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Weblinks
What a fantastic resource. This is a forum where you can submit any really annoying maths questions, and a fellow student or one of the brainy people from Cambridge University will help you solve it. Of course, you can also answer other people's questions yourself, and this really is an excellent way of understanding a topic even more. The specific forums you might want are:
If you are studying for your optional SATs or GCSEs, whatever level you are working at, then this website is a must visit! It is jammed packed full of help, resources and examples, and best of all everything is very clearly arranged and levelled so you are sure to find just the help you need. A truly excellent website!
If you are in Years 7 to 9, or you are doing the Foundation paper at GCSE, then have a look at this excellent Aussie website. It has lots of interactive notes and activities for you to practise your maths and sort any problems out.
They say the old ones are the best, and this is still probably the best one-stop place to call in for notes, examples, interactive solutions and questions for you to have a go at yourself. Each question has its level next to it, and you can even download a big revision check-list to make sure you there will be no nasty surprises in the exam. What more could you want in a website?
Some good old past SATs questions, arranged both by levels and topics, for you to try, and then you can check your answers at the bottom. Practice makes perfect. You must click on the Key Stage 3 section on the left to find them. Lots of good GCSE stuff in the GCSE section as well. Alas, you must pay to access many of the A Level materials, but the sixth form worksheets are free, come complete with answers, and seem very good. An excellent site!
If you are looking for some extra material to help you fine tune those A level maths skills, then this website might just be for you. Loads of questions and worked solutions. It is also one of the few websites around that is particularly good for Further Maths. Give it a go, it just might help!
Lots of topics in the Revision section, and in the Test Yourself bit you can tackle the following topics - Function Machines, Number Properties, Pythagoras, and Quadratics. The questions are of an interactive nature, and are extremely well laid out. There are also some good games, for when you have earned a break from revision.
A lovely set of interactive mind-maps for each A Level and Further Maths module. An excellent way to check you have covered everything and bring all your notes together. Just click on the module you want: C1C2C3C4S1S2D1M1FP1FP2FP3
For A Level Maths, this website is simply brilliant. Pick the topic you need help on from the menu on the left, and then watch all the resources appear. What I like best of all is the variety of resources, from nice little leaflets covering all the main facts, the video tutorials. Brilliant!
Something a bit different. Why not get your school involved in the recycling of old mobile phones and ink cartridges? This website will help you organise everything, and not only will you be doing your bit for the environment, but there is the chance to win prizes for your school as well!
I know that doing a maths puzzle may not seem like the best way to spend 20 minutes, but if you do one puzzle a day off this site, it'll do you a lot more good than reading over your textbook again and again. These puzzles teach you how to think and solve problems, which are essential skills for success at maths. Each puzzle comes complete with a worked answer. This site might just help you enjoy your maths more, so I would give it a go.
If you are serious about revision, then this is the website for you. Not just really good notes on every topic you could ever wish, but the pages are interactive, so you can fill in your answer and it tells you whether you are right... or not. Very good and completely free!
If you are taking your SATs and you want to make sure you get a level 5 (or even a 6!), or you are taking your GCSEs and you want to go from a D to a C, or from an A to an A*, then these free revision programs might be just what you need. It gives a nice focus to your revision at home, and if you put in the work, it will definitely pay off!
If you have trouble with things like using a compass, or measuring angles, or drawing bearings, or constructing triangles, then this might be just what you need. It is a link to a series of short video clips, with commentary, to help you practice these very important skills.
If you are studying for IB maths, and you need some help, then look no further! Lots of really good revision notes, practice test, and information about the IB curriculum. What more could you possibly want?
How about something a bit different! This wonderful site has lots of fancy applets that allow you to revise topics interactively. You can grab shapes and graphs, move them around, and see what happens. Very impressive and well worth a look!
This nice website offers free specimen A level exam papers with written and video solutions to help with your revision. Should you wish to try some more papers, then you need to pay, but if this style of revision works for you then it might be a worthwhile investment. There is also a pretty good maths forum for A Level students to ask questions and help each other. | 677.169 | 1 |
2006-08-01T16:59:02ZFluxBB I went back to college after being out of school for ten years, I knew I had my work cut out for me. I tested into College Algebra, which was essentially the same material as the most advanced maths class I took in high school.
One of the requirements for this course, in addition to the textbook, was a graphing calculator. A TI-86 or equivalent was specified. So, I went calculator shopping.
Being an impoverished college student planning on taking lots of maths classes, I didn't see any sense in purchasing a TI-86 when I could get a TI-89 for about 15% more, and it had many more functions. So, I purchased the TI-89.
That TI-89 has been the best learning aid I could have hoped for. It does pretty much everything I needed to learn to do through multi-variable Calculus and most differential equations that I was expected to learn to solve.
Many times late at night doing homework, when I didn't understand a concept, I could break down a problem to see how solutions changed when I changed a single aspect of the problem, such as changing a constant or an exponent. This allowed me to see many patterns in the problems, and ultimately led to a deeper understanding of the material.
I don't particularly like doing Mathematics, but I am fascinated by all the stuff I can do with Maths. Therefore, I often take the lazy route and use my calculator find my derivatives and anti-derivatives, as well as messy arithmetic. Don't get me wrong, I can do these by hand (most of the time), but I find it's usually a pain, and significantly increases the likelihood of me making a stupid or careless mistake.
Now, I'm the first to admit that if you present me with a complicated integrate by parts problem, I'm going to struggle without my magic box. I don't do such problems by hand often enough to stay current. However, I am also aware that I live in a world where it is becoming increasingly likely that I can find a computer or a calculator before I can find a pencil and a piece of paper.
The vast majority of the Maths instructors I have had have all had a negative view towards calculators. They tend to believe that using a calculator as a learning tool is about teaching keystrokes instead of teaching concepts. I, on the other hand, believe that learning the mathematical concept is wholly different than learning the syntax used to solve the problem, whether it be with a pencil and paper or a calculator.
The calculator does the calculations for me. It does not know how to set up the problem so that I get the correct information as a solution. No matter how many functions my calculator has, if I don't understand the concepts behind those functions, the calculator is useless.
I've had this discussion with every single Maths instructor I've had in college. Only one has come remotely close to agreeing with me, yet not a single one has offered any logical counter argument, usually relying on the outdated "what if you don't have a calculator" response. Yet on occasions where I sought Calculus help from, for example, a Pre-Calculus professor, she was unable to assist me as it had been so long since she had done any Calculus exercises.
I feel we are fast approaching a time when pencil and paper syntax is completely outdated. I also think that many students would understand concepts more easily if teachers embraced calculators in the classroom instead of relying on traditional methods.
How do you feel about calculators in the classroom? Are they common learning / teaching tools in other areas? (I'm in central Florida, USA) | 677.169 | 1 |
Systems of Equations
and Inequalities
Chapter
8
8.1 Systems of Linear Equations in Two variables
a. A solution of the system is an ordered pair that satisfies both equations with the format:
Ax + By = C
b. Solving with the Elimination Method
Pretty stra
Pre-Calculus Summer 2016
Section 2.7
Inverse Functions:
f of g of x is written
f(g(x) or f o g(x)
They give you two equations like: f(x)=6x and g(x)=
Then they will ask you to solve f(g(x) or g(f(x)
Then ask if they are inverses of each other
x
6
How to
Chapter
7
Trigonometry
Continued
7.1 and 7.2 Law of Sines and Cosines
a. The law of sines is used to solve SAA, ASA, and SSA triangles. This can result in one or two triangles.
Remember the sum of all angles in equal to 180
.
b
a
sin A
=
b
sin B
C
A
=
a
Analytical
Trigonometry
Analytical Trigonometry
Chapter
6
1
How to solve:
SECTION 6.1 Verifying Trigonometric
Identities
To verify an identity, show that one side of the
function can be simplified to look like the other side
of the function. In order to d
MATH 135 Advice
Showing 1 to 1 of 1
I recommend Professor Buckler for this course because her teaching style is easily understood. She teaches from the book and, if you complete the practice problems and focus during class, it is easy to build on the knowledge you have learned as the course progresses. I would suggest completing the practice problems the night you do that section and then completing the graded homework before the test. That was what I found to be most effective for making the processes stick in my brain.
Course highlights:
The highlights of this course for me was learning new ways to manipulate numbers in different problems. I liked the way the professor taught the class as I am a visual and audio learner. The main thing I learned in this course was how to calculate derivatives and how a derivative applies to a function.
Hours per week:
9-11 hours
Advice for students:
In this course you need to stay on top of your assignments. It is easy to let some of them slide until the day they are due but it is worth it in the long run and less stressful if you are constantly keeping up with the practice problems.
Course Term:Fall 2015
Professor:Jacqueline Buckler
Course Required?Yes
Course Tags:Background Knowledge ExpectedGreat Intro to the SubjectMany Small Assignments | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
The aim of this book is to provide the reader with a virtually self-contained treatment of Hilbert space theory leading to an elementary proof of the Lidskij trace theorem. The author assumes the reader is familiar with linear algebra and advanced calculus, and develops everything needed to introduce the ideas of compact, self-adjoint, Hilbert-Schmidt and trace class operators. Many exercises and hints are included, and throughout the emphasis is on a user-friendly approach.
Review
"Retherford's book...is like a tour to the top of a mountain. Every step is devoted to the final goal, the trace formula...the book is written in a style which reveals the author's enthusiasm. Thus it could be good prpaganda for functional analysis...this is a remarkable book which forces the student to understand mathematics and to be careful." A. Pietsch, Mathematical Reviews
Synopsis
A virtually self-contained treatment of Hilbert space theory which is suitable for advanced undergraduates and graduate students.
Synopsis
Professor Retherford's aim in this book is to provide the reader with a virtually self-contained treatment of Hilbert space theory, leading to an elementary proof of the Lidskij trace theorem. Many exercises and hints are included, and throughout the emphasis is on a user-friendly approach. | 677.169 | 1 |
The approximation theory is a scope of mathematical analysis, which at its essence, is interested with the approximation of functions by simpler and more easily calculated functions. This theory has widely influenced such other areas of mathematics as orthogonal polynomials, partial differential equations, harmonic analysis, and wavelet analysis. Some modern applications include computer graphics, signal processing, economic forecasting, and pattern recognition. | 677.169 | 1 |
Holt Precalculus: Student Edition 2006
Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem [ are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. The pre-classic Maya and their neighbours had independently developed the concept of zero by at least as early as 36 BCE, and we have evidence of their working with sums up to the hundreds of millions, and with dates so large it took several lines just to represent them.
Groups, rings, linear algebra, rational and Jordan forms, unitary and Hermitian matrices, matrix decompositions, perturbation of eigenvalues, group representations, symmetric functions, fast Fourier transform, commutative algebra, Grobner basis, finite fields. Prerequisites: Math 202A or consent of instructor. Third course in algebra from a computational perspective ref.: McDougal Littell Advanced download here McDougal Littell Advanced Math: Student. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to: (A) extend previous knowledge of sets and subsets using a visual representation notation and scientific notation; and (D) order a set of real numbers arising from mathematical and real-world contexts. (3) Proportionality , cited: Precalculus Ready Reference read online Precalculus Ready Reference (12-pack). There a school studying his ideas grew up there but more than that, Aryabhata set the agenda for mathematical and astronomical research in India for many centuries to come ref.: 365 Addition Worksheets with read pdf 365 Addition Worksheets with 5-Digit,. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language ref.: Pre-Calculus (High School Tutor (Prebound)) The NCTM standards give no indication (beyond four-year intervals) of the sequence with which the content is to be presented and are not helpful to the classroom teacher in designing lessons that meet the standards. The NCTM standards list goals with which no one would be likely to disagree 200 Addition Worksheets with 5-Digit, 3-Digit Addends: Math Practice Workbook (200 Days Math Addition Series) (Volume 29) 200 Addition Worksheets with 5-Digit,.
It was the Egyptians who transformed the number one from a unit of counting things to a unit of measuring things. In Egypt, around 3,000 BC, the number one became used as a unit of measurement to measure length. If you're going to build pyramids, temples, canals and obelisks you're going to need a standard unit of measurement — and an accurate method of applying it to real objects Proficiency in Grammar and Language for CXC theleadershiplink.org. Prerequisite: MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors), MATH-UA 140 Linear Algebra with a grade of C or better , source: 365 Addition Worksheets with 5-Digit, 4-Digit Addends: Math Practice Workbook (365 Days Math Addition Series) (Volume 30) 365 Addition Worksheets with 5-Digit,. This course is a continuation of MAT 225 that deepens a student's understanding of single-variable calculus. Students will learn new techniques of integration, including substitution, integration by parts, partial fractions, and integration tables Research Methods in Psychology download for free travel.50thingstoknow.com. An Honours Year of 48 units of credit must be completed. For information regarding the admission requirements and application process for honours, please see the information under 'Honours'; 5. In addition to the courses required for a student's chosen major, SCIF1121/1131, and honours, students must take 'science' courses so that the major plus SCIF1121/1131, plus Honours year plus 'science' courses total 144 units of credit; 6 200 Addition Worksheets with 3-Digit, 1-Digit Addends: Math Practice Workbook (200 Days Math Addition Series) (Volume 22) It is also of interest to the professional. A classic applied book that is readable and thorough and good to own is: Neter, John, Michael K. Applied Linear Statistical Models,4th ed. Irwin. 1996. 0256117365 1407 pages on linear regression and ANOVA. My favorite text on mathematical statistics is definitely the following. It is a large text with enough material for a senior level sequence in mathematical statistics, or a more advanced graduate sequence in mathematical statistics 500 Subtraction Worksheets read epub travel.50thingstoknow.com. As badly as we wanted new shoes, my in once how easily this over said, skirting sarcasm only by the blandness of his expression. Availability of certified teacher varies by state. Please call to verify the availability of certified teachers. Students like learning math more when they use fantasy sports! Fantasy sports are a hit worldwide with over 30 million participants in the U 365 Addition Worksheets with download epub 365 Addition Worksheets with 4-Digit,.
The Department of Mathematics provides a variety of concentrations leading to Baccalaureate, Masters, and PhD degrees. Our faculty-student ratio is high compared to many universities. The Department of Mathematics is proud of its excellent teaching and active research programs. Several of our faculty members are recipients of the SUNY Chancellor's Award for Excellence in Teaching , e.g. 60 Addition Worksheets with read here National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, Reston, VA, 2000. 101. Ralph Raimi, Standards in School Mathematics, Letters to the Editor, Notices of the American Mathematical Society February 2001. 102. Wilfried Schmid, New Battles in the Math Wars, The Harvard Crimson, May 4, 2000. 103. Hung Hsi Wu, The Mathematics Education Reform: Why you should be concerned and what you can do, American Mathematical Monthly 104 (1997), 946-954 Nelson Probability and Statistics 1 for Cambridge International A Level travel.50thingstoknow.com. There are other situations that incorporate mathematics a little more loosely, but still incorporate them nonetheless. The following situation is an example of ' Expected Value ' McDougal Littell Advanced download online Mandelbrot, Benoit, The Fractal Geometry of Nature (New York: W. A., "The Monster Group and its 196884- Dimensional Space" and "A Monster Lie Algebra?" Chapters 29-30 in Sphere Packings Lattices, and Groups, 2nd edition (New York: Springer-Verlag, 1993), pp. 554-571 , source: Math Made Easy: Combinations read for free Many test takers use exactly the wrong strategy, but you can avoid the pitfalls with the help of our standardized test experts The most efficient and accurate method for picking the right answer make sure you give yourself the best chance of getting the right answer Do you know what a "hedge phrase" is Creative Problem Solving: Multiple Strategies for the Same Answer, Grade 5 To find out more about Aberdeen's course and what it entails, go to High School Diploma. Ketterlin Geller has been living in Kingston, Jamaica with her family since August to support the first months of implementation of the Parent Math Training Pilot project in the island country , source: Algebra: Structure and Method, download online travel.50thingstoknow.com. Students with and without backgrounds in either subject are welcome -- no calculus will be required. In this course, students interested in learning why the calculus is justly described as one of the greatest achievements of the human spirit will find its concepts and techniques made more accessible by being placed in historical context. Beginning with the roots of calculus n the classical mathematics of antiquity, we will trace its development through the Middle Ages to the work of Newton and Leibniz and beyond ref.: 500 Addition Worksheets with 5-Digit, 1-Digit Addends: Math Practice Workbook (500 Days Math Addition Series) (Volume 24) | 677.169 | 1 |
Download
Description/Abstract
This review suggests that using graphing calculators in mathematics education can enable students to approach situations graphically, numerically and symbolically, and can support students' visualisation, allowing them to explore situations which they may not otherwise be able to tackle (and thus perhaps enable them to take their mathematics to a more advanced level). In this way, using graphing calculators can lead to higher achievement among students, perhaps through increased student use of graphical solution strategies, improved understanding of functions, and increased teacher time spent on presentation and explanation of graphs, tables and problem solving activities (compared with students not using such calculators). The impact of the availability of this form of calculator on teaching methods and curricula appears to have been more limited, with teachers reportedly tending to use graphing calculators as an extension of the way they have always taught, rather than provoking any radical change in style of teaching or design of the curriculum.
