial aid options provided by the online school.
Graduate Courses for Teachers
You won't lack in online graduate courses to take once you have decided to earn your Master's Degree as there are quite a lot of offerings on the internet for M.S., M.A. and M.Ed. courses for teachers.
Among the Master of Science courses that you can take are:
- Curriculum and Instruction
- K-12 Special Education
- Educational Leadership
For Master of Arts courses, the following are some of your choices:
- Mathematics Education
- Science Education
- Science Ed. in Chemistry
- Science Ed. in Physics
- English Language Learning
There are also Science Ed. courses in Biological Sciences as well as Science Ed. in Geosciences.
For Master of Education, there are also several online courses to choose from such as Instructional Design and M. Ed. in Learning and Technology.
Bear in mind that the courses listed here have their own areas of study and they may or may not have several domains of study per course.
You should read the course overview and course guide to get more information about them and also to give you an idea if these are what you want to take.
Teaching is a challenging task. Teaching gifted students come with a different set of challenges that only the well-trained and knowledgeable teachers would be able to tackle. Plus, there are certain certifications and degrees that are required to be able to pursue a career in teaching gifted students, who definitely have special needs.
The Gifted, Special and Fragile
Gifted students are classified as such because with the high IQ that they have, they easily get bored with regular flow of classroom discussions. Their minds are advanced and would need a curriculum and a system of teaching that will keep them interested in learning so they continue to do so.
Like special education teachers, teachers who deal with the gifted are required with a special training. Such training will help them know and understand the unique behavioral patters and cognitive abilities of such children. Their skills in ensuring that learning is promoted are further enhanced. That is so they will be able to develop a teaching system that is suitable to the kind of kids they deal with.
Get an Education
The key towards pursuing a career in teaching gifted students is having proper education. Some states may not require an education degree but would ask you to pass licensure exams. There are online courses that will help you through this process.
Teachers of gifted students use different techniques in order to promote learning. At times they are required with individualized instructions that suit their behavior towards learning in general. All these and more are being taught in school.
When choosing a course to become a teacher for the gifted, choose one that will suit your requirements and needs. Make sure that it is worthy of your time, effort, and money. Make sure that it is valuable in a sense that it will be credited for earning your degree or finishing your certification for licensing.
There are a number of courses available for teachers. Each course are directed towards a specific career goal that you may have set previously.
While you are being cautious choosing your course, make sure that you choose an accredited online school as well. You just cannot ignore the possible existence of Internet frauds, which may work towards stealing away either your money or your identity away from you.
As with many other aspects of our modern lives, technology is an important building block in taking a career as a teacher for the gifted. So make sure that you are not left behind.
|
A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered.
MATLAB is presently a globally available standard computational tool for engineers and scientists. The terminology, syntax, and the use of the programming language are well defined and the organization of the material makes it easy to locate information and navigate through the textbook.
1 An Overview of MATLAB
2 Numeric, Cell, and Structure Arrays
3 Functions and Files
4 Decision-Making Programs
5 Advanced Plotting and Model Building
6 Statistics, Probability, and Interpolation
7 Numerical Methods for Calculus and Differential Equations
8 Symbolic Processing
|
Students use problem-solving, estimating, and equation-building skills to develop confidence while conquering the next level of computational math. Realistic challenges engage students to apply basic operations, data interpretation, and geometry skills. Correlated to NCTE, NCTM and IRA Standards
|
No, I'm saying not to take the PHYS 1/2XX courses at all. But if you do, you're doomed either way; trying to prepare now won't help you. Calculus AB material definitely won't help you--you're looking at first year calculus when you need to know tensor products. Not going to happen.
randint wrote:Java's syntax is based on C's, and both are imperative/sequential languages. Beyond that, there is very little similarity between the two languages. You will write code in entirely different ways between the two.
It probably won't benefit you if you tried to learn these languages in advance. You may accidentally miss big concepts or pick up bad habits. They use Scheme in first year because the syntax doesn't pose an obstacle in learning the language, which allows you to tackle really hard problems right away.
randint wrote:MATH 145 is classical algebra. It is a course based in number theory, but also gives a broad introduction to classical algebra, including the study of fields, rings, and groups.
MATH 245 is a very broad, abstract algebra course that covers some fundamental concepts about the structure of vector spaces. Interesting content that we covered includes the Jordan canonical form, the rational canonical form, Markov chains and transition matrices, inner product spaces, dual spaces, quotient spaces, and then some more analysis-oriented stuff like norms (including of matrices), convexity, etc...
randint wrote:I don't know what gr. 12 data management covers. STAT 231 = AP Statistics. STAT 230 = a probability course mostly taught at the high school level, with a little bit of first year calculus background required. A lot of the content in STAT 230 was covered in the stats unit in my grade 12 math courseGet high marks and enjoy life. I did none of the things you just listed and still did well in math. If you really want to do something, co-op employers like independent programming projects. Make a program you'd enjoy makingI did none of these. Didn't take AP math, took a grade 11 Java course in high school. I had a 97 Math (Calc, Alge, CS) average in first term anything could go wrong......?
Stewart (Early Transcedentals) isn't that great of a book, and isn't the one used for MATH 147/148. You'd be better off getting a copy of Spivak.
I recommend that you learn whatever things you're interested in, and take things easy!Basically. In high school, you solve very similar cookie-cutter problems, and you do a lot of calculations/computations/algebraic manipulation. In university, you do proofs, and focus on the understanding of the mathematics, rather than just the doing.
Stewart focuses on the more high school-esque way of teaching math. MATH 147/148 are simply not taught that way. You will learn a large amount of theory that isn't in that textbook.
randint wrote:Uh, basically? Well, I mean, a few people do well, but they're either super geniuses, do really well at memorization, or cheat. The university didn't set things up that way--it's the idiotic physics department. Among other things, they fired a couple of their best lecturers a few years ago.
randint wrote:Well, I found them quite irritating, but on the other hand, it *is* harder to find work without co-op. In co-op, you must do worthless professional development courses and work reports, you pay an extra fee, and you get access to jobmine. Your schedule is far less flexible and you have higher marks requirements, and you are stuck following jobmine regulations if you use it, and get no benefit from your fee if you don't. So yeah.... I dunno. It's your choice. Remember that you can always drop co-op before your 2B term.
randint wrote:If you're not an expert in a field, you'll lack the foresight to understand where you could go wrong. I mean, obviously if you don't know much about C, it is very hard for you to know where things might go wrong.
For instance, there are some subtleties about pointers that are difficult to glean on one's own, especially in terms of appropriateness of use, utility, and the like. It's hard to understand code efficiency in an entirely self-taught manner. In terms of more complex languages, Haskell is extremely difficult to use without instruction, and C++ can be abused so badly without knowing how to properly use it that it's scary.
Familiarizing yourself with the syntax of C is great. Trying to tackle really complex programming challenges at this point might not be appropriate. Of course, this is general advice, so YMMV. Maybe you're a super-genius programmer, waiting to be discovered... don't really see this as an issue. The degree title "Software Engineer" is regulated. That doesn't mean only graduates from an SE program can be SE people. I mean, if I chose to, I could have a job title on my resume saying software engineer (literally, seeing as I'm working for a startup and my coworker/boss told me to choose my job title :P). I have many friends with "Software Engineer, Intern" on theirs. So the fixation on the word seems silly.
broodp4 wrote:I'm not aware of this, but I usually find politics and bureaucracy wasteful and silly.
broodp4 wrote: think financial engineering is a more common term down in the states, as well as for Master's degrees, and is pretty similar to mathematical finance-type programsAnother one, are the pure math and cs proffesors strong enough to answer questions or to solve problems beyond what is taught at the university? Or are they there just to present some basic notions from a book?