Item Type:
Article
Additional Information:
The pagination of this page proof is the same as the published version. | 677.169 | 1 |
Discrete Mathematics
It's not a book to dip into. It's more like an intensive course in book-form. In fact it does seem rather that authors simply combined their lecture notes and wrote them up into book form - eg page 22: "That's probably enough for today. Give these ideas a try and let me know how it goes. And speak up in class if you see that we need to go over something". But that doesn't detract (in fact it makes the author appear more human!) and adds to the atmosphere of it being a taught course not a reference book. | 677.169 | 1 |
PDF (Acrobat) Document File
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5.04 MB | 80 pages
PRODUCT DESCRIPTION
In the unit Introduction to Algebra: Grade 6, we concentrate on two important concepts: expressions and equations. We also touch on inequalities and graphing on a very introductory level. In order to make the learning of these concepts easier, the expressions and equations in this unit do not involve negative numbers (as they typically do when studied in pre-algebra and algebra).
We start out by learning some basic vocabulary used to describe mathematical expressions verbally—terms such as the sum, the difference, the product, the quotient, and the quantity. Next, we study the order of operations once again. A lot of this lesson is review. The lesson Multiplying and Dividing in Parts is also partially review and leads up to the lesson on distributive property that follows later.
Then, we get into studying expressions in definite terms: students encounter the exact definition of an expression, a variable, and a formula, and practice writing expressions in many different ways.
The concepts of equivalent expressions and simplifying expressions are important. If you can simplify an expression in some way, the new expression you get is equivalent to the first. We study these ideas first using lengths— it is a concrete example, and hopefully easy to grasp.
In the lesson More On Writing and Simplifying Expressions, students encounter more terminology: term, coefficient, and constant. In exercise #3, they write an expression for the perimeter of some shapes in two ways. This exercise is once again preparing them to understand the distributive property.
Next, students write and simplify expressions for the area of rectangles and rectangular shapes. Then we study the distributive property, concentrating on the symbolic aspect and tying it in with area models.
The next topic is equations. Students learn some basics, such as, the solutions of an equation are the values of the variables that make the equation true. They use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. I have also included a few two-step equations as an optional topic.
Lastly, in this unit students get to solve and graph simple inequalities, and study the usage of two variables and graphing.
For this item, the cost for one user (you) is $3.65.
If you plan to share this product with other teachers in your school, please add the number of additional users licenses that you need to purchase.
Each additional license costs only $1.25. | 677.169 | 1 |
PERSONALIZED Linear, Exponential, Quadratic PROJECT!
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0.92 MB | 8 pages
PRODUCT DESCRIPTION
This project is all about comparing the three types of functions: Linear, Exponential, & Quadratic. It focuses on graphing, word problems, and key characteristics of each function. A rubric/checklist is included that adds up to 100 points. This is personalized because students will choose the functions they graph and even fill in numbers for the word problems they will solve. The hope is that the personalization will draw students' interest and help avoid cheating! Great review for EOC | 677.169 | 1 |
Category Archives: Mathematics
Overview The SAT Mathematics test includes both multiple-choice questions and questions that require a written numerical answer. Students' mathematical reasoning is assessed in separate areas of number and operations; algebra and functions; geometry and measurement; data analysis; probability and statistics. These areas follow the guidelines set by the National Council for Teachers of Mathematics, and […]
Overview Students taking the SAT Mathematics test are assessed on areas such as number and operations, algebra and functions, geometry and measurement, data analysis, and probability and statistics. The Algebra and Functions strand includes operations on algebraic expressions, solving equations and inequalities, quadratic equations, rational systems of linear equations and inequalities, direct and inverse variation, […]
Overview The SAT Mathematics test questions assess different areas of mathematical knowledge and application such as number and operations, algebra and functions, geometry and measurement, probability and statistics, and data analysis. Data analysis questions address students' ability to interpret information presented in tables, graphs, and charts, recognize change and trends, and analyze change by performing […]
Overview Probability and statistics concepts are tested as one of the strands of SAT Mathematics. Topics include measures of central tendency such as mean, median, and mode; elementary probability; and geometric probability. The test doesn't contain long calculations of standard deviation or other statistical measurements. Why Statistics? Probability and statistics concepts are often combined with […]
Overview Students taking the SAT Mathematics test are assessed on areas such as number and operations, algebra and functions, geometry and measurement, data analysis, and probability and statistics. The Geometry and Measurement strand covers topics such as geometric notation, concepts such as points, lines, and angles in the plane. Geometric figures such as triangles, quadrilaterals, […] | 677.169 | 1 |
High school math homework
Solving Equations with Radicals. Learn arithmetic and geometric progressions. Learn about lines and planes in three dimensions. Learn how to apply various integration techniques. Most of them are capable of. If you want to contact me, probably have some question write me using the contact form or email me on. Solving Equations and Inequalities -. Here you can find every formula you will ever need in your math assignments.
Lessons about operations with matrices and finding inverse of a matrix. This site is designed for high school and college math students. Most of them are capable of. Learn trigonometric formulas and equations. Math Calculators, Lessons and Formulas. Multiplying and Dividing Rational Expressions. Analytic geometry and Linear algebra. Learn the applications of the definite integrals. Here you can find every formula you will ever need in your math assignments. Simplifying Radical Expressions Adding and Subtracting Radical Expressions Multiplying and Dividing Radical Expressions Complex Numbers -. Learn arithmetic operations with polynomials and finding zeros of polynomials. Math lessons section this page contains lessons on learning Algebra, Calculus,. | 677.169 | 1 |
Aie Alg/Trig W/Anl Geom 11e
This Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, Eleventh Edition, effectively prepares students for further courses in mathematics through its excellent, time-tested problem sets. This edition has been improved in many respects, including the addition of technology inserts with specific keystrokes for the TI-83 Plus and the TI-86, ideal for students who are working with a calculator for the first time. The design of the text makes the technology inserts easily identifiable, so if a professor prefers to skip these sections it is simple to do so.
"synopsis" may belong to another edition of this title.
About the Author:
Earl Swokowski authored multiple editions of numerous successful textbooks, including CALCULUS; CALCULUS OF A SINGLE VARIABLE; FUNDAMENTALS OF COLLEGE ALGEBRA; and PRECALCULUS: FUNCTIONS AND GRAPHS, all published by Cengage Learning Brooks/Cole.
Jeffery A. Cole has been teaching mathematics and computer science at Anoka-Ramsey Community College since fall 1981. He started working on the Swokowski series of precalculus texts in 1985 as an ancillary author, and has been a co-author since 1991. His contribution to the Swokowski texts also includes joining the revision team of the calculus text in 1989.
Review:
"There is about the right number of exercises, and there is a nice number of applications each time. The 'stories' attached to the application exercises are well developed and meaningful. I also like the use of illustrations to clarify the exercises. It is very helpful." "The exercises are well chosen and connect well with the text. They also go nicely from easier to harder at basically the right pace." "This text is carefully written. It does not oversimplify, but it is still accessible."
"Many mathematics textbooks are too concerned with rigor at the expense of understanding. This book has an adequate mix of rigor and conceptual development for comprehension." | 677.169 | 1 |
Most Mathematics courses have prerequisites that are listed as part of the course description. You can meet the prerequisite by taking the designated class or by achieving a certain score on the math placement test. If it is unclear that you have met the prerequisite for the class in which you wish to enroll you might be blocked by the system. For more information about how prerequisites work please visit us at the college website: www. canadacollege.edu/registration/index.php
To inquire about placement tests and the assessment process please visit us here: www. canadacollege.edu/assessment/index.php
You may also contact the Cañada College Counseling Office (located in Building 9, Room 139) at (650) 306-3452 for assistance.
Description: This is the first course in a 2-part series covering elementary and intermediate algebra. Topics include the real number system, linear equations, linear inequalities, graphing, systems of equations, integer exponents, polynomials, factoring, proportions, rational expressions, and problem solving. Students who complete this course with a C or better are advised to enroll in MATH 120. Units do not apply toward AA/AS degree innerst@smccd.edu.
These particular sections of MATH 110 and MATH 120 are part of an accelerated algebra sequence for students who want to complete both MATH 110 and 120 in one semester. Enrollment requires permission of the instructor. For further information go to smccd.edu/accounts/innerst. | 677.169 | 1 |
About the Course
Through this course, students will acquire a solid foundation in algebra and trigonometry. The course concentrates on the various functions that are important to the study of the calculus. Emphasis is placed on understanding the properties of linear, polynomial, piecewise, exponential, logarithmic and trigonometric functions. Students will learn to work with various types of functions in symbolic, graphical, numerical and verbal form.
About the Course
the introduction and use of Taylor series and approximations from the beginning;
a novel synthesis of discrete and continuous forms of Calculus;
an emphasis on the conceptual over the computational; and
a clear, dynamic, unified approach.
About this CourseThis course is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems.
About this Course
Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"
The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions.
Probabilistic models use the language of mathematics. But instead of relying on the traditional "theorem – proof" format, we develop the material in an intuitive — but still rigorous and mathematically precise — manner. Furthermore, while the applications are multiple and evident, we emphasize the basic concepts and methodologies that are universally applicable.
The course covers all of the basic probability concepts, including:
multiple discrete or continuous random variables, expectations, and conditional distributions
laws of large numbers
the main tools of Bayesian inference methods
an introduction to random processes (Poisson processes and Markov chains)
We live in a time of unprecedented access to information…data. Whether researching the best school, job, or relationship, the Internet has thrown open the doors to vast pools of data. Statistics are simply objective and systematic methods for describing and interpreting information so that you may make the most informed decisions about life.
About this course
Foundations to Frontiers (LAFF) is packed full of challenging, rewarding material that is essential for mathematicians, engineers, scientists, and anyone working with large datasets. Students appreciate our unique approach to teaching linear algebra because:
It's visual.
It connects hand calculations, mathematical abstractions, and computer programming.
It illustrates the development of mathematical theory.
It's applicable.
In this course, you will learn all the standard topics that are taught in typical undergraduate linear algebra courses all over the world, but using our unique method, you'll also get more! LAFF was developed following the syllabus of an introductory linear algebra course at The University of Texas at Austin taught by Professor Robert van de Geijn, an expert on high performance linear algebra libraries. Through short videos, exercises, visualizations, and programming assignments, you will study Vector and Matrix Operations, Linear Transformations, Solving Systems of Equations, Vector Spaces, Linear Least-Squares, and Eigenvalues and Eigenvectors. In addition, you will get a glimpse of cutting edge research on the development of linear algebra libraries, which are used throughout computational science.
Course DescriptionYou should have good knowledge of linear algebra and exposure to probability. Exposure to numerical computing, optimization, and application fields is helpful but not required; the applications will be kept basic and simple. You will use matlab and CVX to write simple scripts, so some basic familiarity with matlab is helpful. We will provide some basic Matlab tutorials.
Course Summary
In this introduction to computer programming course, you'll learn and practice key computer science concepts by building your own versions of popular web applications. You'll learn Python, a powerful, easy-to-learn, and widely used programming language, and you'll explore computer science basics, as you build your own search engine and social network.
About the Course
A It will give you a better understanding of how computer applications work and teach you how to write your own applications. More importantly, you'll start to learn computational thinking, which is a fundamental approach to solving real-world problems. Computer programming languages share common fundamental concepts, and this course will introduce you to those concepts using the Python programming language. By the end of this course, you will be able to write your own programs to process data from the web and create interactive text-based games.
About this course
Since these courses may be the only formal computer science courses many of the students take, we have chosen to focus on breadth rather than depth. The goal is to provide students with a brief introduction to many topics so they will have an idea of what is possible when they need to think about how to use computation to accomplish some goal later in their career. That said, they are not "computation appreciation" courses. They are challenging and rigorous courses in which the students spend a lot of time and effort learning to bend the computer to their will.
About the Course data
About the Course
Machine leaning
Learning deep learning from basic machine learning course is very necessary, here is some cool machine learning courses you should learn.
Maybe this is the most popular machine learning course on the internet, now is one of the self-faced course on the Coursera and you can learn it any time.
About this Course
Machine learning is the science of getting computers to act without being explicitly programmed. In the past decade, machine learning has given us self-driving cars, practical speech recognition, effective web search, and a vastly improved understanding of the human genome. Machine learning is so pervasive today that you probably use it dozens of times a day without knowing it. Many researchers also think it is the best way to make progress towards human-level AI. In this class, you will learn about the most effective machine learning techniques, and gain practice implementing them and getting them to work for yourself. More importantly, you'll learn about not only the theoretical underpinnings of learning, but also gain the practical know-how needed to quickly and powerfully apply these techniques to new problems. Finally, you'll learn about some of Silicon Valley's best practices in innovation as it pertains to machine learning and AI.
Actually this course is never opened on Coursera, but you can watch it by preview lectures.
About the Course
Machine learning algorithms can figure out how to perform important tasks by generalizing from examples. This is often feasible and cost-effective when manual programming is not. Machine learning (also known as data mining, pattern recognition and predictive analytics) is used widely in business, industry, science and government, and there is a great shortage of experts in it. If you pick up a machine learning textbook you may find it forbiddingly mathematical, but in this class you will learn that the key ideas and algorithms are in fact quite intuitive. And powerful!
Most of the class will be devoted to supervised learning (in other words, learning in which a teacher provides the learner with the correct answers at training time). This is the most mature and widely used type of machine learning. We will cover the main supervised learning techniques, including decision trees, rules, instances, Bayesian techniques, neural networks, model ensembles, and support vector machines. We will also touch on learning theory with an emphasis on its practical uses. Finally, we will cover the two main classes of unsupervised learning methods: clustering and dimensionality reduction. Throughout the class there will be an emphasis not just on individual algorithms but on ideas that cut across them and tips for making them work.
In the class projects you will build your own implementations of machine learning algorithms and apply them to problems like spam filtering, clickstream mining, recommender systems, and computational biology. This will get you as close to becoming a machine learning expert as you can in ten weeks!
Outline
This is an introductory course in machine learning (ML) that covers the basic theory, algorithms, and applications. ML is a key technology in Big Data, and in many financial, medical, commercial, and scientific applications. It enables computational systems to adaptively improve their performance with experience accumulated from the observed data. ML has become one of the hottest fields of study today, taken up by undergraduate and graduate students from 15 different majors at Caltech. This course balances theory and practice, and covers the mathematical as well as the heuristic aspects. The lectures below follow each other in a story-like fashion:
What is learning?
Can a machine learn?
How to do it?
How to do it well?
Take-home lessons.
This course is taught by Geoffrey Hinton since 2012, and definitely first choice of deep learning and neural networks guide:
About the Course
Neural networks use learning algorithms that are inspired by our understanding of how the brain learns, but they are evaluated by how well they work for practical applications such as speech recognition, object recognition, image retrieval and the ability to recommend products that a user will like. As computers become more powerful, Neural Networks are gradually taking over from simpler Machine Learning methods. They are already at the heart of a new generation of speech recognition devices and they are beginning to outperform earlier systems for recognizing objects in images. The course will explain the new learning procedures that are responsible for these advances, including effective new proceduresr for learning multiple layers of non-linear features, and give you the skills and understanding required to apply these procedures in many other domains.
Description: This tutorial will teach you the main ideas of Unsupervised Feature Learning and Deep Learning. By working through it, you will also get to implement several feature learning/deep learning algorithms, get to see them work for yourself, and learn how to apply/adapt these ideas to new problems.
This tutorial assumes a basic knowledge of machine learning (specifically, familiarity with the ideas of supervised learning, logistic regression, gradient descent). If you are not familiar with these ideas, we suggest you go to this Machine Learning course and complete sections II, III, IV (up to Logistic Regression) first.
About the Course
This course covers a broad range of topics in natural language processing, including word and sentence tokenization, text classification and sentiment analysis, spelling correction, information extraction, parsing, meaning extraction, and question answering, We will also introduce the underlying theory from probability, statistics, and machine learning that are crucial for the field, and cover fundamental algorithms like n-gram language modeling, naive bayes and maxent classifiers, sequence models like Hidden Markov Models, probabilistic dependency and constituent parsing, and vector-space models of meaning.
We are offering this course on Natural Language Processing free and online to students worldwide, continuing Stanford's exciting forays into large scale online instruction. Students have access to screencast lecture videos, are given quiz questions, assignments and exams, receive regular feedback on progress, and can participate in a discussion forum. Those who successfully complete the course will receive a statement of accomplishment. Taught by Professors Jurafsky and Manning, the curriculum draws from Stanford's courses in Natural Language Processing. You will need a decent internet connection for accessing course materials, but should be able to watch the videos on your smartphone.
About the Course
Natural language processing (NLP) deals with the application of computational viewpoint CourseAbout the Course
This course provides an introduction to the field of Natural Language Processing. It includes relevant background material in Linguistics, Mathematics, StatisticsMachine learning is everywhere in today's NLP, but by and large machine learning amounts to numerical optimization of weights for human designed representations and features. The goal of deep learning is to explore how computers can take advantage of data to develop features and representations appropriate for complex interpretation tasks. This tutorial aims to cover the basic motivation, ideas, models and learning algorithms in deep learning for natural language processing. Recently, these methods have been shown to perform very well on various NLP tasks such as language modeling, POS tagging, named entity recognition, sentiment analysis and paraphrase detection, among others. The most attractive quality of these techniques is that they can perform well without any external hand-designed resources or time-intensive feature engineering. Despite these advantages, many researchers in NLP are not familiar with these methods. Our focus is on insight and understanding, using graphical illustrations and simple, intuitive derivations. The goal of the tutorial is to make the inner workings of these techniques transparent, intuitive and their results interpretable, rather than black boxes labeled "magic here". The first part of the tutorial presents the basics of neural networks, neural word vectors, several simple models based on local windows and the math and algorithms of training via backpropagation. In this section applications include language modeling and POS tagging. In the second section we present recursive neural networks which can learn structured tree outputs as well as vector representations for phrases and sentences. We cover both equations as well as applications. We show how training can be achieved by a modified version of the backpropagation algorithm introduced before. These modifications allow the algorithm to work on tree structures. Applications include sentiment analysis and paraphrase detection. We also draw connections to recent work in semantic compositionality in vector spaces. The principle goal, again, is to make these methods appear intuitive and interpretable rather than mathematically confusing. By this point in the tutorial, the audience members should have a clear understanding of how to build a deep learning system for word-, sentence- and document-level tasks. The last part of the tutorial gives a general overview of the different applications of deep learning in NLP, including bag of words models. We will provide a discussion of NLP-oriented issues in modeling, interpretation, representational power, and optimization.
Course Description | 677.169 | 1 |
Algebra 2
Tutustu aiheeseen liittyviin aiheisiinFree Math Homework for Algebra 2. This Algebra 2 math homework is aligned with the common core math standards. Can also be used as warm-ups or bell wringers.
Free Math Assessments or Quizzes for Algebra 2. These Algebra 2 Quizzes are aligned with the common core math standards. These Algebra 2 assessments can also be used as quick checks, spiral math review, and progress monitoring.
High School Math Word Wall Ideas
High school Algebra 2 word wall
4 FREE Algebra and Algebra 2 Warm-up Templates
I LOVE warm-up templates! It makes life so much easier to know my warm-up is all set and ready to go. Here are 4 FREE warm-up templates that will work in Algebra 1 and Algebra 2. I use them in my Special Ed Algebra 2 classes. The repeated practice on the skills covered by the templates give my students a ton of confidence and has helped them retain information so much better. All of these templates are available for FREE download in my TpT store. They will also be linked at the bottom of thi
4 Important Things to Include on an Algebra 2 Word Wall
4 important things to add to an Algebra 2 bulletin board. Yes, even big kids like them! This is a snapshot of one part of the word wall in my high school Math classroom. This part is for Algebra 2. I'll have to take photos of the Geometry and Algebra sections too.
Graphing absolute value functions CHEAT SHEET
We started graphing absolute value functions in Algebra 2 this week. At the same time, I got a request for a reference sheet like one I had posted for graphing quadratics. It was great timing. I had one but it wasn't good, so the request was the kick in the pants I needed to make changes. | 677.169 | 1 |
Mathematics
Principles of Algebra
2 or 3 Trimesters
1 or 1.5 Credits Course Description: Principles of Algebra is designed as an intermediate course between Algebra I and Algebra II for students requiring reinforcement of concepts through the use of real world applications. Concepts from both courses are studied with emphasis on strengthening skills for future study. Prerequisite: Algebra I or equivalent
Algebra II
2 or 3 Trimesters
1 or 1.5 Credits Course Description: Algebra II begins with a review of basic terminology, notation, concepts, skills, and applications studied in Algebra I. After the algebraic properties and applications have been explored, the fundamental operations of polynomials are reviewed and extended. Operations with complex numbers, linear, exponential, and logarithmic functions are presented. Equations with singular and multiple variables are solved. Patterns and the continuation of data analysis are explored. Concepts, structure, precision of language, and the inductive reasoning approach are stressed. This course is designed to give a sound basis for further study of more advanced mathematics. Prerequisite: Algebra I
Geometry
2 or 3 Trimesters
1 or 1.5 Credits Course Description: Geometry introduces the study of points, segments, triangles, polygons, circles, solid figures, and their associated relationships as a mathematical system. Emphasis is placed on solving real-world applications of geometric concepts, often using algebraic properties as well. Algebra I and logical reasoning skills are used throughout the course. Topics include but are not limited to similarity and congruence, logical reasoning/writing proofs, analytic and coordinate geometry, comparing and using angle measurements of both two and three dimensional objects. Prerequisite: Algebra I
Intensive Geometry and Algebra (IGA)
1 Trimester
0.5 Credits Course Description: IGA is a course for those students who completed Algebra I, Geometry, and Algebra II. It reviews the concepts of previous Algebra and Geometry courses to prepare the students for the next course in the math sequence. This course is a one-trimester course and students completing this course will receive 0.5 math credit toward graduation. Prerequisite: Algebra II & Geometry
STEM Math
Two Trimesters
1 Credit Course Description: This is a STEM (Science, Technical, Engineering and Math) based Math that will both introduce and reinforce algebraic and geometric concepts in a technical context. Students will be applying the concepts from both classes in order to solve technical problems. Only seniors will be eligible to enroll in this course and must be recommended by their math instructor. Prerequisites: Senior Only - completion of Algebra II and Geometry - Teacher Recommendation Required
Statistics
Two Trimesters
1 Credit Course Description: Statistics is a course for those students who completed Algebra I and Algebra II. It introduces the concepts of using statistical analysis to make informed decisions. This course also introduces the concept of probability. Prerequisites: Algebra II
Trigonometry
Two Trimesters
1 Credit
Weighted Course Course Description: Trigonometry is for those students who have completed Algebra I, Algebra II, and Geometry. This course covers the trigonometric and algebraic functions and their graphs. Due to the difficulty of this topic this is a weighted course. Prerequisites: Algebra II and Geometry
Calculus
Two Trimesters
1 Credit
Weighted Course Course Description: Calculus is a college preparatory course basic to the further study of mathematics, engineering, and the physical sciences. It includes the study of coordinates and related variables, polynomials, and their graphs, differentiation of functions with applications, and integration of functions with applications. Prerequisite: Trigonometry
AP /Advanced Calculus On-Line
Full Year
1.5 Credit
Weighted Course Course Description: AP/Advanced Calculus Part 1-3 is available on-line available to those students that have completed Trigonometry and wish to sit for the AP exam. This course meets the Advanced Placement criteria required by the College Board. Successful completion of both Part 1 and Part 2 of this course is required for students to earn the AP designation. Prerequisite: Trigonometry (92% or higher)
This is a secure Staff Intranet content page and cannot be viewed by the public.
Please contact your Technology Director to gain access to the Staff Intranet
area in order to view this page. | 677.169 | 1 |
Grade : VII
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SWBAT solve some | 677.169 | 1 |
Graphing Reciprocal and Rational Functions Flip Book
Be sure that you have an application to open
this file type before downloading and/or purchasing.
1 MB
Product Description
Graphing Reciprocal and Rational Functions Flip Book
This flip book was created to be used as a stations activity to provide extra practice with graphing reciprocal and rational functions and identifying the following key characteristics: domain, range, x-intercept, vertical asymptote, horizontal asymptote, and hole(s).
There are 8 functions in the book: two reciprocal functions and six rational functions. Print the book for each student and hang the stations up around the room. Students write down the function given at the station, graph it, then identify its characteristics. They rotate through the stations until they have graphed all 8 functions. Directions for printing and answer key | 677.169 | 1 |
Learn More at mathantics.com
Visit for more Free math videos and additional subscription based content!
published:28 Nov 2016
views:5182published:06 Jun 2013
views:821056
published:30 Jan 2015
views:71269published:26 Jan 2013
views:6879509 May 2015
views:48987
This video helps you to understand:
1. what is function..?
2. what is domain and codomain of function..?
3. graph of a function...
published:28 Jul 2015
views:1522415 Jul 2011
views:160872
This video describes how to compose functions, which is used in math. This concept is usually taught in Algebra II and Precalculus.
published:15 Jul 2009
views:114903
An introduction to functions.
More free lessons at:
published:18 Mar 2007
views:864696published:07 Sep 2013
views:522938
Function (mathematics)
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.
Functions of various kinds In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation centuries of sustained inquiryCurricular content
Each state sets its own curricular standards and details are usually set by each local school district. Although there are no federal standards, 45 states have agreed to base their curricula on the Common Core State Standards in mathematics beginning in 2015. The National Council of Teachers of Mathematics(NCTM) published educational recommendations in mathematics education in 1991 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8.
Algebra Basics: What Are Functions? - Math Antics
Learn More at mathantics.com
Visit for more Free math videos and additional subscription based content!
7:57
What is a function? | Functions and their graphs | Algebra II | Khan Academy
What is a function? | Functions and their graphs | Algebra II | Khan Academy42:12
Class 12 XII Maths CBSE Functions 01
Class 12 XII Maths CBSE Functions 01
Class 12 XII Maths CBSE Functions 01
13:09
What is Function ? - Concepts of Function in Mathematics (Introduction & Basics)
What is Function ? - Concepts of Function in Mathematics (Introduction & Basics)Definition of Function in Hindi
Definition of Function in Hindi
This video helps you to understand:
1. what is function..?
2. what is domain and codomain of function..?
3. graph of a function...
12:55
Maths : What is a Function : y=f(x)
Maths : What is a Function : y=f(x)Composition of functions (math)
Composition of functions (math)
This video describes how to compose functions, which is used in math. This concept is usually taught in Algebra II and Precalculus.
9:34
Introduction to functions
Introduction to functions
Introduction to functions
An introduction to functions.
More free lessons at:
5:21
❤︎² How to Find the Inverse of a Function (mathbff)
❤︎² How to Find the Inverse of a Function (mathbff)
❤︎² How to Find the Inverse of a Function (mathbff)
MIT grad shows Function of Function economics blood Out of 12 functional RTA offices under Greater HyderabadCorporation only 4 of them have ...... | 677.169 | 1 |
Elementary Calculus 1 Documents
Showing 1 to 3 of 3
MATH 220
Exam #2 Study Guide
The following is only a guide, not a replacement for studying from your textbook. You must
practice all of these concepts with recommended homework problems and carefully read both the
discussion in the textbook and the exampl
Elementary Calculus 1 Advice
Showing 1 to 2 of 2
I love Maryland. It's so diverse, that I feel like I have the UN right here on campus. The courses are challenging but very relevant and the professors are quite the experts. I love the fact at the Shady Grove campus I have more of one-on-one connection with my professor.
Course highlights:
I usually leave every game with a sore throat because I am able to feed off the energy those I am surrounded by. I am not the biggest sports fan, but when I am at Terrapin game, I am not afraid to support my team.
Hours per week:
6-8 hours
Advice for students:
I have never felt safer in my life. and if something does happen, I know the issue will be taken care of with the utmost care.I've never seen anyone get so excited even though we may not win the game. The Maryland student body are just so enthusiastic.
Course Term:Spring 2013
Professor:AlvinGrenalds
Course Required?Yes
Course Tags:Background Knowledge ExpectedAlways Do the ReadingMany Small Assignments
May 01, 2016
| Would recommend.
Not too easy. Not too difficult.
Course Overview:
He is a tough teacher and doesn't answer questions in class but he is very clear in his teaching and you can self-teach yourself from his numerous provided practice exams. | 677.169 | 1 |
Advanced Mathematical Concepts, (c)2006 provides comprehensive coverage of all the topics covered in a full-year Precalculus course. Its unique unit organization readily allows for semester courses in Trigonometry, Discrete Mathematics, Analytic Geometry, and Algebra and Elementary Functions. Pacing and Chapter Charts for Semester Courses are conveniently located in the Teacher Wraparound Edition.
Advanced Mathematical Concepts lessons develop mathematics using numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator. A full-color design, a wide range of exercise sets, relevant special features, and an emphasis on graphing and technology invite your students to experience the excitement of understanding and applying higher-level mathematics skills. Graphing calculator instruction is provided in the Graphing Calculator Appendix. Each Graphing Calculator Exploration provides a unique problem-solving situation.
Book Description McGraw-Hill Education, 2005682274-N | 677.169 | 1 |
Project #87366 - Calculus 3
28 Mathematics Tutors Online
This week, your Learning Team combines the efforts from previous weeks to form a coherent story of an imaginary space.
Create a 12- to 15-slide Microsoft® PowerPoint® presentation that describes movement in space using some of the mathematical concepts you studied each week. Focus on the appropriate application of the concepts rather than minute detail of numerical coefficients in equations. Complete the following:
Summarize the structure of your presentation in the first slide.
Describe the real-life situation you chose to explore.
Introduce differential equations by describing their relevance to the real-life situation you chose.
Describe at least three concepts from each of the previous weeks that are relevant to describing a complicated space to someone not familiar with the characteristics of that space.
Include at least one math problem from each team member.
Describe what each team member considers the most important concept learned in this course.
Present your project to the class.
For Local Campus students, these are 7- to 12-minute oral presentations accompanied by Microsoft® PowerPoint® presentations.
For Online and Directed Study students, these are Microsoft® PowerPoint® presentations with notes.
I dont need help with the powerpount i just need help making slides that answer the question i just need 10 no more then 12 slides | 677.169 | 1 |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
The graphing calculator simply said is a graphing program. The program graphs functions,
relations, implicit and explicit, parametric and "straight," in two and three dimensions. A good
graphing calculator knows a wide array of mathematical functions already, including all the trig
functions. You communicate to the calculator symbolically in equations and formulas.
Example:
Y=mx + b instead of just y= 3x-1. You can even make parameters into sliders and drag to change
values.
The most fully functional commercial released Graphing Calculator is on both Macintosh and on
Windows it is the 3.2 version. One way of learning how to use a good graphic calculator is just by
messing around with one of the versions offered on the internet by either Macintosh or Windows.
When you have a specific job to do you can read manuals and learn how to use the online graphic
calculator. One of the best methods is to simply go online and read the online help notes, read the
demos and playing with the examples. This of course is for the novice who has already graduated
from college without taking courses on the use of the graphic calculator. Those who have had
training it is very easy for them to use the online methods.
We suggest that you go online and try to mess around with the graphic calculator practicing with
the demos. Use the graphic calculator to do mock projects before you begin to use it for your own
project. You should note that many times errors can be made but with enough practice and
reading the demos it usually will not take long before you can become proficient enough to use the
online version for your projects.
See what I think are the top 10 graphing calculators on the market today, as well as more of my
work all at FINDcollegecards.
Guide
Reviews Goldmine is a growing community of real people happy to share their experiences of products and services. Honest And Impartial Reviews Written By Real People Like You.Check more reviews in : | 677.169 | 1 |
Synopsis
Edexcel GCSE Maths, Foundation Student Book has been created by experts to help deliver exam success in Edexcel's new Maths GCSE. Written for Foundation tier students, the book focuses on developing students' fluency in key mathematical skills and problem solving using carefully chosen examples and extensive practice. Powered by MyMaths the book links directly to the ever popular web site offering students a further source of appropriate support. | 677.169 | 1 |
MSM Mathematics: Bk. 4X
Description
The "MSM Mathematics" series offers an integrated and comprehensive assessment for GCSE mathematics. It provides a one-book-per-year mathematics course. There are worked examples and numerous graded exercises. The maths is set in the context of everyday life, involving investigations and project work, to provide approaches to all kinds of mathematical problem solving. The writing team has organized the mathematics covered by the National Curriculum into a series of topic-based sections within each book. Mathematical knowledge and skills are developed in line with current practice in maths teaching. The "MSM" series comprises course books at all levels. Books 1 and 2 provide maths for all abilities at Key Stage 3. Students of average ability can continue with the "x" series - books 3x, 4x and 5x. The "w" series provides support for students having difficulty with the maths covered in the books 1, 2, 3x-5x. The material in the "w" books is organized in the same sequence as the main course, but concentrates on the development of basic concepts for those students experiencing difficulties.
The "y" series caters for more able students, providing maths for top grades of GCSE and preparation for Sixth-Form work leading up to Levels 9-10 at Key Stage 4.show more | 677.169 | 1 |
That brings up the question of what criteria to apply, and the example I give is: introduces the concept of "function" with some examples that include operating with strings (characters) or other non-numeric inputs.
One hallmark of CS is its algorithms are often only "semi- numeric" i.e. numbers enter into it, but so do other types of object. Tradebooks like 'Godel, Escher, Bach' make it clear that mathematics is not strictly confined to "numbers" where concepts such as "function" are concerned.
I'm imagining a listing of mathematics textbook titles with several columns of criteria, either with a check or an X, measuring how friendly this textbook's way of teaching mathematics is to computer science, meaning it provides smooth segues, jumping off points, topics, for going back and forth between them etc. What else?
Another criterion: programming and programming languages are at least mentioned, perhaps only in a side bar. Actually using a programming language would be another column. And so on. | 677.169 | 1 |
third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.
--This text refers to an out of print or unavailable edition of this title.
--This text refers to an out of print or unavailable edition of this title.
Editorial Reviews
From the Back Cover
Master linear algebra with Schaum's—the high-performance study guide. It will help you cut study time, hone problem-solving skills, and achieve your personal best on exams and projects!
Students love Schaum's Outlines because they produce results. Each year, hundreds of thousands of students improve their test scores and final grades with these indispensable study guides.
Get the edge on your classmates. Use Schaum's!
If you don't have a lot of time but want to excel in class, this book helps you:
* Use detailed examples to solve problems * Brush up before tests * Find answers fast * Study quickly and more effectively * Get the big picture without poring over lengthy textbooks
Schaum's Outlines give you the information your teachers expect you to know in a handy and succinct format—without overwhelming you with unnecessary jargon. You get a complete overview of the subject. Plus, you get plenty of practice exercises to test your skill. Compatible with any classroom text, Schaum's let you study at your own pace and remind you of all the important facts you need to remember—fast! And Schaum's are so complete, they're perfect for preparing for graduate or professional exams.
Inside, you will find: * A bridge between computational calculus and formal mathematics * Clear explanations of eigenvalues, eigenvectors, linear transformations, linear equations, vectors, and matrices * Solved problems that relate to the field you are studying * Easy-to-understand information, perfect for pre-test review
If you want top grades and a thorough understanding of linear algebra, this powerful study tool is the best tutor you can have!