What's available for a first-year to take besides CS 145 and MATH 145 and MATH 147 ? You said PMATH 352 and what's next? I would like to take more pmaths or second year courses.
Does anyone get OSAP and is in a possesion of a car? Do they give u less money because u own a car?
For a career in software engineering/programming, which is the better program overall, Software Engineering or Computer Science coop at Waterloo? Assuming I take all the advanced courses in CS and mathYou are eligible for all pmath courses as a CS major. You don't have to declare anything more. (Remember that the converse does not apply, so don't drop out of CS.)
xtremepi wrote:
Another one, are the pure math and cs proffesors strong enough to answer questions or to solve problems beyond what is taught at the university? Or are they there just to present some basic notions from a book?xtremepi wrote:
What's available for a first-year to take besides CS 145 and MATH 145 and MATH 147 ? You said PMATH 352 and what's next? I would like to take more pmaths or second year courses.
Take some arts courses in your 1A at least. There are, quite literally, no other options. And remember that you need 5.0 (10 courses) of non-math credit to graduate.
In your 1B term, some things open up to you... such as overrides into MATH 249, PMATH 352, etc. Not necessarily recommended, you'll see when you get there.
xtremepi wrote:
Does anyone get OSAP and is in a possesion of a car? Do they give u less money because u own a car?1. Is it better to declare CS major in 2B? (CS245, CS246 are for Honours Mathematics Students) for the purpose of reducing tuition?
2. How feasible is my first year plan? It looks like this:
Semester 1A: MATH 145 MATH 147 CS 145 PHYS 121 + 131L MTHEL 131
Semester 1B: MATH 146 MATH 148 CS 146 PHYS 122 + 132L + 124
3. If I do not declare CO (or any math) major, I could just take random CO, PMATH, AMATH coursesWhat happens when they can't answer specific questions?( i mean here math questions) Do they just ignore it? This might be a silly question but, you know, I wanna know if almost 1 k per course is worth it.
greygoose wrote:
In your 1B term, some things open up to you... such as overrides into MATH 249, PMATH 352, etc. Not necessarily recommended, you'll see when you get there.
It is "recommended" since that's my goal. I've been given credits for MATH 135 and MATH 137, doesn't that make me a 1b student when talking about math? If I go as a regular student I will be extremely bored and I will waste a lot of money and time. Doesn't matter why... I would like to take 4 math courses per term and deal with the non-math at the end of my graduation because I loose interest in most of the things when doing math. I know people say I'm narrow, but actually it's totally different imo.
greygoose wrote:I know that u have to declare anything of value that u hold. I live in Kitchener on a bus route, but that's not the problem, the problem is when I have to be on the other side of the tri-city area for some personal reasons and I have to switch 2-3 buses and wait in a bus station for dozens of minutes or when I need to get out of the tri-city area. What other costs besides the insurance, gas, and parking near my home do you know that I might have? Also how's the parking at UW? Do I have to pay for it?I have a couple of questions:
1. What exactly do you mean by personal development and academic career? Do you learn more (about programming/compsci) in CS(all advanced courses) than in SE?
2. Is it true that SE students tend to get better jobs and why?
3. Is there a point3. Is there a pointSome big software developers or engineers claim that you don't really need a master's degree or a phd to work in the software engineering industry. Here , the most valuable assets are the internships. These big software developers work at google, two of them refused google and started their own project in Canada named Sumify. If you think that's pretty logical, you don't really need so much school to get in SE jobs, research is way more intense in studying and mastering the "art".
|
Math 101: Calculus with Problem Solving. This course provides an introduction to the two fundamental notions of calculus: the derivative and the integral. The five days a week meeting format allows for review of pre-calculus topics as needed. Admission to Math 101 is by placement via Carleton Placement Exam #1 (CP#1) only. After completion of Math 101 students may enter Math 121.
Math 111: Calculus 1. A first introduction to the calculus that develops the derivative and the integral. Designed for students with little or no previous exposure to calculus. Placement is through Carleton Placement Exam #1 (CP#1) or Carleton Placement Exam #2 (CP#2). Upon successful completion of Math 111 students may enter Math 121.
Math 211: Multivariable Calculus. Develops the calculus in two or more dimensions. Successful completion of Math 121, an AP BC score of 4 or 5, or a placement via Carleton Placement Exam #3 (CP#3) is required. Upon successful completion of Math 211 students may take Math 232.
AP Credit
Calculus AB: Score of 4 or 5: take Math 121. Score of 3: take Carleton Placement Exam #2 (CP#2) to determine which Calculus class you should enroll in. If you successfully complete Math 121 with a grade of C- or better, then you will receive 6 credits which count toward the mathematics major and graduation requirements.
Calculus BC: Score of 4 or 5: take Math 211. After successfully completing Math 211 with a grade of C- or better, you will receive 12 credits which count toward a mathematics major and graduation requirements (for Math 111 and 121).
Other Options into Mathematics Courses
Math 115 or Math 215: Introduction to statistics and probability. See the statistics section below.
|
Calculus
posted on: 10 Dec, 2011 | updated on: 15 Oct, 2012
Calculus! many students get fear with this branch of mathematics as it is consider as one of the difficult branch of Math. The word Calculus is derived from Latin word, which means calculating something and used for counting anything. Calculus covers a wide area of modern mathematics it mainly focuses on limits, functions, derivatives, integrals, and infinite series. Calculus plays a very important role in modern mathematics. It has two branches Differential Calculus and Integral Calculus. Both the branches play a vital role when we deal with Calculus. Calculus has lots of story from its beginning to present. The modern use of calculus began in Europe during the 17th century. Calculus have various applications where they are used, differential calculus persist:-velocity, acceleration optimization and Slope of the curve while integral calculus contain:-area, volume, center of mass, work, pressure and Arc Length. Calculus provides tools to solve many problems one of the problem is paradox which comes in limit and infinite series. Calculus was initially started to calculate or manipulate small type of quantities. If we see the fundamental theorem of calculus then we find that the differential and Integration operation are opposite to each other or we can say that inverse to each other. Calculus have so many applications, and is widely use in different fields wide variety of fields like computer science, statistics, economics business, physical science etc.
|
Tackle your math homework with Microsoft Mathematics
Have trouble with math? Me, too. I absolutely loved science classes as a student. But, there's a reason I didn't grow up to wear a white lab coat—my grasp of the required mathematical concepts was always a bit shaky. I wish I had Microsoft Mathematics when I was in school. Microsoft Mathematics is designed to help you understand and improve your math skills. Download a free copy and discover how it can help you visualize math concepts and aid you when you are studying.
Microsoft Mathematics 4.0 includes a full-featured graphing calculator. It's designed to work just like those expensive handheld calculators and it's optimized for algebra, trigonometry, statistics, and calculus. Additional math tools help you solve systems of equations, evaluate sides and angles in a triangle, and convert from one system of units to another. It also has a quick reference library of formulas commonly used in mathematics, chemistry, and physics.