About the Author
Seymour Lipschutz, Ph.D. (Philadelphia, PA), is presently on the Mathematics faculty at Temple Univeristy. He has written more than 15 Schaum's Outlines. Marc Lipson, Ph.D. (Philadelphia, PA), is on the mathematical faculty of the University of Georgia. He is co-author of Schaum's Outline of Discrete Mathematics.
--This text refers to an out of print or unavailable edition of this title.
Top Customer Reviews
This book is an incredible value! It covers all major topics in the field of linear algebra, in fact, many more topics than some more expensive texts. This book has many theorems with proofs as well as example problems and applications. My only complaints about this book are that the coverage of theory is a bit terse, and the proofs to many of the theorems are found in the exercises section at the end of the chapter, rather than immediately after the theorem. The terseness means that, without prior exposure to linear algebra, it would be a bit difficult to teach yourself strictly from this book. But with a little bit of background, this book should be no trouble at all. A "real" textbook would get a lower rating for these types of complaints, but since this book costs less than $13, there is no reason not to get it.
I have learned more about Linear Algebra from this single book than all my other assorted (and fairly expensive!) texbooks combined; and, for that matter, more than my college class on the subject too. I'm now studying Differential Geometry and I always have this book nearby (in addition to two other Shaum's Outlines: that of Vector Analysis and Differential Geometry). Every textbook has something to offer - at least I usually give them the benefit of the doubt ;-). But for gaining a "working knowledge" of abstract mathematical concepts, there's nothing so enlightening than studying the details of problem solving performed with delightful clarity by an expert in the field. In this book you can become the apprentice to Seymour Lipschutz as he reveals the mysteries that often form clouds in the minds of students of Linear Algebra. Frequently, when working in this text, I would suddenly understand something that seemed so complex or obscure in my other LA texts. Actually, I could then see what those other texts were attempting to convey but because of their particular styles, they were just a bit "pedagogically challenged" :-)
Anyway, for me, this book is worth more than any pricey textbook on the subject. Just buy it and use it! You'll begin to see how cool Linear Algebra really is. Beware - mathematics at this level and beyond can be very addictive!
i'm currently taking linear algebra from one of the authors of this book. the first day of class the instructor passed out a corrections sheet for the book. even with the corrections sheet, i found many wrong answers in the answer sections. the pros - low cost compared to a normal textbook - easy to understand explanations - concise - lots of problems the cons - many, many wrong answers in the answer key means that you'll spend a lot of time checking your math with a calculator.
This Schaum's outline is particularly good for upper level undergraduate students that are learning about applied linear algebra. If you are a student of pure mathematics, or a graduate student taking numerical linear algebra this is probably not the book for you. The book excels particularly at students trying to learn techniques that will help them solve the kind of linear algebra problems that you run into when learning computer graphics programming techniques - dot products, cross products, the intersection of two planes, the equation of a line perpendicular to a plane, the equation of a plane given two lines that form a plane, etc. The book starts at the beginning equating vectors with matrices. It then goes on to show the reader the algebra of matrices, including how to know if a matrix is invertible and what it means if it is not. It also covers solution of linear systems of equations, determinants, a little about LU decomposition, eigenvalues, and eigenvectors. Throughout the book the geometric implications of operations are stressed. The one negative point of this book is that there are a moderate number of typos. Many are in places where it is obvious what is meant, but some are in the solutions to the solved problems and could cause problems for a novice to the subject.
For graduate students looking for a Schaum's outline that is more about large systems of matrices and the techniques and algorithms that are used on them, I suggest you look at "Schaum's Outline of Matrix Operations" by Bronson.
First of all I would like to clarify that there is one more book in schaum's series on linear algebra by one of these authors and is called '3000 solved problems in linear algebra'. Dont get confused, I havent read that book. This book is an excellent book for both theory and problems. It has numerous solved examples ...and moreover the examples are very cleverly chosen. After this you have hell lot of problems of varied difficulties to test yourself. The best point is that hints are given for some important and difficult problems! Only trouble I got during my undergraduate was that I found that this book was a bit to elementry for a bit advanced topics like matrix diagonalization and other things. For that one has to get some other advanced book ....like that by Hoffman. Second deficiency is, probably because this book intends to be mainly a problem book, it doesnt introduce the student to dazling applications (!) of linear algebra ... linear algebra is an extremely nice portion of mathematics and it has equally amazing applications. I feel in undergraduate courses student should get an exposure of its applications also. The best example of that being the Quantum Mechanics. So some science/math students might go for a good quantum mechanics book after going through this excellent problems book. (one on Heisenberg's apporach and not Schroodinger's) Best of Luck Mukul | 677.169 | 1 |
School maths fails to prepare future physicists and engineers
Many physics and engineering academics feel that new undergraduates in their subjects are entering university ill-prepared for their courses, and not achieving their full potential, because of a lack of fluency in maths.
A new report, Mind the Gap: Mathematics and the transition from A-levels to physics and engineering degrees, prepared for the Institute of Physics (IOP) by EdComs, suggests that exams and specifications have weakened the crucial relationship between maths and the physical sciences.
Gathering the opinions of both physics and engineering academics, and first- and second-year undergraduates in physics, engineering and computer science, the report highlights many academics' belief that current maths and physics provision at A-level leads to students learning by rote rather than developing their own independent techniques.
As one engineering academic said in interview, "Deep down, the problem is, mathematics is a language that they don't speak because they are not taught to speak it…. You can imagine when you present physics material, which is all equations, they just go bonkers.
"You need to have competence in mathematics to explain the concepts. They say the equations are so difficult but they don't get the point that it is not the equations that are difficult; it is the concept that is difficult. You can harness extremely complicated concepts into one equation, this is the power of mathematics. They don't seem to get that because they are not being taught in that way."
A physics student, also interviewed for the report, agreed, "The lack of any proper maths at A-level physics meant that I felt quite overwhelmed and had to learn the skill of deriving physical meaning from maths, something I'm still having to pick up on."
There was close to unanimous agreement, as 92% of the academics contacted agreed, that the lack of fluency in maths would have a detrimental effect on the prospects of the young physical scientists.
A physics academic said, "If they haven't really got to that level of fluency of understanding what somebody else is writing, let alone writing it themselves, yes, they are at a serious disadvantage."
The report is based on an online survey of around 400 undergraduates and 40 academic physicists and engineers, along with a series of one-to-one interviews with academics and physics and engineering undergraduates.
More than half of the academics contacted asserted that their first year undergraduates were not very/not at all well prepared to cope with the maths content of their degrees and, although only a fifth of the students felt mathematically ill-prepared for their courses, many of the students' comments from interviews acknowledged a gulf between the maths they were taught at school and their degree's requirements.
One engineering academic said, "They don't usually admit that they've got a problem. They don't quite understand what problem they've got. They know they are not quite understanding it but they can't pinpoint where the problem lies."
A common complaint from both academics and students was the treatment of maths and physics in school as two distinct subjects; there is seen to be minimal crossover in terms of syllabus content, when in reality there should be a great deal.
Elizabeth Swinbank, Chair of the IOP's Maths in Physics working group, said, "The Institute of Physics has, for some time, been concerned that physics and mathematics A-levels do not provide sufficient mathematical preparation for those who continue with their physics at university either within a physics degree or in other cognate disciplines.
"It established a 'Maths in Physics' working group to review this concern and to establish some hard evidence relating to it."
Philip Diamond, Associate Director of Education and Planning at IOP, said, "The Institute will be discussing the implications of the report with the Government and the examination agencies."
In response to the publication of the report, the Schools Minister Nick Gibb commented, "We need to ensure that our curriculum and qualifications are robust and rigorous and that they keep pace with the demands of employers and universities. This research reflects widespread concerns that A levels are still not preparing students sufficiently well for the study of a science degree course at university, with insufficient maths preparation in science in particular.
"We will look at this report carefully but our reforms to date are designed to address some of these very serious concerns. We are overhauling the National Curriculum so teaching focuses on the core, essential knowledge that students need for further study. We will set out proposals shortly to put universities at the heart of developing A levels in the future. And we want to attract the brightest and best science and maths graduates into teaching with bursaries of up to £20,000 – to inspire future generations of undergraduates."
Our 48 special-interest groups are at the heart of the physics community. We caught up with the Nuclear Industry Group earlier this year to learn out more about what they do. Find out more at iop.org/groups | 677.169 | 1 |
MEN OP MATHEMATICS
Pz-Q
the second of which is to be applied to the x in the first. We get
qS)z -r W -r qS)
~~ (rP -r sR)z -r (rQ - sS) '
Attending only to the coefficients in the three transformations
we write them in square arrays, thus
\P
and see that the result of performing the first two transforma-
tions successively could have been written down by the follow-
ing rule of 'multiplication'.
\P
r
" \\R SJJ jjrP + sR rQ + sS
where the rows of the array on the right are obtained, in an
obvious way, by applying the roars of the first array on the left
onto the columns of the second. Such arrays (of any number of
rows and columns) are called matrices. Their algebra follows
from a few simple postulates, of which we need cite only the
following. The matrices
are equal (by
B|
*[|"»[IC D\
definition) when} and only when, a = A, b = B, c = C,d = D.
The sum of the two matrices just written is the matrix
a b
C d-t
The result of multiplying
i mb
by m
c d
u
The rule for 'multi-
(any number] is the matrix {(
ijmo /rtiijj
plying', x, (or 'compounding') matrices is as exemplified for
p q\ JiP Q ,
!» ;« «; above,
r Sj {{12 5^
A distinctive feature of these rules is that multiplication is
not commutative^ except for special kinds of matrices. For
example, by the rule we get
3 _
Pq + Qs
412 | 677.169 | 1 |
MATH2021
Introduction to Applied Mathematics
Section 2 : Oscillations (Part 2)
1
MATH2021
The pendulum
A bob (mass) able to swing on a rod (pic. from wikipedia)
Balance of the forces due to motion and gravity tells us
that (t) satisfies the differential e
MATH2021
Introduction to Applied Mathematics
Section 1 continued : Stability and bifurcation
1
MATH2021
Recall from last week
Concept of a model
Succinct description in the language of mathematics
What is important and what isnt important?
Variables and p
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2021 Introduction to Applied Mathematics 2017
Laboratory 1
Questions marked ? are more difficult, perhaps conceptually or algebraically.
1. How might we deduce (an accurate appro
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1720 Mathematics Fundamentals
Semester 2, 2015
Solutions to Problem Set 3a
B. Modelling using Simultaneous Equations
In each case, formulate an appropriate set of simultaneous eq
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1720 Mathematics Fundamentals
Semester 2, 2015
Problem Set 6 Solutions
A. Evaluation using indices
A. Evaluation using indices
1. Evaluate each of the following expressions.
1
1
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1720 Mathematics Fundamentals
Semester 2, 2015
Problem Set 7
A. Logarithmic and exponential forms
1. Write the following logarithmic equations in exponential form.
1
(a) log10
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH0700 Preparatory Mathematics
Practice Test 10700 Preparatory Mathematics
Practice Test 21720 Mathematics Fundamentals
Semester 2, 2015
Problem Set 5 Solutions
There are often several different (but equally efficient) ways to solve index problems. Only one version is
The University of Western Australia
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1720 Mathematics Fundamentals
Semester 2, 2015
Solutions to Problem Set 2a
Algebra: solving equations
(a)
2x 5
=
3 (Note that its easier to remove the 5 first rather than the 2) | 677.169 | 1 |
Foundations of Mathematics and Pre-Calculus 10
Introduction:
This course introduces students to the mathematical understandings and critical-thinking skills further developed in the Foundations of Mathematics and Pre-calculus pathways. Topics include algebra, measurement, number and relations and functions. The seven mathematical processes (communication, connections, mental mathematics and estimation, problem solving, technology and visualization) are interwoven throughout the mathematical topics.:.
Course Layout:
Topic
Learning Activities
(% of Learning Activities)
Course Completion Milestones
Unit 1
Real Numbers & Radicals
Unit Lessons (2)
Practice Activities (2)
Unit 1--Send-In Assignment (6)
Unit 1—10%
Unit 2
Measurement
Unit Lessons (2)
Practice Activities (2)
Unit 2--Send-in Assignment (6)
Module 1 Exam Units 1/2 (5)
Unit 2--15%
Unit 3
Exponents
Unit Lessons (2)
Practice Activities (2)
Unit 3--Send-in Assignment (6)
Unit 3--10%
Unit 4
Polynomials
Unit Lessons (2)
Practice Activities (2)
Unit 4--Send-in Assignment (6)
Module 2 Exam Units 3/4 (5)
Unit 4--15%
Unit 5
Linear Systems
Unit Lessons (2)
Practice Activities (2)
Unit 5--Send-in Assignment (6)
Unit 5--10%
Unit 6
Linear Relations
Unit Lessons (2)
Practice Activities (2)
Unit 6--Send-in Assignment (6)
Module 3 Exam Units 5/6 (5)
Unit 6--15%
Unit 7
Linear System
Unit Lessons (2)
Practice Activities (2)
Unit 7--Send-in Assignment (6)
Unit --10%
Unit 8
Right Angled Trigonometry
Unit Lessons (2)
Practice Activities (2)
Unit 8--Send-in Assignment (6)
Module 4 Exam Units 7/8 (5)
Unit 8--15 40%
Exams: 60 | 677.169 | 1 |
Prentice Hall
Supplemental materials are not guaranteed for used textbooks or rentals (access codes, DVDs, workbooks).
For courses in undergraduate Combinatorics for juniors or seniors. This carefully crafted text emphasizes applications and problem solving. It is divided into 4 parts. Part I introduces basic tools of combinatorics, Part II discusses advanced tools, Part III covers the existence problem, and Part IV deals with combinatorial optimization | 677.169 | 1 |
Three streams are available: a Specialist Stream, Mathematics, and a Further Stream. Each is designed to prepare students for further study of Mathematics during the VCE years. At the start of the year, each student is placed in a course based on their level of performance during the Year 9 Mathematics course.
Course Objectives
Students will be able to:
Show understanding of the fundamental concepts involved in each topic.
Make appropriate use of technology, including scientific and computer algebra system (CAS) calculators.
Apply mathematical skills to practical situations.
Content – Specialist Stream
This course is for able students of Mathematics. Students who are accelerating in VCE Mathematical Methods 1 & 2 are expected to take this concurrently with their VCE study.
The topics of Geometry, Rational and Irrational Numbers, Indices and Logarithms, Linear Relationships, Quadratic Relationships, Trigonometry, Probability, Statistics, Polynomials and Measurement are studied. Students will also be shown how to use a CAS calculator and given extension work where appropriate.
Content – Mathematics
This course is for students who are of standard Year 10 ability in Mathematics. It covers a broad range of topics necessary for preparation for Units 1 & 2 of VCE Mathematical Methods or VCE General Mathematics.
The topics of Geometry, Financial Mathematics, Algebra, Linear Relationships, Non-Linear Relationships, Trigonometry, Probability, Statistics, and Measurement are studied. Students will also be shown how to use a CAS calculator.
Content – Further Stream
This course is for students who find Mathematics difficult. It does not cover the breadth of the other two Year 10 Mathematics courses and students are not required to study topics to the same depth as in the other courses. The course will sufficiently prepare students who are planning to study Units 1 & 2 VCE General Mathematics.
Topics covered are Linear Relationships, Measurement, Trigonometry, Algebra, Statistics, Geometry, Probability and Financial Mathematics. Students will also be shown how to use a CAS calculator. | 677.169 | 1 |
Description:
About this title:
Synopsis: The Big Ideas Math student edition Algebra 1, Geometry and Algebra 2 textbooks mirror the pedagogical philosophy that made the middle school Big Ideas Math books so successful. Each lesson begins with an Essential Question, followed by Explorations. Once the inquiry section is completed, students begin the direct instruction Lesson, helping them to reason and make sense of their answers based on the knowledge they gained during discovery.
Book Description Big Ideas Learning. Hardcover. Book Condition: Very Good. 160840840X very good condition, pages are clean and free of markings, minimal wear to corners and edges, ships same day or next. Bookseller Inventory # 425000025
Book Description HOUGHTON MIFFLIN HARCOURT, 2014 160840840X-VG
Book Description Big Ideas Learning, 2014. Book Condition: Good. 1st Edition. N/A. Ships from Reno, NV. Former Library book. Shows some signs of wear, and may have some markings on the inside. Bookseller Inventory # GRP95484898
Book Description HOUGHTON MIFFLIN HARCOURT. Hardcover. Book Condition: Very Good. 160840840X160840840XZ2 160840840XGOA | 677.169 | 1 |
The best way to describe Algebra is to say it's a way to traverse a mathematical puzzle even if you do not have all the pieces. This year we'll use algebra to solve an equation, define a function, and simplify a complicated expression.
Video Support
Home
Welcome to Mr. Graves Website.
I teach Math here at East. This year I teach Geometry, Algebra 2 Concepts, Pre-Calculus and Calculus. My goal for this website is to provide calendar information, contact information and forms. Education is a dynamic process which makes any website a work in progress. Please feel free to send me feedback at any time.
Graves
ISD 709 Staff: Two tabs will appear if you are logged into google and have permissions: Staff Help and Teacher Workshop. Email me if you need help. | 677.169 | 1 |
Precalculus: Enhanced with Graphing Utilities (5th Edition)
Author:Michael Sullivan - Michael Sullivan III
ISBN 13:9780136015789
ISBN 10:136015786
Edition:5
Publisher:Pearson
Publication Date:2008-02-14
Format:Hardcover
Pages:1200
List Price:$198.67
 
 
These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilities gives students a model for success in mathematics. | 677.169 | 1 |
MATH Documents
Showing 1 to 30 of 2,516
VECTOR OPERATIONS
MATH23-1
CALCULUS 3
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Perform dot product and cross product.
Apply the concepts of dot product to get the angle between
two vectors.
Apply cross products in variou
Confidence Intervals and
Sample Size
McGraw-Hill, Bluman, 7th ed., Chapter 7
1
Overview
Introduction
Confidence Intervals for the Mean When
Is Known and Sample Size
Confidence Intervals for the Mean When
Is Unknown
Confidence Intervals and Sample Size for
VECTOR OPERATIONS
MATH23
MULTIVARIABLE CALCULUS
Week 2 Day 2
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Perform dot product and cross product
Apply the concepts of dot product to get the angle between
two vectors
Apply cro
4/29/2016
IntroductiontoQuiz:Aerodynamics|Coursera
Introduction to Quiz: Aerodynamics
This document contains all the information you need to solve the quiz called 'Aerodynamics'. Please calculate your
answers to the questions at the bottom of this page. Y
Wind Turbine Terminology and Components
Morten Hartvig Hansen
Learning objectives
After this lecture you should be able to:
List the main components of the commercial horizontal-axis wind
turbine
List the main degrees of freedom of the commercial horizo
disp("Made by:")
disp("Correia, Feliciana Maria")
disp("Reyes, Carl Samuel")
disp("This function will solve for the roots of your given system of linear equations.")
disp("Please enter the coefficients in matrix form. Denote also how many equations you ha
disp("This program shall compute the number of total seats in a given place.")
disp("Please input the number of seats on the last row, number of rows and the difference per row respectively in matrix form.")
function [seats, first] = series(x,y,z)
first =
disp("Made by:")
disp("Correia, Feliciana Maria")
disp("Reyes, Carl Samuel")
disp("This program shall compute the equation of the line using method of least squares.")
disp("Please input points in two separate matrices. X first and then Y and the number o
disp("This function will solve for the area of the base of a cylinder given any height and lateral surface area.")
disp("Enter your lateral surface area and height respectively. Separate using commas and brackets ")
function [Area, radius] = cylinder(x, y
LIMITS
OF
FUNCTIONS
DEFINITION: LIMITS
The most basic use of limits is to describe how a
function behaves as the independent variable
approaches a given value. For example let us
2
examine the behavior of the function f ( x ) x x 1
for x-values closer and
TOPIC
LENGTH OF AN ARC
and
AREA OF SURFACE OF REVOLUTION
LENGTHS OF CURVES
To find the length of the arc of the curve y = f (x) between x =
a and x = b let ds be the length of a small element of arc so
that:
2
ds
dy
2
2
2
ds dx dy thus 1
dx
dx
LENGT
DIFFERENTIAL CALCULUS
Linearization and Differentials
MATH21: DIFFERENTIAL CALCULUS
LINEARIZATION
As
you can see in Figure 3.50, the tangent to the curve
lies close to the curve near the point of tangency. For a
brief interval to either side, the -values
MATH23X EXIT EXAM REVIEW MATERIAL
TRIGONOMETRY
1. The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of
100 meters from its base is 45 degrees. If the angle of elevation of the top of the complete
pillar at the sam
COURSE OUTCOME 3
ANTIDERIVATIVES
OBJECTIVES:
At the end of the lesson the students are
expected to:
know the relationship between differentiation
and integration;
identify and explain the different parts of the
integral operation; and
perform basic integr | 677.169 | 1 |
Software Listing: Physics
The perfect solution for physics students and teachers everywhere, Physics 101 SE is the premier physics calculation tool, allowing you to focus on physics and not mathematical busywork by working with over 125+ equations and other features such as:
Physics games bring a lot of fun to us, and our mission is to gather the best of them in one place to help you find them. Here at phyfun.com, online flash games are mostly for free and they can be a lot of fun, not to mention totally captivating and addictive. Play online physics games at phyfun.com, we also provide free solution, online faqs, cheats, walkthroughs for review. Welcome to us and enjoy!. Physics Games - Free Physics-based Games on Phyfun.com. Physics Games are games based on physics engine. New physics-based games with video walkthrough, solution and cheats added every day..
Crayon Physics Deluxe lets you draw objects on the screen by clicking and dragging your mouse, or by drawing with the stylus of a tablet PC, as in this video. The objects you scrawl become part of the game world. The goal is to create objects that propel a crudely drawn ball toward a crudely drawn star. There is no single correct way to scoot that ball around; the fun is in exploring the options. Within seconds of hitting start, you're furiously scribbling blocks and ramps and wedges and seesaws, whatever it takes to reach the goal. Some players may get sidetracked creating hilariously inefficient Rube Goldberg devices.
Examples used in the December 2010 webinar titled "Teaching Math and Physics with Symbolic Math Toolbox". View the recorded webinar here:
Topics include:
* Using symbolic computation for common tasks such as solving, simplifying, and plotting equations, and performing other calculations such as derivatives, integrals, limits, and inverses
* Creating interactive animations to demonstrate concepts during class
* Developing curriculum materials and homework assignments using the MuPAD notebook interface.
Designed for physicists, this book covers a variety of numerical and analytical techniques used in physics that are applied to solving realistic problems.
For a full book description and ordering information, please refer to
MATLAB in Physics is four lecture series in MATLAB that is offered to first year physics undergraduate students.
The aim of this lecture series is to introduce students to computational methods in MATLAB to solve problems arising in physics that cannot be solved analytically. No prior knowledge of MATLAB is required. Themes of data analysis, visualization, modeling and programming are explored throughout these lectures..
OBJ To Convex Physics is a lightweight and easy to use application designed to help you convert geometry OBJ files into convex meshes, which you can use in physics libraries.
OBJ To Convex Physics only runs using the command prompt and it won't pose any problem to those who are familiar with the console. Just specify the input and output file names and let the program do the rest.
Virtual physics is the collection of programs simulating physical phenomena. They can be used as demonstration tools at school or for individual studies and experiments at home.
See what you can never see in nature!
The most exiting feature of Virtual Physics is that you can perform and observe many experiments you can never see either in nature or in the school laboratory, eg. stars moving on their orbits or the motion of the molecules of gas.
Preparing an experiment in a virtual laboratory you can start with the sample settings supplied with the program or set the parameters from scratches
Then you can run the simulations and observe the result in the windows.
Optical Physics Experiments Analyzer is a handy, Java based interactive user interface designed to demonstrate the use of free technologies in the development of tools to aid the learning of Physics. .
An Intuitively Easy-To-Use collection of -at least- 142 interactive Physics simulation modules that allow the user to simulate and visualize Physics experiments. Useful from High School to University. PVL may be used (1) during classes, to support the teacher's explanation and to make classes more interesting, (2) in the library by students, to review the topics covered in classes (3) at home by students to run experiments with the student's data, even extreme experiments, which result in a better grasping of studied topics. The teacher may also use the PVL to prepare classes and suggest experiments.
This is a command line based Physics Simulator that can export data to a spreadsheet program. Awesim Physics can be used to simulate everything from a ball being thrown upwards to the obits of planets in a solar system..Easy Tweaker is one of the many thousands of applications out there aiming to assist frustrated Windows users with their slow, sluggish and bogged up systems. Easy Tweaker is more than just one application however. It contains a number of different | 677.169 | 1 |
Math 6
All students study the mathematical strands recommended by the California Framework: Number, Measurement, Geometry, Patterns and Functions, Statistics and Probability, Logic and Language, and Algebra. Focus is placed on problem solving, conceptual understanding, building connections throughout the strands, and developing computational skills. Students solve problems, communicate ideas in both oral and written form, analyze and organize information, evaluate alternative mathematical approaches, and interpret results. Students use mental math, estimation, paper and pencil computation, manipulatives, calculators, computers, and other technology as appropriate. Students are heterogeneously grouped in sixth grade and work both independently and in cooperative group situations. Instruction is differentiated to appropriately challenge students. Extra challenge and enrichment opportunities are available to all students. The curricular materials for math include The Connected Mathematics Series (published by Pearson Prentice Hall) and Concepts and Skills Course 1 (published by McDougal Littell)
Math 7:
Students will follow the 7th grade Common Core Content Standards. Seventh grade mathematics focusses on the application of ratios and proportional reasoning. Students develop an understanding of and apply proportional relationships. They develop a further understanding of operations with rational numbers and work with expressions and linear operations. Students solve problems involving scale drawings and informal geometric constructions. They work with two- and three-dimensional shapes to solve problems involving area, surface area, and volume. They draw inferences about populations based on samples. Students will solve problems, communicate ideas in both oral and written form, analyze and organize information, evaluate alternative mathematical approaches, and interpret results. The Mathematical Practice Standards will be used to build on the student's ability to make sense of a variety of problems.
Math7A:
This is the first year in a two-year accelerated course. Students will cover all of 7th grade Common Core standards, with the addition of some Standards for 8th grade, allowing students to cover 3 years of material in 2 years (7th and 8th grade). These additions from 8th grade include having students develop the concept of a function and use functions to describe quantitative relationships as well as analyze two- and three-dimensional space and figures using distance, angles, similarity and congruence. They grasp the concept of a function and use functions to describe quantitative relationships. Students learn and apply the Pythagorean Theorem. The Mathematical Practice Standards will be used to build on the student's ability to make sense of a variety of problems. Students are expected to maintain a B- or better for placement into the Algebra 8 course in 8th grade.
Math 8:
Students will follow the 8th grade Common Core Standards. Eighth grade mathematics focusses on the algebra of linear relationships. Students formulate and reason about expressions and equations, including modeling an association in bivariate data with a linear expression and solving linear equations and systems of equations. They grasp the concept of a function and use functions to describe quantitative relationships. Students analyze two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and learn and apply the Pythagorean Theorem. Students continue to use and improve their skills in problem?solving, communicating mathematical ideas in oral and written form, analyzing complex situations, organizing information, using computers and graphing calculators, and working independently and in groups. The Mathematical Practice Standards will be used to build on the student's ability to make sense of a variety of problems.
Algebra 8:
This is the second year in a 2-year accelerated course. This course completes the 8th grade Common Core Content Standards in addition to the Algebra Content Standards. Students deepen and extend their understanding of linear and exponential relationships by contrasting them with each other and by applying models to data. Students engage in methods for analyzing, solving, and using quadratic functions. Students analyze two- and three-dimensional space and figures using distance, angle, similarity, and congruence. In addition to analyzing bivariate data, students will also focus on data distribution and correlation. The focus is on solving linear, quadratic and exponential functions. The Mathematical Practice Standards will be used to build on the student's ability to make sense of a variety of problems. Students are expected to maintain a B- orbetter for placement into Geometry in 9th grade. Pre-requisite: Math 7A or satisfactory completion of the Summer Bridge to Algebra Course.
Math Placement in the Secondary Grades
AMC8
Open to all students, held during WIP (runs a few minutes after school). Stay tuned to KJLS and math classrooms for information and sign-ups. Sign-ups take place about 2-3 weeks prior to the competition. Held in November. Sponsored by the JLS Math Teachers
AMC10
By invitation only. Open to the top 10 scores from the AMC8 competition. Held in Feb. Sponsor: Elizabeth Fee
Math Madness
Club and weekly competitions against other middle schools around the country. Computer based and sponsored by AMC. Open to all students. Must sign up prior to competing. Sponsor is Elizabeth Fee.
Math Counts
10 students compete live against other schools. Try outs take place in December. There are weekend obligations with this competition. Stay tuned to KJLS for information.
BAMO - Bay Area Math Olympiad
Bay Area Math Olympiad. Students have up to 4 hours to work through a set of questions. Competition takes place on a Thursday in Feb. Stay tuned to KJLS for information. | 677.169 | 1 |
Includes: Guided Notes that build a deep understanding of each concept, and plenty of practice problems. A wide variety of question types are included in each lesson to ensure that students are able to think about the concepts on multiple levels | 677.169 | 1 |
Function Family Tree Lesson Plan
Be sure that you have an application to open
this file type before downloading and/or purchasing.
26 KB|3 pages
Product Description
This lesson plan is designed to be a series of lesson that teach students about parent functions and different characteristics of specific types of functions. The lesson plan ends with a performance assessment in which students create a photo album or portfolio that represents each "family" of functions and their branches. This is ideal for Algebra 2 or PreCalculus students, but could be adapted for Algebra 1 and/or Geometry. | 677.169 | 1 |
Math 1350 Mathematics for Teachers I Information
Catalog Description This course is intended to build or reinforce a foundation in fundamental mathematics concepts and skills. It includes the conceptual development of the following: sets, functions, numeration systems, number theory, and properties of the various number systems with an emphasis on problem solving and critical thinking.
Prerequistes MATH 1314 OR placement by testing;
College level readiness in reading and writing
Required Materials
Textbook: Billstein, Libeskind, Lott; A Problem Solving Approach to Math for Elementary Teachers, 12th ed.; Pearson Required: Students must buy an access code to MyMathLab, an online course management system which includes a complete eBook; students will first need a Course ID provided by the instructor in order to register; online purchase of MyMathLab access at hard copies of access codes available with ISBN: 9780321199911 Hardbound text (optional), ISBN: 9780321987297 Hardbound text + free MyMathLab access, ISBN: 9780321990594
Calculator:
Graphing calculators may be required for some assignments/assessments at the discretion of the instructor. TI 83, TI 84 or TI 86 series calculators recommended.
Calculators capable of symbolic manipulation will not be allowed on tests. Examples include, but are not limited to, TI 89, TI 92, and Nspire CAS models and HP 48 models.
Neither cell phones nor PDA's can be used as calculators. Calculators may be cleared before tests. | 677.169 | 1 |
Algebra III
September 11, 2015Uncategorized
Grading – Homework: Each student will start each 9 weeks off with a 100 homework average. Each time the student does not attempt the assignment they will lose 5 points. This homework grade will count TWICE in the 9 weeks average. – Graded Assignment: The assignments will be given after a lesson has been taught and gone over. These will be done with NO assistance in compliance of the honor pledge. Each will count ONCE in the 9 weeks average. – Quizzes: They will be given after lessons have been taught and homework also given. They will be announced. Each will count ONCE in the 9 weeks average. – Tests: They will cover several quizzes. They will also be announced. Each will count TWICE in the 9 weeks average.
The link to the curriculum guide is:
9/8 &9/9: Rules and Procedures Numbers and their Properties (Translating and Order of Operations) Homework: Worksheet and get rules and procedures sheet signed | 677.169 | 1 |
Mathemat. Paperback. Condizione libro: new. BRAND NEW, The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965-1996, Anthony Gardiner, B9780198501053
Descrizione libro Paperback. Condizione libro: New. Not Signed; obviou. book. Codice libro della libreria ria9780198501053_rkm | 677.169 | 1 |
Pearson Education Mymathlab Program Review
Mymathlab is loved by teachers- that is lazy teachers who dont want to actually do their job.If your professor uses MML let them know exactly how worthless of a teacher they are.
They love it because they dont have to create homework or grade. MML does all the work for them. Meanwhile its the students who suffer. You WILL spend HOURS on homework each day because of this program.
You will have the correct answer but accidentally forget a decimal or a negative sign or something *** and- even though you do know the correct answer and have the correct answer written down on your scratch paper- it will mark you wrong and make you restart. It will take hours to complete about 20 questions. Hours that are precious to hard working students- especially college students. All because the professor is lazy and would rather be working on their own research projects than TEACH.
Also, it costs about 100 dollars for a year subscription. Which is useless because math classes last one semester. And if you buy it and use the access code and activate it you cannot get a refund under any circumstances. I once tried to get a refund and emailed Pearson about it and never got a response from the MML customer service.
In summary, MML eats your time and will have you throwing your computer out the window.
Having to type in math answers leads to SIGNIFICANT error in your answers as opposed to just writing out your answer on paper and working out a problem that way.Professors need to stop being lazy and write out their own homework and grade by hand! | 677.169 | 1 |
Topic outline
Introduction
Welcome to Applied Algebra! This page is designed to help parents and students find important resources such as digital book access, course syllabus, lesson plans and video resources. This course will cover chapters 1, 3, 4, 5, 6, 7, 8, 9, and 10.
Radicals
Course Contacts
Teacher
Students must have successfully completed Pre-Algebra or Algebra to enroll in Applied Algebra. Any student who has completed Geometry or Higher Algebra must check with the Applied Algebra instructor before enrolling in these classes. Videos demonstrating math in the work place, hands-on labs, and application problems will be used during each course. Students will use problem solving skills for application in Agriculture and Agribusiness; Business and Marketing; Home Economics; Health Occupations; and Industrial Technology. Topics that will be covered include formulas, solving equations, nonlinear functions, graphing data and inequalities, probability and statistics. | 677.169 | 1 |
This book provides mathematics teachers with an elementary introduction to matrix algebra and its uses in formulating and solving practical problems, solving systems of linear equations, representing combinations of affine (including linear) transformations of the plane and modelling finite state Markov chains. | 677.169 | 1 |
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this file type before downloading and/or purchasing.
1 MB|7 pages
Product Description
Algebra that Functions
Linear Functions Math Project Summer Jobs
This math project is a fun way for your students to apply mathematical reasoning of linear functions in a real-world situation that is important to them! Your junior high or high school students will enjoy making choices about summer jobs based on their personal needs and wishes. This engaging and relevant task is the perfect end of year math project. A rubric and a peer evaluation form is included for assessing the project. My students absolutely love this math project!