One of my favorite features is step-by-step solutions, which are provided for many types of problems. They show you the individual steps to a solution, with basic explanations to help you understand how to solve the problem. It's great when you get stuck and you need someone to explain it to you.
For visual thinkers, the most satisfying (and fun) feature is creating graphs. You can easily plot complex equations, as well as view, rotate, and even animate graphs. That helps you visualize concepts and solutions, not just calculate the answers. You can plot in 2D or 3D, and choose Cartesian, polar, cylindrical, and spherical coordinates, which makes this a suitable tool for multivariable calculus.
If you install this on a Tablet PC, Microsoft Mathematics includes handwriting recognition to help you enter expressions with Ink. And you can save your worksheets for later review, or export them to Microsoft Word. (Want to use the computation and graphing features of Microsoft Mathematics in Word documents or OneNote notebooks? Download the free Microsoft Mathematics Add-in for Word and OneNote.)
For struggling students of all ages, Microsoft Mathematics 4.0 can make math studies less frustrating—and maybe even a little bit fun. And if you need more help with a particular math concept, check out the hundreds of free math and science tutorials from the Khan Academy.
Thanks for the great comment, GoodThings2Life. I think this is a pretty nifty program myself. And how awesome is it that it's free? Every math student should know about this great resource, so please spread the word!
This is a truly great program for performing mathematics on the computer, and the best thing is the add-in for Word and OneNote taking advantage of the equation editor built into Office! I have been looking for something like this program for years, I am glad that you guys kept the program around. I was afraid that it would have gotten killed with the rest of the student package. For me it makes the most sense to have it completely integrated into Office as the add-in does.
Where are all the videos explaining how to do it in video tutorials. Great to get it but lousy if you are trying to learn the calculator in applying it to the book or any other way. simple and easy so you get a good footing, unlike the falls of other calculators out their in the physical form.. alot of time teachers learn one and you have another and have no idea how to teach it to you. can't all live in a perfect world where we can afford on expensive on then have another. Not trying to sound negative but hey lets have a sense of being ablel to be problems solvers. you expect smart people to know how to use tools if they are going to teach us.
|
The Pearson History teacher companion for Year 10 (NZ Year 11) makes lesson preparation and implementation easy by combining full student book pages with a wealth of teacher support to help you meet the demands of the Australian History Curriculum. You'll find clear curriculum links to e...
Achievement Standard 1.2 (AS91008) asks you to: Demonstrate geographic understanding of population concepts. Using this resource, you will first be introduced to the concept of population and then progress to taking a deeper look at population in both New Zealand and Monsoon Asia, specifically study...
Pearson History 10 Value Pack supplies both the Pearson History 10 Student Book and Pearson History 10 Activity Book.
By purchasing a value pack, you will save compared to purchasing these two books separately.
Science Directions Workbook NCEA Level 1 provides homework, content coverage and practical work for NCEA Achievement Standard 1.5 Demonstrate Understanding of Aspects of Acids and Bases.
This full colour workbook contains one Achievement Standard.
On the purchase of a class set of any of the ...
Sigma Statistics Workbook has been completely updated to reflect the current requirements of Mathematics and Statistics in the New Zealand Curriculum. It contains a huge array of exercises that are linked to the corresponding Sigma Statistics. This makes it easy for teachers to choose, and students ...
Sigma Statistics is a brand-new statistics-focussed textbook following in the tradition of Sigma Mathematics.
The sections on Achievement Standards 3.10, 3.11 and 3.12 provide ground-breaking coverage of the ultimate stages of the statistics enquiry cycle approach. The emphasis is on statistical inf...
|
Personal tools
Sections
MATLAB
MATLAB is a high-level language and interactive environment for numeric computation.
MATLAB is high-level language and interactive environment for numeric computation created by Cleve Moler in the 1970s. It was initially designed to give students access to the LINPACK and EISPACK linear algebra libraries without having to learn to code in Fortran. MATLAB has now evolved into one of the most widely used numerical computation tools for research and data visualization in math, engineering and science.
MATLAB contains a sophisticated suite of tools for solving differential equations. Though MATLAB has a learning curve for students to overcome, since it is so widely used it can be beneficial to students to learn how to use MALAB.
There are many books on how to use MATLAB to solve differential equations, but perhaps the most authoritative is Solving ODEs with MATLAB, by Shampline, Gladwell and Thompson.
|
matrices
Course: CS 2130, Fall 2009 School: East Los Angeles College Rating:
Word Count: 2410
Document Preview capital letters to denote complete matrices1. For example: 11 4 A = 7 2 10 4 7 5 -6
The individual entries in a matrix are called the elements of the matrix. We denote the element in row i, column j of the matrix A by aij, that is we use the corresponding lower case letter with two subscripts denoting the row (i) and column (j) numbers. Thus a11 = 11, a21 = 7, a12 = 4, a23 = 5. If A is an n n matrix, we say that A is a square matrix of order n. A 1 n matrix is called a row vector of order n. An m 1 matrix is called a column vector of order m. When referring to the elements of a row vector the row number is (of course) always 1 and so is often omitted. Thus the element in column j of a row vector V is often denote as vj rather than v1j. Similarly the element in row i of a column vector U is usually denoted by ui rather than ui1 as the column number is always 1. and U = [1 2 3 4] 3 V = 4 5 is a row vector of order 4 and u1 = 1, u2 = 2, u3 = 3, u4 = 4.
is a column vector of order 3 and v1 = 3, v2 = 4, v3 = 5.
If the elements of a matrix are all integers we speak of a matrix over Z (the integers). If the elements of a matrix are all real numbers we speak of a matrix over R (the Reals). If the elements of a matrix all belong to the integers modulo n we speak of a matrix over Zn. Two matrices A and B are equal if and only if they are the same size that is each has the same number of rows and columns that is both are m n for some integers m and n and corresponding elements are equal that is aij = bij for 1 i m and 1 j n Matrix Addition and Subtraction We can add or subtract matrices A and B only if they are the same size. The element in row i column j of the sum of two matrices is the sum of the corresponding elements of the two matrices. Thus A and B are m n matrices, their sum C = A + B is also m n and cij = aij + bij
1
for 1 i m and 1 j n.
In hand-written text matrices are usually denoted by underlining the capital letter with a curly line.
A Barnes 2000
1
CS126/L13
Similarly if A and B are m n matrices, their difference D = A B is also m n and dij = aij bij For example 11 4 A = 7 2 10 4 and 7 5 -6 for 1 i m and 1 j n. 3 7 B= 2 4 1 2 2 8 -8 5 3 then 2 8 11 12 C= A+B= 9 2 2 11 2 -4
and
14 3 D = A - B = 5 6 9 6
Scalar Multiplication Multiplication of a matrix A by a scalar a (a single number) produces a matrix of the same size as the original whose elements are the corresponding elements of A each multiplied by a. Thus if A is an m n matrix then B = aA is also m n and bij = aaij Example 3 1 2 If A = 6 5 4 3 6 then 3A = 18 15 9 12 for 1 i m and 1 j n. .
Matrix Multiplication Suppose A and B are m p and q n matrices respectively. Then we can form the matrix product AB if and only if p=q, that is if the number of columns of A is equal to the number of rows of B. We say A and B are conformant for the product AB. In this case C = AB is an m n matrix and the element in row i, column j of the product matrix C = AB is given by
p
cij = aik bkj = ai1b1 j + bi 2 b2 j + ai 3b3j ... + aipb pj
k =1
Note that to obtain the element in row i column j of AB we multiply corresponding elements in the ith row of A and the jth column of B and then add up these p multiples. The number of rows in the product matrix AB is the number of rows in A and the number of columns is the number of columns in B. Thus, for example, given that 3 7 5 4 2 3 4 6 7 A= and B = 2 4 3 then C = AB = 1 2 5 4 9 9 1 2 2 Note that since A is 2 3 and B is 3 3, AB is 2 3 and, for example, the element in row 1 column 2 is formed from row 1 of A and column 2 of B: 2 (7) + 3 4 + 4 2 = 6 Similarly the element in row 2 column 3 is formed from row 2 of A and column 3 of B: (1) (5) + (2) 3 + 5 2 = 9.
A Barnes 2000
2
CS126/L13
Note that in the above example BA does not exist, since B is 3 3 and A is 2 3. We say A and B are not conformant for the product BA. Note also that square matrices are conformant for multiplication if and only if they have the same size. Even if A and B are square matrices of the same size it is NOT usually the case that AB = BA. For example, if 1 2 5 6 A= and B = 7 8 3 4 19 22 then C = AB = 43 50 23 34 and D = BA = 31 46
We say that in general matrix multiplication is not commutative. In a few special cases we can find square matrices A and B such that AB = BA, in this case we say A and B commute. The Zero Matrices The zero matrix Om n is the m n matrix with all its elements equal to zero Thus the 2 2 and 3 2 zero matrices are 0 0 0 0 O2 2 = O32 = 0 0 0 0 0 0 If the size of the matrix is clear from the context the subscripts are omitted and the zero matrix is represented simply by O. Let A be any m n matrix, then the following properties of the zero matrix are fairly obvious: A + O m n = Om n + A = A AOnp = Om p OqmA = Oqn
The Identity Matrices The identity matrix In is a square matrix of order n with ones on the main diagonal and zeros elsewhere. Thus the 2 2 and 3 3 identity matrices are 1 0 I2 = 0 1 1 0 0 I 3 = 0 1 0 0 0 1
If the size of the matrix is clear from the context the subscript is omitted and the identity matrix is represented simply by I. If A is any 2 3 matrix then AI3 = I 2 A = A. Thus multiplying a matrix the by identity matrix of the appropriate size leaves the matrix unchanged. For example 3 2 1 2 1 0 0 4 2 0 1 0 = 5 1 0 0 1 3 2 4 5 4 2 3 1 0 2 3 and = 0 1 1 2 5 1 2 4 5
More generally we find that if A is m n then AIn = Im A = A. In particular if A is square of order n we have AIn = I n A = A. Thus any square matrix commutes with the identity matrix of the same size.
A Barnes 2000
3
CS126/L13
Powers of a Square Matrix Let A be a square matrix of order n. Then we can form the matrix product AA which is also n n; this product is denoted by A2. Similarly we can form the products: A3 = A2A, A4 = A3A, ......., Am = Am1 A for any integer m1
where we define A1 = A and A0 = In. We saw above matrix multiplication is not normally commutative and however it can be shown that any two powers of a square matrix A always commute, thus: A3 = A2A = AA2 and more generally Am = Am1 A = AAm1 37 54 and also C = A2 A = 81 118 for any integer m1. Thus any two powers of a matrix commute. For example, if 1 2 7 10 2 A= then A = 15 22 3 4 37 54 thus B = AA2 = 81 118
The Transpose of a Matrix The transpose of an m n matrix A is the n m matrix formed from A by interchanging the rows and columns of A. The transpose of A is denoted by AT (or sometimes by A'). Thus, for example, if
Suppose that A is m p and that B is p n, then A and B are conformant for the product AB which is m n . Note that BT is n p and that AT is p m and therefore BT and AT are conformant for the product BTAT which is n m. In fact the following result holds: (AB)T = For example, if 1 2 A= and 3 4 3 5 1 B= 1 8 0 1 then A = 2
T
BTAT 3 4 3 1 and B = 5 8 1 0
T T
1 2 3 5 1 5 21 1 thus AB = = 3 4 1 8 0 13 47 3 3 1 1 B A = 5 8 1 0 2
T T
5 13 and (AB) = 21 47 1 3
whereas
5 13 3 T = 21 47 = (AB) 4 1 3
Laws of Matrix Algebra In the laws that follow, a and b are scalars (single numbers) and A, B and C are matrices of an appropriate size such that the matrix sums and products indicated exist. Commutative Law for Addition A+B= B+A 4
A Barnes 2000
CS126/L13
NO commutative Law for Multiplication Associative Law for Addition Associative Laws for Scalar Multiplication Associative Law for Matrix Multiplication Distributive Laws (Matrix) (Scalar) Properties of Zero Matrix Properties of the Identity Matrix Scalar Multiplication Properties
AB BA (A + B) + C = A + (B + C) a(AB) = (aA)B = A(aB) (ab)A = a(bA) (AB)C = A(BC) (A + B)C = A(B + C) = a(A + B) = (a + b)C = AC + BC AB + AC aA + aB aC + bC
O + A= A+ O = A OA = O AO = O IA = A 1A = A AI = A 0A = O
Symmetric and Anti-Symmetric Matrices If A is a square matrix with A = AT, we say that the matrix A is symmetric. If A is a square matrix with A = AT, we say that the matrix A is anti-symmetric (or skewsymmetric). The symmetric part of a square matrix A is the matrix (A + AT). The anti- (or skew-) symmetric part of a square matrix A is the matrix (A AT). Not surprisingly the symmetric part of a square matrix is symmetric! This follows since (A + A T)T = (AT + (A T)T) = (AT + A) = (A + A T)
Similarly the anti-symmetric part of a square matrix is anti-symmetric! The proof is left as an exercise. For example, if 1 2 A= 3 4 then 1 1 2.5 (A + AT ) = 2 2.5 4 and 1 0 -0.5 (A A T ) = 0 2 0.5
Matrices in Ada There is no direct support for matrix algebra in Ada. However it is relatively easySRT210Week ThreeWeek OverviewChange Management Revision Control Time ManagementChange management is an organized effort to implement changes to a systemChange ManagementTypically, change management involves the following elements: Th
CS2130 Programming Language ConceptsUnit 11 More on Function and Procedure Abstractions Defining New Operators Some languages, for example Ada and C+, allow new overloadings of existing operator symbols to be defined. For example in Ada:TYPE Vector
CS2130 Programming Language ConceptsUnit 2 Concurrent Programming So far in this degree programme you have mainly considered sequential programs in which statements are obeyed in a single thread of control, that is where only one instruction sequenc
CS2130 Programming Language ConceptsUnit 16 Encapsulation and Abstraction Client/Server Model Most modern programming languages provide a means of grouping related services together in some program entity. This entity is variously called a package,
CS2130 Programming Language ConceptsUnit 10 Function and Procedure AbstractionsExpressions & Commands Revisited Recall that an expression is anything that can be evaluated to produce a value whereas a command modifies program state (the internal s a" > double we would define#include <math.h> Concurrent
|
Introduction to Combinator introduction to the main concepts of combinatorics, features fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations.is the ideal text for advanced undergraduate and early graduate courses in this subject. The Second Edition contains over fifty new examples that illustrate important combinatorial concepts and range from the routine (i.e. special kinds of sets, functions, and sequences) to the advanced (i.e. the SET game... MORE, the Gitterpunktproblem, and enumeration of partial orders). The tables and references are been updated throughout, reflecting advances in Ramsey numbers and Thomas Hales' solution of Kepler's conjecture). In addition, many exciting new computer programs and exercises have been incorporated to help readers understand and apply combinatorial techniques and ideas. The author has now made it possible for readers to encode and execute programs for formulas that were previously inaccessible, allowing for a deeper, investigative study of combinatorics. Each of the book's three sections, Existence, Enumeration, and Construction, begin with a simply stated first principle, which is then developed step-by-step until it leads to one of the three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Polya's graph enumeration formula, and Leech's 24-dimensional lattice. Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, enabling readers to build confidence and reinforce their understanding of complex material.