This end of year math project is a great way to end the school
year for your students and for you!
This product is a paid digital download from my TpT store Algebra that Functions and it is for use in one classroom only. This product*Math Project
*Writing Linear Equations
*Solving Linear Equations
*Word Problems
*Real World Application
Go to your My Purchases page (you need to login). Beside each purchase you'll see a Provide Feedback button.
Be the first to know about my sales, freebies and product launches! CLICK ON THE GREEN STAR next to my store logo to become a follower. You will receive updates about my store. | 677.169 | 1 |
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Unformatted text preview: Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Practice problems for the final Important notes • The final will be 2 hours long . • The final will be a closed-book exam . • All of the course material will be covered (except parts that were explicitly described as enrichment material, e.g., the proof that e is irrational), with an emphasis on material covered after the midterm. • The practice problems in this problem set are designed to aid you in studying and reviewing important parts of the course material. They are designed to be in some cases slightly more difficult, and to take longer to solve, than actual exam questions. • The practice problems do not cover all the course topics. Topics that are not covered by this problem set may still appear on the final! • Solutions to this problem set will be posted on the course web page later this week. (An email announcement will be sent.) • It is also recommended to go over the homework and its solutions (as well as the textbook and class notes) as preparation for the final. One final recommendation for the exam • When writing a proof, use words to explain the logic of what you are doing — don't just write formulas or equations (like you may be used to doing in other math classes). In other words, remember the following rules: Proof 6 = formulas Proof = formulas + explanation 1 Math 25: Advanced Calculus...
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced. | 677.169 | 1 |
97801304316aging the Mean Math Blues
"A supplemental book" for courses in Study Skills. This text is designed so that reluctant and anxious math students learn current and relevant cognitive therapy and math study skill techniques. A broad variety of strategies journaling, self-assessment, goal setting, math exercises, questionnaires, webbing, etc. are designed to actively assist the student in pushing past their individual barriers to master math. Along the way, basic math exercises are introduced so that students can practice newly learned skills | 677.169 | 1 |
Math 121: Topics for Exam I / 100 Name: The following instructions will appear on the exam: Show all of your work and justify your answers completely! You are being graded on the process and well as the answer. Make sure your work is clearly organized and legible. Leave your answers in exact form , this means √ 2 , π and fractions, not crazy decimals. You may leave your answers unsimplified. You may not use a calculator. Here is a (mostly?) complete list of topics that will appear on the first exam. (1) Sequences (a) least upper bounds and greatest lower bounds (b) monotonicity (c) limits: the definition and limit arithmetic
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This note was uploaded on 11/16/2011 for the course MUSIC Music 101 taught by Professor Veroni during the Spring '11 term at Antelope Valley College. | 677.169 | 1 |
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
Practical Mathematics 0: Beginning with Numbers
Learn Natural Integers by Practice
3.3 only a computer and a way to download the free open source software Scilab
Description
That math course is an introduction to the first tool for mathematics: the simpler numbers that are the natural integers. These are the numbers we use to count, and they are introduced in a progressive way, beginning with counting on our fingers, and ending with the biggest numbers we can imagine. By the end of that course, you will be able to manipulate the natural integers through they concrete representations, the decimal and the exponential notations. You will also be ready for the rest of our series about Practical Mathematics, beginning with ''Practical Mathematics I: The Numbers''. To enter that course, you will need only the knowledge of a child of sixth degree. There are two main lectures, one with the beginning of the counting operation, and the second one that leads you to count ever further and that formalizes the addition.
Who is the target audience?
You need only the knowledge of a child of sith degree to enter the course
In that introductary lecture, the content of the course is announced, as well as the fact that it is a prequel of a series about "Practical Mathematics".
Introduction
02:30
+–
The Natural Integers
2 Lectures
23:15
We begin this mathematics course with the first mathematical
activity: counting. In mathematicians' wording, it mean exploring the
positive integers.
It is done progressively. We start from counting on the fingers to count the population of the World.
Scilab is used as a tool to count further than by heart.
About Counting
11:54
We count further and further in order to get all the natural integers,
including 0 and all the great numbers, even greater than we can imagine.
Then we define the addition by the means of counting process.
Counting Further and Adding
11:21
+–
Conclusion
2 Lectures
04:16
This is the recapitulative conclusion of the course, that opens to the rest of the series about Practical Mathematics.
Conclusion
02:01
This a sample of the Promo video, so that you may download it… and shareit!
She obtained the French highest degree to teach mathematics at undergraduate level. This means that she is very accurate in mathematics and in teaching them to anyone from the beginners to the undergraduate level.
She then turned to an Engineer's career for about 30 years.
She became an expert in R&D, especially when using applied mathematics and scientific programming in high level languages such as Matlab and Scilab.
She notably worked on satellites guidance, shuttle accosting and reentry .She applied her exoertise to implementing various complex algorithms such as Kalman filters and fuzzy logic.
After that, she worked in the railway industry, on automatic urban transportation systems for Paris and New York. After various R&D projects including error correcting Viterbi encoding and decoding as well as formal method based B language, she became an expert in safety analysis involving many specialized sharp inductive and deductive approach, including probability calculations.
She then founded Mathedu with her husband, a Researcher in Control Science.
Mathedu aims to teach mathematics from a practical point of view.
The idea is to let the students be in action with a very pragmatic approach, using its computer with Scilab installed as a laboratory.
Then and only then, the link with theory is done, in a very progressive way.
Learning maths with us will let you find the subject easy, so that you will no more understand why mathematics were so hard to understand before… | 677.169 | 1 |
gcse international mathematics extended standards
1.
IGCSE INTERNATIONAL MATHEMATICS –CORE / EXTENDED STANDARDSAims 1. acquire a foundation of mathematical skills appropriate to further study and continued learning in mathematics; 2. develop a foundation of mathematical skills and apply them to other subjects and to the real world; 3. develop methods of problem solving; 4. interpret mathematical results and understand their significance; 5. develop patience and persistence in solving problems; 6. develop a positive attitude towards mathematics which encourages enjoyment, fosters confidence and promotes enquiry and further learning; 7. appreciate the beauty and power of mathematics; 8. appreciate the difference between mathematical proof and pattern spotting; 9. appreciate the interdependence of different branches of mathematics and the links with other disciplines; 10. appreciate the international aspect of mathematics, its cultural and historical significance and its role in the real world; 11. read mathematics and communicate the subject in a variety of ways.Assessment objectives 1. know and apply concepts from all the aspects of mathematics listed in the specification; 2. apply combinations of mathematical skills and techniques to solve a problem; 3. solve a problem by investigation, analysis, the use of deductive skills and the application of an appropriate strategy; 4. recognise patterns and structures and so form generalisations; 5. draw logical conclusions from information and understand the significance of mathematical or statistical results; 6. use spatial relationships in solving problems; 7. use the concepts of mathematical modelling to describe a real-life situation and draw conclusions; 8. organise, interpret and present information in written, tabular, graphical and diagrammatic forms; 9. use statistical techniques to explore relationships in the real world; 10. communicate mathematical work using the correct mathematical notation and terminology, logical argument, diagrams and graphs; 11. make effective use of technology; 12. estimate and work to appropriate degrees of accuracy.Graphics calculator requirements Sketch a graph. Produce a table of values for a function. Find zeros and local maxima or minima of a function. Find the intersection point of two graphs. Find mean, median, quartiles. Find the linear regression equation.Curriculum content 1 Number – Core
2.
1.7 Equivalences between decimals, fractions, ratios and percentages 1.8 Percentages including applications such as interest and profit 1.9 Meaning of exponents (powers, indices) in w Standard Form a x 10n where 1 = a < 10 and n ?w Rules for exponents1 Number – Extended 1.6 Absolute value 1.7 Equivalences between decimals, fractions, ratios and percentages 1.8 Percentages including applications such as interest and profit 1.9 Meaning of exponents (powers, indices) in n Standard Form a x 10n where 1 = a < 10 and n ?w Rules for exponents 1.10 Surds (radicals), simplification of square root expressions Rationalisation of the denominator2 Algebra – Core curriculum 2.1 Writing, showing and interpretation of inequalities, including those on the real number line 2.2 Solution of simple linear inequalities 2.3 Solution of linear equations 2.4 Simple indices – multiplying and dividing 2.5 Derivation, rearrangement and evaluation of simple formulae 2.6 Solution of simultaneous linear equations in two variables 2.7 Expansion of brackets 2.8 Factorisation: common factor only 2.9 Algebraic fractions: simplification addition or subtraction of fractions with integer denominators multiplication or division of two simple fractions or a simple quadratic sequence2 Algebra – Extended curriculum 2.1 Writing, showing and interpretation of inequalities, including those on the real number line 2.2 Solution of linear inequalities Solution of inequalities using a graphics calculator 2.3 Solution of linear equations including those with fractional expressions 2.4 Indices 2.5 Derivation, rearrangement and evaluation of formulae 2.6 Solution of simultaneous linear equations in two variables 2.7 Expansion of brackets, including the square of a binomial
3.
2.8 Factorisation: common factor difference of squares trinomial four term 2.9 Algebraic fractions: simplification, including use of factorisation addition or subtraction of fractions with linear denominators multiplication or division and simplification of two fractions 2.10 Solution of quadratic equations: by factorisation using a graphics calculator using the quadratic formula, a quadratic sequence or a cubic sequence Identification of a simple geometric sequence and determination of its formula 2.13 Direct variation y ?x, y ? x2, y ? x3, y ? x Inverse variation y ? 1/x, y ? 1/x2, y ? 1/ x Best variation model for given data3 Functions – Core curriculum 3.1 Notation Domain and range Mapping diagrams 3.5 Understanding of the concept of asymptotes and graphical identification of simple examples parallel to the axes 3.6 Use of a graphics calculator to: sketch the graph of a function produce a table of values find zeros, local maxima or minima find the intersection of the graphs of functions 3.8 Description and identification, using the language of transformations, of the changes to the graph of y = f(x) when y = f(x) + k, y = f(x + k)3 Functions – Extended curriculum 3.1 Notation Domain and range Mapping diagrams 3.2 Recognition of the following function types from the shape of their graphs: linear f(x) = ax + b quadratic f(x) = ax2 + bx + c cubic f(x) = ax3 + bx2 + cx + d reciprocal f(x) = a/x exponential f(x) = ax with 0 < a < 1 or a > 1 absolute value f(x) = | ax + b | trigonometric f(x) = asin(bx); acos(bx); tanx 3.3 Determination of at most two of a, b, c or d in simple cases of 3.2 3.4 Finding the quadratic function given vertex and another point, x-intercepts and a point, vertex or x-intercepts with a = 1. 3.5 Understanding of the concept of asymptotes and graphical identification of examples 3.6 Use of a graphics calculator to: sketch the graph of a function produce a table of values find zeros, local maxima or minima find the intersection of the graphs of functions 3.7 Simplified formulae for expressions such as f(g(x)) where g(x) is a linear expression 3.8 Description and identification, using the language of transformations, of the changes to the graph of y = f(x) when y = f(x) + k, y = k f(x), y = f(x + k) 3.9 Inverse function f –1 3.10 Logarithmic function as the inverse of the exponential function y = ax equivalent to x = logay Rules for logarithms corresponding to rules for exponents Solution to ax = b as x = log b / log a.Curriculum content 4 Geometry – Core
4.
Angle sum of a triangle, quadrilateral and polygons Interior and exterior angles of a polygon Angles of regular polygons 4.5 Similarity Calculation of lengths of similar figures 4.6 Theorem of Pythagoras in two dimensions Including: chord length and its distance of a chord from the centre of a circle distances on a grid 4.7 Vocabulary of circles Properties of circles tangent perpendicular to radius at the point of contact tangents from a point angle in a semicircle4 Geometry – Extended Angle sum of a triangle, quadrilateral and polygons Interior and exterior angles of a polygon Angles of regular polygons 4.5 Similarity Calculation of lengths of similar figures Area and volume scale factors 4.6 Theorem of Pythagoras and its converse in two and three dimensions Including: chord length and its distance of a chord from the centre of a circle distances on a grid 4.7 Vocabulary of circles Properties of circles: tangent perpendicular to radius at the point of contact tangents from a point angle in a semicircle angles at the centre and at the circumference on the same arc cyclic quadrilateral5 Transformations in two dimensions – Core Curriculum 5.1 Notation: Directed line segment AB ; component form 5.4 Transformations on the cartesian plane translation, reflection, rotation, enlargement (reduction) Description of a translation using the Notation in 5.16 Mensuration – Core. 6.3 Circumference and area of a circle Arc length and area of sector 6.4 Surface area and volume of prism and pyramid (in particular, cuboid, cylinder and cone) Surface area and volume of sphere 6.5 Areas of compound shapes5 Transformations and vectors in two dimensions – Extended curriculum 5.1 Notation: Vector a; directed line segment AB ; component form 5.2 Addition of vectors using directed line segments or number pairs Negative of a vector, subtraction of vectors Multiplication of a vector by a scalar 5.3 Magnitude | a | 5.4 Transformations on the cartesian plane: translation, reflection, rotation, enlargement (reduction), stretch Description of a translation using the Notation in 5.1 5.5 Inverse of a transformation 5.6 Combined transformations6 Mensuration – Extended 6.3 Circumference and area of a circle Arc length and area of sector
5.
6.4 Surface area and volume of prism and pyramid (in particular, cuboid, cylinder and cone) Surface area and volume of sphere 6.5 Areas and volumes of compound shapes7 Co-ordinate geometry – Core lines 7.6 Equation of a straight line as y = mx + c or x = k 7.8 Symmetry of diagrams or graphs in the cartesian plane8 Trigonometry – Core curriculum 8.1 Right-angled triangle trigonometry 8.7 Applications: three-figure bearings and North, East, South, West problems in two dimensions compound shapes9 Sets – Core). 9.2 Sets in descriptive form { x | } or as a list 9.3 Venn diagrams with at most two sets 9.4 Intersection and union of sets7 Co-ordinate geometry – Extended and perpendicular lines 7.6 Equation of a straight line as y = mx + c and ax + by = d (a, b and d integer) 7.7 Linear inequalities on the cartesian plane 7.8 Symmetry of diagrams or graphs in the cartesian plane8 Trigonometry – Extended curriculum 8.1 Right-angled triangle trigonometry 8.2 Exact values for the trig ratios of 0°, 30°, 45°, 60°, 90° 8.3 Extension to the four quadrants i.e. 0–360° 8.4 Sine Rule 8.5 Cosine Rule 8.6 Area of triangle 8.7 Applications: three-figure bearings and North, East, South, West problems in two and three dimensions compound shapes 8.8 Properties of the graphs of y = sin x, y = cos x, y = tan x9 Sets – Extended) 9.2 Sets in descriptive form { x | } or as a list 9.3 Venn diagrams with at most three sets 9.4 Intersection and union of sets10 Probability – Core curriculum 10.1 Probability P(A) as a fraction, decimal or percentage Significance of its value 10.2 Relative frequency as an estimate of probability 10.3 Expected number of occurrences 10.4 Combining events: the addition rule P(A or B) = P(A) + P(B) the multiplication | 677.169 | 1 |
FSc Math Book2, CH 6, LEC 1: Introduction to Conics
This video lecture from Conic Sections (F.Sc. second year Mathematics) covers: Introduction to conic sections. Find more e-learning material and educational video lectures in Urdu at maktab.pk. These videos are free to use for promotional and commercial purpose by keeping the credits to Maktab | 677.169 | 1 |
Format of THEA Math
Types of Questions Asked in THEA Mathematics
The THEA stands for Texas Higher Education Assessment. It
is a
standardized test that is developed for those school students of the
state of Texas, who are going to enter any college or university in the
state of Texas. This test is a basic requirement for getting admission
into any degree or diploma course at the college or university level.
There are three sections in the THEA, all of which are equally
important. These three sections are: the Reading test, the Writing test
and Mathematics.
It is very difficult
to master all these three sections for a student; hence, they have
certain weak and strong areas among these.
Just like every other Texan school-student, you too must be
having a
strong or a weak area among the test sections. If that area is Math,
then this article is definitely going to help you. So, read this
article to improve your knowledge on
this segment.
Structure of Math
Section
Since the THEA is standardized, its three sections
are similar.
In other words, there is a fixed structure or format for every section.
Given below are a few important points you must know about Mathematics:
There are approximately 50 multiple-choice questions in this section. These
questions are taken from the areas of fundamental mathematics, algebra,
geometry and problem solving.
You are tested on your ability to perform mathematical
operations and solve problems.
Appropriate formulas that are
used in Mathematics
will be provided to you in the form of a reference sheet.
You will be allowed to use a calculator. If you are taking
the
Computer-based test, an online calculator will be provided to you. If
you are taking the paper-based Quick-test, you will be allowed to bring
your own calculator. However, only simple calculators are allowed in
the test.
Questions in this
Test
The questions that appear in
Math section
assess you on the basis of four main skills, which are described below:
Fundamental Mathematics
You are tested on your ability to solve word-problems
by using
integers, fractions, decimals and various units of measurement of
quantities.
You are also tested on your ability to solve similar
word-problems by using concepts of data-interpretation and
data-analysis.
Algebra
You need to solve equations and word-problems with one
or two variables.
You will be required to identify graphs, co-ordinates,
inequalities, slopes, intercepts etc.
You need to solve problems that test concepts of
geometry or revolve around geometrical figures.
Problem Solving
In these types of questions, you need to solve problems
by using
reasoning skills or mathematical skills or a combination of both.