Featuring a modern approach, Introduction to Combinatorics, Second Edition illustrates the applicability of combinatorial methods and discusses topics that are not typically addressed in literature, such as Alcuin's sequence, Rook paths, and Leech's lattice. The book also presents fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations.
Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, allowing readers to build confidence and reinforce their understanding of complex material. In addition, the author successfully guides readers step-by-step through three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Pólya's graph enumeration formula, and Leech's 24-dimensional lattice. Along with updated tables and references that reflect recent advances in various areas, such as error-correcting codes and combinatorial designs, the Second Edition also features:
Many new exercises to help readers understand and apply combinatorial techniques and ideas
A deeper, investigative study of combinatorics through exercises requiring the use of computer programs
Over fifty new examples, ranging in level from routine to advanced, that illustrate important combinatorial concepts
Basic principles and theories in combinatorics as well as new and innovative results in the field
Introduction to Combinatorics, Second Edition is an ideal textbook for a one- or two-semester sequence in combinatorics, graph theory, and discrete mathematics at the upper-undergraduate level. The book is also an excellent reference for anyone interested in the various applications of elementary combinatorics.
MARTIN J. ERICKSON, PhD, is Professor in the Department of Mathematics at Truman State University. The author of numerous books, including Mathematics for the Liberal Arts (Wiley), he is a member of the American Mathematical Society, Mathematical Association of America, and American Association of University Professors.
|
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
For three-semester undergraduate-level courses in Calculus.
This text combines traditional mainstream calculus with the most flexible approach to new ideas and calculator/computer technology. It contains superb problem sets and a fresh conceptual emphasis flavored by new technological possibilities. The Calculus II portion now has a new focus on differential equations.
Table of contents
1. Functions, Graphs, and Models.
Functions and Mathematical Modeling. Graphs of Equations and Functions. Polynomials and Algebraic Functions. Transcendental Functions. Preview: What Is Calculus?
2. Prelude to Calculus.
Tangent Lines and Slope Predictors. The Limit Concept. More about Limits. The Concept of Continuity.
Riemann Sum Approximations. Volumes by the Method of Cross Sections. Volumes by the Method of Cylindrical Shells. Arc Length and Surface Area of Revolution. Force and Work. Centroids of Plane Regions and Curves.
Introduction. Infinite Sequences. Infinite Series and Convergence. Taylor Series and Taylor Polynomials. The Integral Test. Comparison Tests for Positive-Term Series. Alternating Series and Absolute Convergence. Power Series. Power Series Computations. Series Solutions of Differential Equations.
12. Vectors, Curves, and Surfaces in Space.
Vectors in the Plane. Three-Dimensional Vectors. The Cross Product of Vectors. Lines and Planes in Space. Curves and Motions in Space. Curvature and Acceleration. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates.
Offers students the opportunities to focus on and study these challenging topics in separate chapters—and test their knowledge of them in separate unit tests.
Approximately 7000 total problems and interesting applications—Covers all ranges of difficulty, highly theoretical and computationally oriented problems.
Encourages students to learn by doing.
Technology projects—Features icons that take users to Maple/ Mathematica/MATLAB/Calculator resources on the CD-ROM.
Gives students the opportunity to apply conceptually based technology following key sections of the text.
320 Section-ending Concepts: Questions & Discussion.
Serves students with a basis for either writing assignments or class discussion.
Small optional section of matrix terminology and notation in the multivariable portion of the text.
A lively and accessible writing style.
Helps students feel comfortable with the topics covered, and their ability to master them.
Most visual text on the market.
Highlights are hundreds of Mathematica and MATLAB generated figures
|
Core (General Education) Skill Objectives:
1. Thinking Skills:
(a) Students will use reasoned standards in solving problems and presenting arguments.
2. Communication Skills: Students will ...
(a) ... read with comprehension and the ability to analyze and evaluate.
(b) ... listen with an open mind and respond with respect.
(c) ... access information and communicate using current technology.
3. Life Value Skills:
(a) Students will analyze, evaluate and respond to ethical issues from an informed personal value system.
4. Cultural Skills: Students will ...
(a) ... understand culture as an evolving set of world views with diverse historical roots that provides a framework for guiding, expressing, and interpreting human behavior.
(b) ... demonstrate knowledge of the signs and symbols of another culture.
(c) ... participate in activity that broadens their customary way of thinking.
5. Aesthetic Skills:
(a) Students will develop an aesthetic sensitivity.
Specific Course Goals:
1. From the perspective of mathematical content, this course should allow the student to expand and apply skills and knowledge gained in the first semester of Calculus to the topics of integration and applications of integration.
2. The student will gain knowledge and skills, and the ability to apply these, to a variety of situations which might be encountered in the world of mathematics, science, or engineering.
3. The student will further improve his/her ability to communicate mathematical ideas and solutions to problems.
4. The student will improve her/his problem-solving ability.
5. From a most general perspective, the student should see growth in his/her mathematical maturity. The three-semester sequence of calculus courses form the foundation of any serious study of mathematics or other mathematically-oriented disciplines.
General Education Course Objectives:
1. Thinking Skills: Students will ...
(a) ... understand the basic concept of the integral as the limit of a sum.
(b) ... understand the Fundamental Theorem of Calculus and learn how to use it to evaluate definite integrals.
(c) ... explore differentiation and integration formulas for a variety of functions, including exponential and logarithmic, trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic functions.
(d) ... explore a variety of integration techniques, such as integration by parts, partial fractions, various substitution methods, and methods of approximation for integrals which have no easy anti-derivative.
(e) ... investigate a wide variety of applications of integration, including solving simple differential equations, finding areas and arc lengths, and finding volumes and surface areas of solids of rotation.
(f) ... broaden their ability to work with functions by exploring parametric and polar functions, including graphing, and differentiation and integration applications.
(g) ... explore conic sections, including their treatment in polar form.
2. Communication Skills: Students will ...
(a) ... collect a portfolio of their work during the course and write a reflection paper.
(b) ... do group work (labs and practice exams) throughout the course, which will involve both written and oral communication.
(c) ... use technology (graphing calculators and Maple V) to solve problems and communicate solutions and explore options.
(d) ... improve their ability to write logically valid and precise mathematical proofs and solutions.
3. Life Values Skills: Students will ...
(a) ... develop an appreciation for the intellectual honesty of deductive reasoning.
(b) ... understand the need to do one's own work, to honestly challenge oneself to master the material.
4. Cultural Skills: Students will ...
(a) ... develop an appreciation of the history of Calculus and the role it has played in mathematics and in other disciplines.
(b) ... learn to use the symbolic notation correctly and appropriately.
5. Aesthetic Skills: Students will ...
(a) ... develop an appreciation for the austere intellectual beauty of deductive reasoning.
(b) ... develop an appreciation for mathematical elegance.
Course Philosophy and Procedures:
I am a firm believer in the notion that you as the student must be actively engaged in the learning process and that this is best accomplished by your DOING mathematics. My job here is not to somehow convince you that I know how to do the mathematics included in the course, but to help guide YOU through the material; I am to be a "guide on the side, NOT a sage on the stage". I may not always say things in the way which best leads to your full comprehension, but I will try to do so and I ask that you try to see your learning as something YOU DO rather than as something I do to you. PLEASE feel free to come see me to discuss questions or concerns you may have during the course; I will not bite! PLEASE ask questions in class if you have them; the only dumb question is the one you fail to ask!
Let me therefore urge you to make it a regular part of your day to try working the homework problems. There will never be enough time for us to go through every listed problem in class, and it is unrealistic to think that you will be able to find the time to work through every listed problem, but you should at least spend some time thinking about virtually every problem, and working the more interesting or challenging to completion. The daily Homework assignments will not generally be collected or graded. They are intended to structure your learning so that you regularly challenge yourself to see that you understand the material we are looking at. The important thing is that you at least look at all the assigned problems. You should also feel free to work other problems if you deem it necessary for your comprehension of the content.
In general, I think students can benefit greatly by working together on problems. While there is some danger of the "blind leading the blind" syndrome, or of students deceiving themselves into thinking they understand the material better than they actually do, for the most part grappling with ideas and trying to explain them to another student or learning to listen to others explain an idea is a wonderful way to see that you really do understand the material. In class we will spend an occasional day working on a group "lab", and we sill also typically have a group "practice exam" before the individual exams, and I also encourage you to find a "learning group" outside of class.
I will be asking you to keep a PORTFOLIO of your work. This portfolio will be collected twice during the semester, once upon our return from our mid-semester long weekend, on Monday 26 October, and again at the end of the course, on Friday 11 December. Each of these portfolios should be a representative collection of your work during that half of the semester; each collection you turn in should include 5 problems, written up in a finished form, along with a brief discussion of why you chose to include that particular problem. These problems should represent work you are proud of, or problems which brought you to a breakthrough point. Each of these portfolios of your work will be worth 40 points, 8 points per problem. Presumably you will hand in 5 sheets of paper, each of which will include a nicely organized solution to the problem, which might be from homework, or from a lab or worksheet, of from an exam, along with that reflection mentioned above.
I use a rather traditional Grading Scale: A = 90% (or better), AB = 87%, B = 80%, BC = 77%, C = 70%, CD = 67%, D = 60%. I do try to come up with a mix of points so that exams are not weighted too heavily; about half the points during the semester will come from individual, in-class exams - the rest will come from group labs, take-home problems, group practice exams, journals, and portfolios.
It is rarely much of a problem at the level of a calculus course, but it remains important that students turn work in in a timely manner, so that they do not get behind. Consequently, Late Assignments will be penalized 20% of the possible points for each class period late, up to a maximum of three periods. Assignments more than three days late will not be accepted.
Americans with Disability Act:
If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski in MC 320 (796-3085) within 10 days to discuss your accommodation needs.
|
Welcome to the Algebra II/Trigonometry portion of the site!
On this and the following pages, we'll try to clear up some common
problems people have with intermediate algebra (Algebra II) and
trigonometry (which ought to have a class of its own just to learn
how to pronounce it). Everything from solving basic equations
to the trigonometric ratios are covered.
After each section, there is an optional (though highly recommended)
quiz that you can take to see if you've fully mastered the concepts. And don't forget to visit the
message board and the
formula database.
|
This "blog" is a running summary of what we are doing in class, Math 307-101,
Fall (i.e., Winter Term I) 2011.
Begin Wednesday, September 7:
Chapter 4 of "Google's PageRank and Beyond." Note that Chapter 15
is a useful summary of all the mathematics used in the book, although
sometimes it is written in a "heavy-handed" fashion.
- Explain the idea behind the hyperlink matrix, H. Note that we will
allow repeated links and self-links, at least in certain situations.
- Review the idea of a stationary distribution of a matrix
- KEY EXAMPLE: Two state Markov chains, such as PageRank with two
sites or, equivalently, people moving in and out of California,
or, equivalently, people switching from left to right lanes,
etc. We get a 2 x 2 matrix
[ a 1-a ]
[ ]
[ 1-b b ]
to describe the transition probabilities. Applying the chain
"repeatedly" amounts to taking powers of the matrix, and the
limit is (usually)
[ p q ]
[ ]
[ p q ]
where p+q=1 and q(1-b) = p(1-a), i.e., p = (1-b)/(2-a-b)
and q=(1-a)/(2-a-b).
The steady state distribution is
[ p q ]
- Try an example:
Initial population [ 12 12 ]
Chain:
[ 3/4 1/4 ]
[ ]
[ 1/2 1/2 ]
Get sequence [ 12 12 ] -> [ 15 9 ] -> [ 15.75 8.25 ]
-> [15.9375 8.0625 ] -> can you guess the pattern?
Can illustrate most of basic linear algebra using this example:
solving equations (for the limit), eigenvectors/values,
powers of matrices, power method, etc.
- e.g. assume there is a limit [ V P ] to the above, starting from
[ 12 12 ]. Solve for V,P: V+P = 24, V = (3/4)V+(1/2)P,
P = (1/4)V+(1/2)P. Get V+P = 24 and V = 2P . So V=16, P=8.
In general, if V+P is whatever, we still have V=16, P=8.
- [ 16 8 ] is an eigenvector in the above, with eigenvalue 1.
[ 1 -1] is an eigenvector with eigenvalue 1/4 . This explains
the class formula for the iterates: V = 8 + 16 (1/4)^x, where x
is the iteration number ( [12 12] is iteration 1, [15 9] is
iteration 2, etc. )
- Give the general procedure for matrices with simple eigenvalues:
Solve for eigenvalues of M with characteristic equation:
det(M - I lambda ) = 0; lambda are the eigenvalues.
Then find eigenvectors. Then write initial condition in terms
of eigenvectors. This gives the formula for iterates of a
vector under repeated multiplication my M.
- General facts about Markov matrices:
Definition: non-negative entries, all row sums = 1
Def: "Irreducible" if can get from any state to any other.
Def: "Aperiodic" if it is of period 1 (G.C.D. of cycle lengths),
i.e., some power of the matrix has all positive entries.
(See IV.5.2 of Notes, conditions (3') and (4'))
Remark: Notes IV.5 sometimes confuses row and columns on page 161.
Thm: If a Markov matrix is irreducible, there is a unique (up to
scalar multiplication) eigenvector of eigenvalue 1 with all
non-negative entries. When normalized to sum of entries one,
it is a stochastic vector called the "stationary distribution."
- See also Markov Chain entry in Wikipedia.
- Similar example: Notes IV.3.1 Fibonacci Numbers
More theory: Notes IV.5 Markov Chains
(Attention: Notes IV.5 has column vectors instead of row vectors;
the text and in class we use row vectors and matrices multiply
on the right!)
- Explain how to use 2 x 2 Markov chains to reason about two populations
of websites, or Americans (in various states) and Canadians (in various
provinces), etc. This is the idea of refining Markov chains, not
covered in the text (as far as I know).