Scoring Information of Mathematics
The number of questions that you answer correctly in the Math
section is used to calculate your scaled scores. These scaled scores
range from 200 to 300 and in order to pass this section, you need to
score at least 230 points.
If you do not pass in
this section,
you may retake the test after a gap of minimum 14 days from the
previous test date. Also note that you may register yourself for a
second attempt within the 14-day period; however, you cannot retake the
test until the 15th day from the previous test date. In the second
attempt, you may try only the Math section if you have been able to
clear the others.
Hence, it can be concluded that the Math section
evaluates you on a variety of skills. Also, there are a number of
different questions that appear in this test section. All you need to
remember is the passing scores you need and the areas where you need to
be strong. Hence, don't take thetest lightly and instead
start preparing for it right away! After all, to become successful, you need
to pass every section of it with flying colors! | 677.169 | 1 |
Contest Preparation
Computer Programming
Need Help?
Need help finding the right class? Have
a question about how classes work?
Click here to Ask AoPS!
Group Theory Seminar
This 7-week seminar is an introduction to the basics of group theory. In this seminar we will focus on defining and understanding groups: how they are constructed and how they interact with the rest of the world. This seminar is not intended to be a complete group theory course (which is a core component of a college undergraduate mathematics curriculum), but rather is intended to give strong high school students a first look at the most important concepts and examples of this very vast subject.
Diagnostics
Schedule
AoPS Holidays
There are no classes May 29, July 4, September 4, November 18 - 26, and December 21 - January 3.
Lessons
Lesson 1
Symmetry
Lesson 2
Examples of Groups
Lesson 3
Cyclic Groups
Lesson 4
Abelian Groups
Lesson 5
Group Actions
Lesson 6
Orbits and Burnside
Lesson 7
Polya Counting and Cayley Graphs
This has probably been the best AoPS class I've had. It was the most challenging, but the instructor was a great teacher. He gave good explanations and I was able to understand the concepts, which has been a challenge for me with geometry. Overall, it was a great class. | 677.169 | 1 |
Evaluating Algebraic Expressions Powerpoint
Presentation (Powerpoint) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
0.11 MB | 16 pages
PRODUCT DESCRIPTION
Use this step-by-step powerpoint to help teach your kids about basic algebra. The powerpoint is broken into two sections. First, the kids will learn what a variable is and how to solve an expression when the variable is given to them. This part of the powerpoint also covers how use variables to complete a table.
The second half of the powerpoint explains how to solve for a variable.
Both forms of algebra are modeled step by step. Then, there are sample problems for the students to try. The answers to these problems can appear with the click of your mouse.
I use this with my own students, and the breakdown of the steps really helps them understand | 677.169 | 1 |
.07
What Calculus Expert helps people like you achieve
I get a lot of students telling me that hard math subjects FINALLY clicked for them after they watched my videos and read through the non-gibberish course notes.
I'm addicted to seeing people uncover those "Ah-ha!" moments, and I've made sure that every single lesson will deliver that same feeling of immense relief, giving you the easiest experience with math that you've ever had.
Yes, the easiest experience with math you've ever had.
(Except for maybe those subtraction "problems" you solved in Kindergarten by eating Skittles.)
Math is hard. That's no kind of understatement.
So I teach you how to learn math in a non-math way your brain can actually handle.
I might not use the exact same wording as your professor, but I'm guessing you don't care. (I'm also guessing that's why you're here in the first place.)
Inside of Calculus Expert, I teach you the techniques I learned for myself that made math easy for me. Everything's still 100% technically correct and sound and will get you the right answer… it's just an approach that looks at math from a non-number crunching perspective | 677.169 | 1 |
A lively invitation to the flavor, elegance, and power of graph theory This mathematically rigorous introduction is tempered and enlivened by numerous illustrations, revealing examples, seductive applications, and historical references. An award-winning teacher, Russ Merris has crafted a book designed to attract and engage through its spirited exposition,... more...
This adaptation of an earlier work by the authors is a graduate text and professional reference on the fundamentals of graph theory. It covers the theory of graphs, its applications to computer networks and the theory of graph algorithms. Also includes exercises and an updated bibliography. more...
Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures ? like roads, computers, telephones ? instances of abstract data structures ? like lists, stacks, trees ? and functional or object oriented programming. In turn, graphs are... more...
Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects. more...
In its second edition, expanded with new chapters on domination in graphs and on the spectral properties of graphs, this book offers a solid background in the basics of graph theory. Introduces such topics as Dirac's theorem on k-connected graphs and more. more...
The field of geometric graph theory is a fairly new discipline. This contributed volume contains twenty-five original survey and research papers on important recent developments in geometric graph theory written by active researchers in this field. more... | 677.169 | 1 |
St John's College - Mathematics . . .
The St John's Mathematics Department is presently in a strong position. Classes are streamed from Year 9 which allows teachers to focus more specifically on the needs of the class in front of them. St John's top students do very well. Year 13 students have gained several Bursary Scholarships (top 2% in the country) in the past 5 years and external pass rates are among the top schools in the region. Many of these students move on to mathematical based courses at university such as engineering, architecture, surveying, computing and medicine and other health sciences. Middle stream students move through the college's mathematical pathway on the B line and end their time with the numeracy skills needed to enter the trades and the armed services. The lower streamed students are well catered for with class sizes being reduced to a very comfortable number. This gives 'strugglers' every opportunity to gain their 10 basic numeracy credits and move in to careers. Whatever the need of any young Hawkes Bay student, St John's College has the mathematics courses and expertise to cater for his needs.
STAFF All of the senior teachers at St John's have a long and proven record of teaching young Hawkes Bay boys.
Mr Grant McFarland (HOD) has taught at the college for several years and teaches all levels and is currently the Calculus teacher. In past years he has been Dean of Colin and has coached both the 1st XV rugby and the 1st XI cricket teams.
Mr Michael Oliver has rejoined the St John's teaching staff after a stint in the South Island and other Hawkes Bay schools. He has experience as a HOD and is a highly regarded Statistics teacher. Mr Oliver is involved in football and cricket.
Elaine Rogers has been a consistent performer at St John's for several years. Presently she is teaching Statistics, but has the ability to also teach Calculus.
Part time teachers include:
Mr Johnny Greaney
Mr Chris Hansen
Mr Chris Owen
Mr Matthew Scott
Mr Ian Smith
SENIOR PROGRAMME H.O.D. Mr G. McFarland
Course Entry Requirements An achievement equivalent to the following or at the discretion of the HOD Year 11 An adequate standard in Year 10 Mathematics Year 12 12 MAC 4 passes in Level 1 Mathematics or approval of HOD. 12 MAT 4 passes in Level 1 including a merit pass in Algebra 1.2 or the approval of the HOD. Year 13 Maths with Calc. Five passes in 12 MAT including a merit in 2.2, 2.6, 2.7, or the approval of the HOD. Maths with Stats. Four passes in 12 MAT including 2.2, 2.6, 2.11, 2.12 the approval of the HOD. 13 MAC Four passes in 12 MAC or the approval of the HOD. A Graphic Calculator is required for Senior A Mathematic.
LEVEL 1B MATHEMATICS (11 MXB) This course will suit any student who has struggled with Mathematics in the junior school. The course involves helping students gain confidence with their Mathematics so that they can pass the 10 basic numeracy standards that all students need to pass to gain NCEA Level 1. Last year the pass rate was 98%, so success is possible for any student with a desire to do well. Additional credits are available for those who do well. AS Number | Achievement Standard Title | Credits | Assessment Method US 26623 | Use Number to solve problems | 4 | Internal US26627 | Use Measurement to solve problems | 4 | Internal US26626 | Use Statistics to solve problems | 3 | Internal 1.9 | Transformational Geometry (Microsoft Publisher) | 3 | Internal US18743 | Using Spreadsheets (Microsoft Excel) | 3 | Internal 1.41 | Use Microsoft Office software to plan a school trip | 4 | Internal
LEVEL 2B MATHEMATICS AND COMPUTING (12 MAC) This course is a combination of Mathematics and Computing skills. The technological developments over the last 20 years have meant that these subjects have become very closely linked. The course is suitable for Mathematics students who will struggle with the heavy algebra content of the 2A Mathematics course. The course includes practical Mathematics topics that past students have found relevant for careers in the trades, the armed forces and in office jobs. It also contains work involving Microsoft applications (Word, Excel, Access, Publisher), webpage design using html/css, Photoshop and Robotics. A student can NOT take both 12 Mathematics and Computing (12 MAC) and 12 Digital Technology (12DT). AS Number | Achievement Standard Title | Credits | Assessment Method US2784 | Dealing with Money using Microsoft Excel | 3 | Internal 2.4 | Trigonometry used in the Construction Industry | 3 | Internal 2.5 | Networks used to find Shortest Routes (e.g. GPS navigation systems) | 2 | Internal 2.42 | Evaluate Website techniques | 4 | Internal 2.43 | Design a Website using html5/css3 and Photoshop | 3 | Internal US2786 | Organising your Business with Microsoft Access | 3 | Internal 2.10 | Conduct Experiments with Statistical Data | 3 | Internal Preparation for the Army/Navy Mathematics Exam Design, build and use Robots | 677.169 | 1 |
Real and Complex Fourier Series
References
Subject Index
Preface
First of all let us believe and recall the neglected fact in our life that "the mother of all human science is Mathematics". Dear students, readers, researchers and all those who respect mathematics, we introduce this book which may be helpful for them. Most of the researchers directed their own research work to the numerical analysis, due to the rapid communications and the advances in the computer programming. In the present book, we introduce some mathematical concepts that are widely important and can be considered on the basis of the numerical analysis. We presented the topics in a simple way of presentation and no proofs for theorems, because we decided to introduce the scientific material without complications related to those working on pure mathematics. Our main research work is the computational engineering and applied mathematics and numerical analysis; therefore, the present book can serve as the base of a future textbook in numerical analysis. We hope and pray that this work will last forever. | 677.169 | 1 |
Calculus: Limits by Delta Epsilon Notes plus Worksheets
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0.22 MB | 12 pages
PRODUCT DESCRIPTION
Calculus LIMITS
This resource contains 13 Limits by Definition problems from Calculus 1 using the basic Delta Epsilon method to find delta, given epsilon. Three of the problems included have fully typed solutions, ready to give out to students as examples or project on the whiteboard. There are also two practice sheets of five problems each with answer keys included.
This resource is appropriate for all first semester Calculus classes; AP, Dual Enrollment, and Honors | 677.169 | 1 |
Transformations with Rigid Motion Common Core Geometry
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5.91 MB | 40 Pages pages
PRODUCT DESCRIPTION
This unit includes 7 complete days of teaching Transformations with Rigid Motion in Common Core Geometry. This includes rotations, reflections, translations, transformation symmetry, sequence of rigid motion and all homeworks to go all along with lessons. A review sheet and quiz are also included | 677.169 | 1 |
Basic Mathematics Covers basic mathematic techniques, including graphs, multiplication table, decimals, perimeters, fractions, exponents, geometry and area of shapes, along with a selection of calculators and math resources.
Wessa Mathematical statistics and forecasting calculators including an equation plotter, multiple regression modeling, and a variety of descriptive statistics from variability and concentration to percentiles and correlation.
Mega Converter 2 Discover how many seconds old you are, the difference between a gallon in the USA and a gallon in the UK, how many nanometers in an inch, how many quarts in a chaldron and more. Handy scientific calculator is also available. | 677.169 | 1 |
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
Introduction to Advanced Mathematics
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
Answer questions concerning basic algebra, coordinate geometry, polynomials, uncertainty, indices, and using the language of advanced mathematics. Students will be able to understand the C1 module of A-Level Maths, especially the difficult OCR MEI Examination syllabus.
Students should be familiar with the content of GCSE or equivalent level mathematics.
Description
Do you want to gain confidence and fluency in maths, to the equivalent of AS-level in the UK? It is also appropriate for anyone making the transition to more comlicated maths. Through this course of videos, I will illuminate various topics in core mathematics.
Learn, Reinforce and Master Advanced Maths Techniques with Theo
Make algebra much less confusing
Deal with any sort of coordinate geometry with ease
Learn how to sketch any curve given its equation
Transition smoothly from GCSE-equivalent level maths to maths appropriate for any AS or A2 maths exam board
What really makes this course different is that you can see me the entire time. Too many maths course have zero human interaction, they simply display a blank screen on which the instructor writes. Not so here. I will interact with you by looking directly into the camera, as you would expect from a private tutor in a classroom setting.
I base my lessons on the OCR MEI exam topics, which in turn are covered in all the other exam boards. So you can be sure that what you will learn here, you can apply on exam day. Let me help you become a real success at maths, and open doors for yourself in terms of exam grades and getting a really good career in the years to come.
A complete summary of the skills and techniques needed to ace the C1 - Introduction to Advanced Mathematics module at A-level, for the guidance-seeking student to the adult brushing up on once-known skills. The course will take to complete from 2 to 4 hours. The course is structured very conveniently to ensure smooth transition from one topic to another.
Who is the target audience?
These videos are for students taking A-level maths or a similar level of maths, who are unsure of techniques that they need to answer problems correctly. You will enjoy these videos as I explain mathematical concepts in a personal, face-to-face way as you would with your own teacher. These videos are not appropriate for anyone below A-level standard, but they may also be handy for people who need to brush up their maths skills.
Carfax Education, an international group, originating from Oxford and headquartered in London, specializes in providing guidance and support to individuals and institutions who seek to access the best educational opportunities available in Britain, Switzerland, the USA, the UAE, and Azerbaijan. Carfax opened its first office in Azerbaijan in 2012 and plans to expand to other countries throughout the Caspian Sea region. | 677.169 | 1 |
Mathematics
The course covers pure mathematics which extends algebra, trigonometry and transformations covered at GCSE. New topics such as logarithms, differentiation and integration are included. Applications of mathematics to model situations and to solve problems in a variety of contexts are learnt in the statistics and mechanics part of the course.
Specific requirements
A minimum of 5 GCSEs, 2 B grades and 3 C grades, including English.
Grade 7 or A preferred. Students with grade 6 or B may be considered according to their GCSE profile.
Please click here to see grades conversions in line with the new grading system.
Where does it lead?
Mathematics is a subject which can be studied in order to develop high levels of numeric and problem solving skills. Studying mathematics opens up a wide range of career paths. Students with a mathematics A level often go to work in areas such as engineering, architecture, accounting, business, pharmacy, statistics, medicine and dentistry.
This is a very valuable option for students considering degrees in Mathematics, Computing, Physics, Engineering or Economics.
"I studied Economics with Maths, Further Maths and History at Cadbury College, achieving grades A*AAA. I am now studying a degree in Economics at The University of Bristol." | 677.169 | 1 |
Mathematics
Unit outlines
The mathematics program at Campbell High School aims to promote a positive attitude towards mathematics and its relevance to students both personally and in their community. The program gives students the opportunity to develop a sound understanding of mathematics and its processes by following the Australian Curriculum.
Year 7
A common course of study is provided to students in year 7. Year 7 classes follow a common program with common assessment items. The year 7 courses cover the basic skills of mathematics and enable students to be numerate in our society as well as preparing them for further high school mathematics studies.
Years 8 and 9
A common course of study is provided to students in years 8 and 9, which is structured to accommodate the range of students in each cohort. Extension classes exists in year 8 and 9 to extend gifted students. The year 8 and 9 courses cover the basic skills of mathematics and enable students to be numerate in our society as well as preparing them for further mathematical enquiry.
Year 10
In year 10, all students will study a common core program. This program provides students with the opportunity to study mathematics in different levels of depth. Students can follow a 10 or 10A program of study which cover topics which prepare students for tertiary level mathematics at college. The 10A program leads to a further study of mathematics as needed in heavily mathematics based college courses such as Physics, Chemistry and Mathematical Methods to Specialist Mathematics. High achieving students have an opportunity to be selected for the 10A course.
Mathematics Tutorials
A formalised mathematics tutorial session is available to all students through the Enrichment program. It is designed to give those students who require extra help in mathematics the opportunity to work through problems with a teacher in a smaller group.
Mathematics Competitions
Students at Campbell High School have the opportunity to compete in a number of national mathematics competitions including the Computational and Algorithmic Competition, ICAS Mathematics Competition and the Australian Mathematics Competition. Students who have an interest in the creative understanding of mathematics can also join the mathematics and engineering section of the Tournament of Minds.
Mathematics Challenge for Young Australians
Students also have the opportunity to participate in the Mathematics Challenge for Young Australians offered by the Australian Mathematics Trust. For a small fee, students are given mathematical theory and problems based outside the curriculum designed to extend and challenge their understanding of mathematical concepts. | 677.169 | 1 |
Buy New
$39.9539.95This lesson teaches students how to solve equations that contain polynomials. Students are taught how to properly factor the equation and set the factors equal to zero in order to obtain the solution. Numerous examples are presented to gain practice with this material. | 677.169 | 1 |
Well, it is September (one of my favorite months of the year) and I hope that everyone has had a positive and successful return to school! If you are a teacher, I know first hand how exciting it is to meet your new students and planning all of the wonderful activities for the year.
If you are a student, it's exciting to see all of your friends again, but also I know deep down, you are thinking about all of the wonderful things you are going to learn!
.
Whatever you do this year, stay positive! Your goals are attainable, if you put your mind to it! Best of luck in your studies!
As always, please feel free to contact me if you have a question or comment!
I wish you well in your Algebra studies!
All the best,
Karin
Here's a Guaranteed Way to Ace Your Solving Equations Unit!
Typically, most Algebra 1 curriculum begins with a week of reviewing integers, algebraic expressions, and the distributive property. These are all the basic skills needed for success in Algebra 1.