- Explain how the power method works in the 2 x 2 case.
------------------------------
Around Sept 19: Have yet to describe the power method and eigenvalues.
- On Sept 16 explained the notion of refining a Markov chain: say have
Markov chain on people from Vancouver, Toronto, Montreal. We can
refine the Vancouver state to "UBC" and "non-UBC" people from
Vancouver, and get a related Markov chain. More precisely, assume
the sum of the transition probabilities to UBC and non-UBC states
is the same as the original probability to Vancouver, and the
transition probabilities from UBC and non-UBC are both identical to
those in the original from Vancouver; then the sum of the stationary
distribution of UBC and non-UBC equals that of the original
distribution of Vancouver, and all other stationary distributions
are the same. In class we did something like:
[ 0,1/2,1/2 ; 1,0,0 ; 1,0,0 ] (in Matlab/Octave notion) can be refined
to [ 0,0,1/2,1/2; 0,0,1/2,1/2; .7,.6,0,0 ; .3,.4,0,0 ].
- Another refinement example: there are n websites, and each website
has k1 links to the first website, k2 links to the second, etc.
If k = k1 + ... + kn, then we see that the i-th website has PageRank
ki/k. Note that we could refine the i-th website into ki individual
websites with the same result.
- Another refinement example: n websites all point to one another
(including themselves). One website, called A, decides to m websites
such that the new websites only point to themselves and A. How does
A's PageRank go up? Three mega-states: (1) new, (2) A, and
(3) original excepting A. We get matrix:
[ m/(m+1),1/(m+1),0 ; m/(m+n),1/(m+n),(n-1)/(m+n) ; 0,1/n,(n-1)/n ]
We get: pi1 = pi2 (m+1)m/(m+n), pi3 = pi2 n(n-1)/(m+n).
Hence pi = [ (m+1)m , m+n , n(n-1) ] / ( (m+1)m+m+n+n(n-1) ).
- Study power method on bad Markov chains: not irreducible or periodic.
Then study power method on two by two, say symmetric for simplicity:
[ a, 1-a ; 1-a , a ]. What happens when a is near 1? Near 0?
-------------------------------
Around October 2: The Power Method for PageRank.
Power method interpretation:
We have G = alpha S + (1-alpha) E, where E = (1/n)e e^T is the matrix
whose entries are all 1/n, and S is a Markov matrix.
The power method starts with a stochastic vector, v, and considers
v^T, v^T G, v^T G^2, v^T G^3,...
and hopes that this converges to the stationary distribution pi^T.
For any stochastic w we have
w^T G = w^T [ alpha S + (1-alpha) E ] = alpha w^T S + (1-alpha) e^T/n
where e is the all 1's vector, since w^T E = e^T/n for any stochastic w.
So
w^T G^2 = alpha^2 w^T S^2 + alpha (1-alpha) e^T/n S + (1-alpha) e^T/n ,
and more generally
w^T G^k = alpha^k w^T S^k + (1-alpha) e^T/n (I + alpha S + alpha^2 S^2 + ... +
alpha^(k-1) S^(k-1) ) .
Hence the limit as k -> infinity (assuming that 0 <= alpha < 1 ) is
(1-alpha) e^T/n (I + alpha S + alpha^2 S^2 + ... )
= (1-alpha) (sum from k=0 to infinity of e^T/n alpha^k S^k ).
In other words, the k-th step Markov process when we start from a random
website is (e^T/n) S^k; these vectors are summed with weights
(1-alpha) alpha^k . The sum of the weights on step k and beyond is
alpha^k. Hence if alpha = .85, then the sum of the first 20 terms
accounts for "all but 4 percent" of the PageRank.
One may also understand the weights of e^T/n S^k for k=0,1,2,..., i.e.,
the weights (1-alpha) alpha^k as the weights of the geometric distribution;
in other words, the weight (1-alpha) alpha^k represents the probability
of seeing exactly k tails if we repeatedly flip a coin until he get a
heads, provided that the probability of heads is (1-alpha) (and therefore
the probability of tails is alpha).
Another way to understand the effects of damping is on the eigenvalues.
We claim that if S has eigenvalues lambda_1=1, lambda_2, ..., lambda_n,
then G has eigenvalues 1, alpha lambda_2, alpha lambda_3, ... alpha lambda_n.
The first eigenvalue of 1 comes with any Markov chain; to see the other
eigenvalues, assume that S has distinct eigenvalues; if v_i^T S =
v_i^T lambda_i, i.e., v_i is the left eigenvalue corresponding to lambda_i,
and if i > 1, then v_i^T e = 0, i.e., the sum of v_i's components is zero;
too see this, note that (lambda_i v_i^T) e = (v_i^T S) e = v_i^T (S e) =
v_i^T e, and hence (lambda_i - 1) v_i^T e = 0. But then v_i^T E = 0, and so
v_i^T G = v_i^T alpha S + v_i^T (1-alpha) E = alpha lambda_i v_i^T .
Hence replacing S with G has the effect of multiplying all the higher
eigenvalues by alpha. In particular, even if S has many eigenvalues of
absolute value 1 (i.e., S is periodic or reducible), then G won't be
(if 0 <= alpha < 1) and all of G's eigenvalues except 1 will be bounded
in absolute value by alpha.
-------------------------------
Teleportation: (See Section 5.3 of text): We have set
G = alpha S + (1-alpha)E with E = e e^T / n. Now we generalize this:
we set E = e v^T, where v is a stochastic vector which we will specify.
It represents how a "random jump" is taken. All our formulas generalize
easily to this case; for example, our formula for the PageRank power
method becomes
w^T G^k = alpha^k w^T S^k + (1-alpha) v^T (I + alpha S + alpha^2 S^2 + ... +
alpha^(k-1) S^(k-1) ) ,
and PageRank converges to
PageRank(S,v,alpha) = (1-alpha) v^T (I + alpha S + alpha^2 S^2 + ...)
= (1-alpha) v^T (I - alpha S)^(-1).
The text calls I - alpha S the fundamental matrix. The text shows that
if S and v are fixed, then (see Theorem 6.1.3)
d PageRank(alpha)
PageRank'(alpha) = ----------------- = - v^T (I-S) (I-alpha S)^(-2) ,
d alpha
where ' denotes the derivative with respect to alpha.
Of course, this could be expressed as a power series, and this will be
more useful to us:
PageRank'(alpha) = -v^T + (1-2 alpha) v^T S + (2 alpha - 3 alpha^2) v^T S^2
+ (3 alpha^2 - 4 alpha^3) v^T S^3 + ...
------------------------------------------
Around Oct 12:
Let us give a nice consequence of the above formulas in terms of L^1 norm:
THE L^1 NORM: || pi ||_1 = | pi_1 | + | pi_2 | + ... + | pi_n |.
The L^1 norm of a vector is the sum of the absolute values of its components.
E.g. if u is any stochastic vector, then || u ||_1 = 1.
It satisfies properties that any "norm" is supposed to have:
(1) || u || = 0 iff u = 0
(2) For any vector, u, and scalar, beta, we have || beta u || =
| beta | || u ||
(3) For vectors u and w we have || u+w || <= || u || + || w ||.
There are two other popular norms:
THE MAX NORM: || pi ||_max = max( |pi_1|,|pi_2|,...,|pi_n| ).
THE L^2 NORM: || pi ||_2 = sqrt( |pi_1|^2 + ... + |pi_n|^2 ).
Claim: If A is a Markov matrix, then for any v we have || v^T A || <= || v^T ||
provided that the norm || . || being used is the L_1 norm.
Norms are useful for measuring what happens when you truncate power series.
Notice that || PageRank ||_1 = 1 since PageRank is stochastic. Now we claim
that
|| PageRank'(alpha) ||_1 <= 2/(1-alpha).
This follows since
|| PageRank'(alpha) ||_1 <= || v^T || + | 1-2 alpha | || v^T S || +
| 2 alpha - 3 alpha^2 | || v^T S^2 || + ...
since v^T, v^T S, etc. are stochastic, this is
<= 1 + | 1-2 alpha | + | 2 alpha - 3 alpha^2 | + | 3 alpha^2 - 4 alpha^3 |
+ ...
Call this infinite sum f(alpha). One can show that f(alpha) <= 2/(1-alpha).
This is not a good estimate for alpha close to 1, where
f(alpha) is roughly c/(1-alpha), where c = 1 - 1/2.718281828...
= .63212055... (the 2.718281828... is the "e" of calculus).
Now let || || be the L_1 or Max norm. Then we have
|| v^T G - w^T G || = || alpha (v^T -w^T ) S || <= alpha || v^T - w^T ||.
provided that v and w are stochastic vectors. In particular, for w = pi,
the stationary distribution we have
|| v^T G - pi^T || <= alpha || v^T - pi^T ||.
Hence
|| v^T G^k - pi^T || <= alpha^k || v^T - pi^T ||.
-------------------------------
Around November 7: Start Least Squares and Curve Fitting.
See notes, Chapter III.1 (with background from Chapter II.1 and II.2 to
be able to speak of "column spaces").
Given a matrix, A, and a vector, b, we speak of a "best solution to Ax=b"
as any x that minimizes || Ax-b ||_2 (the 2-norm of Ax-b). All such x
satisfy the "normal equations"
A^T A x = A^T b.
The value of Ax is called "the projection of b onto the column space of A,"
since it is the orthogonal projection of b onto the subspace spanned by the
columns of A. We claim that for every b and A there is at least one solution,
x. This will require some notions from Chapter II of the notes--dimension,
bases, and the column space (see below).
We can apply this to linear regression: given data (x_1,y_1), ... (x_n,y_n),
we can find the "best" alpha and beta to model this data with the line
y = alpha + beta x,
meaning that we find the alpha and beta that minimize
sum from i = 1 to n of ( y_i - alpha - beta x_i)^2 .
Using least squares gives the equation
[ n sum_i x_i ] [ alpha ] = [ sum_i y_i ]
[ ] [ ] = [ ]
[ sum_i x_i sum_i x_i^2 ] [ beta ] = [ sum_i y_i x_i ]
which are the usual equations for linear regression. Note that it is
not obvious that these equations have a solution in alpha and beta,
nor is it obvious when these solutions are unique.
Similar curve fitting can be done with any variable modelled by functions
of other ("explanatory") variables.
If A^T A is invertible, then the normal equations A^T A x = A^T b have
the unique solution: x = inverse(A^T A) A^T b, and the projection of
b onto the column space of A (the space of all vectors of the form Ay
over all y) is
A x = A inverse(A^T A) A^T b.
The "projection matrix of A", A inverse(A^T A) A^T, can be understood
in terms of Gram-Schmidt orthonormalization: given vectors v_1,v_2,...
we define a sequece of orthonormal vectors u_1,u_2,... (orthonormal
means that the dot product u_i . u_j is 1 if i=j and 0 if i is not j)
via
u_1 = v_1 / || v_1 ||,
u'_2 = v_2 - (u_1 dot v_2) u_1 = v_2 - projection onto u_1 of v_2
u_2 = u'_2 / || u'_2 ||
u'_3 = v_3 - (u_1 dot v_3) u_1 - (u_2 dot v_3) u_2
= v_3 - proj onto u_1 of v_3 - proj onto u_2 of v_3
u_3 = u'_3 / || u'_3 ||
etc.
Then u_1, u_2, ... are orthonormal vectors with the same linear span
as v_1,v_2,... Note that if the v_i are linearly dependent, then
some of the u_i will be zero and they will be omitted from the
Gram-Schmidt construction.
This means that A has the same column space as Q, the matrix whose
columns are u_1,u_2,... But Q^T Q = I, and so the projection matrix
for Q (which is the same as for A) is simply
Q Q^T b = u_1 (u_1 . b) + u_2 (u_2 . b) + ...
This formula for a projection is extremely simple (and useful in
practice) (although it requires you to apply Gram-Schmidt).
The Gram-Schmidt process gives a decomposition
A = Q R ,
with A and Q as before and R an upper triangular matrix. This is known
as the QR-decomposition or QR-factorization (see the Wikipedia page on
the QR-decomposition).
The Gram-Schmidt process also shows that if U and W are subspaces of
n-dimension space (real or complex) and U is a proper subspace
of W, then there is a vector in W that is orthogonal to all the
vectors in U.
The normal equations always have a solution; this is equivalent to saying that
the column space of A^T A is that of A^T, and we can see that this is true, for
otherwise there would be an element of the column space of A^T that is
orthogonal to the column space of A^T A; i.e., there would be a y with A^T y
nonzero but (A^T A)x . A^T y = 0 for all x, or, in other words, A x . (A A^T)y
= 0 for all x; taking x = A^T y would show that A A^T y . A A^T y = 0 so A A^T
y = 0; but then y . A A^T y = 0 so A^T y . A^T y = 0 so A^T y = 0, a
contradiction.
------------------------------------------------------------------
Here are some random notes:
First Day of Class: Wednesday, September 7, 2011
Last Day of Class: Friday, December 2, 2011
Thanksgiving: Monday, October 10, 2011 (2nd Monday in October)
Midterm: Wednesday, November 2, 2011
Remembrance Day: Friday, November 11, 2011
Final Exam Period: December 6 to December 20
|
MATH& 153Calculus III•
5 Cr.
Department
Division
Emphasizes the study of infinite sequences and series including power series. Topics include plane analytic geometry, graphing in polar coordinates, and an introduction to vectors. Fulfills the quantitative or symbolic reasoning course requirement at BC. Recommended: MATH& 152.
Outcomes:
After completing this class, students should be able to:
to understand the polar coordinate system and plot points and graphs of functions in that system
to apply the ideas of the derivative and integral in the polar coordinate system
to calculate rates of change, slopes of tangent lines and areas in polar coordinates
to understand parametric equations and plot points and graphs of functions in that system
to apply the ideas of the derivative and integral in parametric equations
to calculate rates of change, slopes of tangent lines and areas for functions given as parametric equations
to determine the equations of lines and other simple forms in parametric equations
to use parametric equations to model rotational and other motion
to describe the conic sections as loci of points and to give the equations of the conic sections as functions in rectangular coordinates and polar coordinates and as parametric equations
to describe the meaning of a sequence of numbers
to graph sequences and determine the convergence or divergence of common sequences
to describe the meaning of an infinite series of numbers, the meaning the partial sums of a series, and the meaning of convergence for an infinite series
to recognize a geometric series and determine the value of an infinite geometric series
to determine whether some classes of series converge or diverge by selecting and applying the appropriate convergence tests:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.