During this week make sure that you (or your child) really focuses on memorizing their integer rules! I've had students who were able to complete the steps to solve equations, but always had an incorrect answer because they couldn't add or subtract integers!
After this brief review, most teachers typically move right on to solving equations. This is also an important chapter in Algebra 1, as you will utilize this skill throughout the entire curriculum.
This is also, often times, a breaking point for a lot of students. If students have difficulty with this first chapter, many times they just give up for the rest of the year!
So, I've come up with a "Solving Equations Study Guide" to help students solve equations!
What is the Solving Equations Study Guide?
The Solving Equations study guide is a 7 step process that helps guide students through each step of solving an equation. This is a great resource for students who have trouble remembering "what to do next!"
This original Solving Equations Study Guide can only be found in My latest ebook:
Within the ebook you will find over 100 pages of examples, practice problems, quizzes, and a test for the entire Solving Equations unit. The best part about the book, is you get an answer key for every problem with every step necessary to solve the problem! So, if you are considering a tutor (for $30-60 an hour), you may want to reconsider! With this Ebook, you can teach yourself, or your child how to solve equations!
The Solving Equations Unit is typically about 12-15 school days. So, that's about 3 weeks long! Let's say you get a tutor at $30 an hour, once a week. That's $90! $90 and your tutor may not have time to cover all the topics that you need help with! So, what should you do? Tutor yourself, or your child! Just think of how much fun you can have learning together!
I am running a back to school special! You'll save 50% if you order by September 8th! If you are going to be studying equations, try it out! There is a risk-free guarantee - so if it doesn't work for you, let me know and I'll return your money! | 677.169 | 1 |
Excel HSC General Maths QS
Excel HSC General Mathematics Quick Study is the perfect tool for studying and revising on the go! This app is designed specifically for the HSC General Mathematics course. There are two parts to the app: 1. HSC study cards • There are 134 study cards to revise. • All the Core topics (Measurement; Algebraic Modelling; Probability; Financial Mathematics; Data Analysis) are covered. • Special revision features include: - bookmarks—you can bookmark each card with a green, yellow or red bookmark depending on how well you know each card - revision notes—you can type in your own revision notes to customise your revision for each card or topic. 2. Quick Quiz • There are 553 questions in total. • You can take a randomly generated quick quiz of 10 questions from the topic of your choice, or from all topics combined. • Each question in the quiz is marked instantly for you, with the correct answer highlighted so you can learn from your mistakes. • You are given a score out of ten and your percentage mark at the end of each quiz. • You are also given a comprehensive summary of all your results for each topic including your percentage improvement, which helps you keep track of your progress. • You can take the quizzes as many times as you like until you get the consistently high percentage score you want. Excel Quick Study apps are a convenient and efficient way to revise—any time, anywhere!TechnicalPlease note: due to the large number of different devices supporting Android 2.2 and up, not all brands/models have been tested. Your feedback is appreciated: please email us to report any errors or bugs and we will do our utmost to fix them and to support other | 677.169 | 1 |
Visualization and mathematics have begun a fruitful
relationship, establishing links between problems and solutions
of both fields. In some areas of mathematics, like differential
geometry and numerical mathematics, visualization techniques are
applied with great success. However, visualization methods
heavily rely on mathematical concepts.
Applications of visualization in mathematical research and
the use of mathematical methods in visualization have been topic
of the international workshop
VisMath 95 in
Berlin. The workshop was met with great approval by
mathematicians and computer graphic experts. It was considered a
link between the fields "Visualization in Mathematics" and
"Mathematical Methods in Computer Graphics".
This book contains a selection of contributions of this
workshop which treat topics of particular interest in current
research. Experts are reporting on their latest work, giving an
overview on this fascinating new area. The reader will get
insight to state-of-the-art techniques for solving visualization
problems and mathematical questions.
We hope that the research articles of this book will
stimulate the readers' own work and will further strengthen the
development of the field of Mathematical Visualization. | 677.169 | 1 |
A Level Further Mathematics (Legacy)
What is Further Mathematics?
A level Further Mathematics builds upon those skills acquired while studying Mathematics. It is, in particular, a qualification which both broadens and extends the topics covered in AS/A level Mathematics.
What sort of student does it suit and what will you get out of the course?
This is a challenging course for those with a real interest in and aptitude for the subject. The training in logic that the course provides is appropriate to most subjects and the course supports most careers in the fields of Mathematics, Science and Engineering, Computing, Accountancy and Economics. It equips you with skills such as logistic analysis and deduction, data handling, mathematical modelling and problem solving, all of which can be applied in almost any field of work.
AS Level
MPW approach to AS study
The AS Further Mathematics course is taught separately from the A level Mathematics course and is delivered systematically, unit by unit, with much interactive discussion in our small groups. Regular homework, timed assignments and practice examination questions are all analysed in detail in lessons so that students become thoroughly familiar with the application of all the mathematical concepts involved.
Unit 6 (M1) 16.67% 1h 30m exam Mathematical models in mechanics; vectors in mechanics; kinematics of a particle moving in a straight line; dynamics of a particle moving in a straight line or plane; statics of a particle; moments.
Reading List
Author
Title
Publisher
Pledger (ed)
Modular Maths Edexcel Further Pure Mathematics 1
Heinemann
A2 Level
MPW approach to A2 study
The A2 Further Mathematics course is also taught separately from the A level Mathematics course and ultimately results in A level qualifications in Mathematics and Further Mathematics. The course is followed in the same systematic way as is required at AS level. The intellectual demands of A2 level work, however, mean that students must expect to work extremely hard. Breadth of study, depth of analysis and exam technique are crucial to achieving top grades. | 677.169 | 1 |
Torrent Files List
Torrent description
Math You Can Really Use—Every Day skips mind-numbing theory and tiresome drills and gets right down to basic math that helps you do real-world stuff like figuring how much to tip, getting the best deals shopping, computing your gas mileage, and more. This is not your typical, dry math textbook.
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Advanced Placement Courses
Mathematics
Lourdes High School emphasizes the synthesis of faith, culture, and life in accordance with Christian values. We believe that the learning of mathematics enables students to see the patterns and relationships in God's creation. We enable students to grow and mature in taking responsibility for their own education. We ensure all students acquire sufficient and appropriate mathematical literacy for present and future needs through a variety of experiences. We believe students should be taught to be lifelong learners as well as be able to apply and analyze life experiences from a mathematical perspective.
Mathematics Department Curriculum Summary We offer a comprehensive variety of mathematics courses at each grade level. Students follow a sequence of study starting with algebraic concepts and finishing in calculus. Our freshman students are typically placed in Algebra I, Algebra II, or Honors Algebra II based on past performance, testing results, and teacher recommendations. Our sophomore, junior, and senior students are placed in appropriate academic courses based on their achievement through each successive year. | 677.169 | 1 |
The content in this course showed me how math is used in everyday life when people post examples in the discussion questions. I can use math to solve daily problems like how much money I am going to spend on something or cooking problems where I need to use proportions. Linear equations help me when I need to calculate my expenses for a long term, and I can also use system of equation when needing to see how many adults come to a theater compared to kids so we can use the info and set up a
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Unformatted text preview: system of equation. I can apply this is my personal life when cooking and need to use proportions to solve for the right amount of some sort of ingredient to be used in a recipe. I used the MyMathLab study guide to do practice problems and help prepare myself for the quizzes and tests, and upcoming final. This guide let me prepare and take on my anxiety of quizzes. Using that guide, it made me more used to the type of questions and everything would be then quite easy....
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This note was uploaded on 04/20/2011 for the course MATH 116 taught by Professor Mcmillian during the Spring '09 term at University of Phoenix. | 677.169 | 1 |
Synopsis
Complete Mathematics Practice Book 2 Practice Book by David Bowles, Andrew Manning, Shaun Procter-Green, David Pritchard
Complete Mathematics is the only course designed especially for Independent and Grammar Schools. This Key Stage 3 course lays the foundations for good teaching and learning of mathematics from the first year of secondary school through to GCSE and beyond. The full content of the new Programmes of Study are covered and reinforced through practice of essential mathematical skills and processes. Each chapter covers a particular attainment target (Number, Algebra, Geometry and Measures or Statistics) and they are presented in a possible teaching order. This Practice Book covers work at levels 6 to 7 and supports the Year 8 Pupil's Book. There is an accompanying website featuring Personal Tutor examples. | 677.169 | 1 |
Introduction to Linear Algebra, 5/e is a foundation book that bridges both practical computation and theoretical principles. Due to its flexible table of contents, the book is accessible for both students majoring in the scientific, engineering, and social sciences, as well as students that want an introduction to mathematical abstraction and logical reasoning. In order to achieve the text's flexibility, the book centers on 3 principal topics: matrix theory and systems of linear equations, elementary vector space concepts, and the eigenvalue problem. This highly adaptable text can be used for a one-quarter or one-semester course at the sophomore/junior level, or for a more advanced class at the junior/senior level.
"synopsis" may belong to another edition of this title.
Product Description:
This edition contains a new section covering elementary vector-space ideas, for example subspace, basis, and dimension, introduced in the familiar setting of R. | 677.169 | 1 |
COURSES
2. IIT JAM MATHS
2.9. MODULAR COURSES
There are many students who have need guidance in some specific topics only. To cater those students,we have modular courses available in video courses, test series,study material.One can buy video lectures,books and test series of specific modules from our e-commerce website | 677.169 | 1 |
Seventh Grade Essential Standards
7th Grade Essential Standards
The seventh grade Mathematics courses are intended to serve as transitional courses, expanding on the work accomplished at the elementary schools while developing solid foundations in pre-algebraic and geometric topics, with emphasis on: ratios and proportions, the number system, expressions and equations, basic geometry, and introductory probability and statistics. The topics covered in the seventh grade Mathematics courses form the basis for much of the high school curriculum. Seventh grade Mathematics courses are leveled (Modified, Regular, and Honors) in order to differentiate student academic needs, but at their core all levels will be founded on the following essential standards: | 677.169 | 1 |
Passage to Abstract Mathematics Passage to Abstract Mathematics facilitates the transition from introductory mathematics courses to the more abstract work that occurs in advanced courses. This text covers logic, proofs, numbers, sets, induction, functions, and morematerial whichMore...
Passage to Abstract Mathematics facilitates the transition from introductory mathematics courses to the more abstract work that occurs in advanced courses. This text covers logic, proofs, numbers, sets, induction, functions, and morematerial which instructors of upper-level courses often presume their students have already mastered but are in fact missing from lower-level courses. Students will learn how to read and write mathematicsespecially proofsthe way that mathematicians do. The text emphasizes the use of complete, correct definitions and mathematical syntax | 677.169 | 1 |
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GeoGebra is a dynamic mathematics software that joins geometry, algebra, and calculus. Two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in thegeometry window and vice versaMen of War: Vietnam, a real time strategy game where the gameplay consists of you moving around half a dozen soldiers either on the side of Vietnam or America army (initially locked out) trying to complete various missions before the enemy …
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Find the Error - Solving Inequalities
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0.53 MB | 5 pages
PRODUCT DESCRIPTION
This is a very good way to have students analyze mistakes which creates a better understanding of the concept. I created this for my students to find the mistake in problems "a student" had solved incorrectly. I used common errors when creating the solutions to help them see the mistakes that are commonly made.
I had the students work alone to solve the problems first, and then share with their groups. If there was any discrepancy, they looked in the answer folder on their tables to see the correct answer. I rotated around monitoring and helping as individuals or groups needed help. It was terrific to see the discussions that were going on! This really worked | 677.169 | 1 |
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Synopsis
Pass Mathematical Literacy Grade 12 by Paul Douglas Carter
PASS Mathematical Literacy provides a comprehensive overview of the curriculum to help you prepare for the final exam. This contains: * summary notes that follow the exam structure * typical exam questions and memoranda * useful hints and tips to help you pass your exam Grade 12 Mathematical Literacy in a nutshell! | 677.169 | 1 |
The ABC news unit intends to offer viewers a different kind of look at the places to which reporters travel – a 360-degree shot made possible through virtual reality. Viewers will be prodded to check out a panoramic, practically three-dimensional view related to the report that can be accessed anywhere on mobile or desktop by visiting by downloading a virtual-reality app from Jaunt,a developer of virtual-reality hardware and software, for iOS and Android devices.
Read Introduction to Mathematical Programming: Applications Book Download Free Donwload Here Feature * Authors Wayne Winston and Munirpallam Venkataramanan emphasize model-formulation and model-building skills as well as interpretation of computer software output. Focusing on deterministic models, this book is designed for the first half of an operations research sequence. A subset of Winston's best-selling OPERATIONS RESEARCH, INTRODUCTION TO MATHEMATICAL PROGRAMMING offers self-contained chapters that make it flexible enough for one- or two-semester courses ranging from advanced beginning to intermediate in level. The book has a strong computer orientation and emphasizes model-formulation and model-building skills. Every topic includes a corresponding computer-based modeling and solution method and every chapter presents the software tools needed to solve realistic problems. LINDO, LINGO, and Premium Solver for Education software packages are available with the book. Donwload Here Download Introduction to Mathematical Programming: Applications and Algorithms Download Introduction to Mathematical Programming: Applications and Algorithms PDF Download Introduction to Mathematical Programming: Applications and Algorithms Kindle Download Introduction to Mathematical Programming: Applications and Algorithms Android Download Introduction to Mathematical Programming: Applications and Algorithms Full Ebook Download Introduction to Mathematical Programming: Applications and Algorithms Free
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Read: Computer Simulation and Data Analysis in Molecular Book Download Free Donwload Here Feature * This book provides an introduction to two important aspects of modern bioch- istry, molecular biology, and biophysics: computer simulation and data analysis. My aim is to introduce the tools that will enable students to learn and use some f- damental methods to construct quantitative models of biological mechanisms, both deterministicandwithsomeelementsofrandomness;tolearnhowconceptsofpr- ability can help to understand important features of DNA sequences; and to apply a useful set of statistical methods to analysis of experimental data. The availability of very capable but inexpensive personal computers and software makes it possible to do such work at a much higher level, but in a much easier way, than ever before. TheExecutiveSummaryofthein?uential2003reportfromtheNationalAcademy of Sciences, "BIO 2010: Transforming Undergraduate Education for Future – search Biologists" [12], begins The interplay of the recombinant DNA, instrumentation, and digital revolutions has p- foundly transformed biological research. The con?uence of these three innovations has led to important discoveries, such as the mapping of the human genome. How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative, and biological research has become critically dependent on concepts and methods drawn from other scienti?c disciplines. The connections between the biological sciences and the physical sciences, mathematics, and computer science are rapidly becoming deeper and more extensive. Donwload Here Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) PDF Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) Kindle Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) Android Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) Full Ebook Download Computer Simulation and Data Analysis in Molecular Biology and Biophysics: An Introduction Using R (Biological and Medical Physics, Biomedical Engineering) Free | 677.169 | 1 |
Math Courses, Block 1
At Proof School we begin the school year by studying a selection of mathematical topics that are not commonly addressed in the standard secondary school curriculum.
These topics, which are sometimes featured at math circles or summer programs, expose students to the breadth of this majestic subject. In the process, students come to understand that mathematics consists of so much more than the narrow sequence of algebra, geometry, precalculus, and calculus that typically defines the high school math experience.
Courses in block one generally fall into one of two categories. Our problem solving courses present a wide variety of engaging topics, with a focus on developing strategies for tackling difficult problems, learning methods of proof, and making progress in mathematical writing skills. Courses in discrete math address counting techniques, graph theory, and related areas. We are also offering one more utilitarian course this year, an Algebra 1 Lab to provide an on-ramp for students gearing up for Algebra 2 next block. We're pleased that the background of our math faculty, the structure of our days and year, and the enthusiasm of our students allow us to run such rich and thought-provoking math courses.
Algebra 1 Lab with John DeIonno In this course, we will explore applications that rely only on skills learned in Algebra 1. We will solve geometry problems using equations of lines, fit curves to data, use piecewise-linear functions to compare insurance and cell phone plans, and use linear programming to find optimal production schedules. There will also be purely mathematical investigations, such as using visual means to divide polynomials and finding connections between Pascal's triangle and n-dimensional cubes and simplexes. In addition we will reinforce algebra skills in areas of quadratic equations, graphing, and simplifying expressions with fractions.
Problem Solving 1 with Sam Vandervelde This course is designed for students with minimal mathematical problem solving background. Students will learn foundational methods such as formulating a simpler question, finding a unifying pattern, or working backwards. They will develop strategies for making progress on problems when the best approach is not clear. Particular emphasis will be placed on working through a broad selection of carefully chosen problems. Specific topics will include basic probability, Pigeonhole Principle, logic, and tiling problems.
Problem Solving 2 with Susan Durst This class will be an exploration of problem solving techniques, with an emphasis on proof writing. We will practice working through the complete arc of solving a math problem, articulating our reasons for believing that our solution is correct, and transforming our intuition into a rigorous proof. The particular problems that we work with will come from a variety of mathematical subject areas, including game theory, logic, and topology.
Enumerative Combinatorics with Sachi Hashimoto This course is a formal introduction to counting techniques, with a twin emphasis on computational problems and the methods of combinatorial proof. Students will explore counting principles, binomial coefficients, permutations and combinations, combinatorial identities, inclusion-exclusion, finite probability, and graph theory. Along the way, students will hone proof techniques such as mathematical induction, recursion, and proof by bijection.
Topics in Algebraic Combinatorics with Austin Shapiro In this course, students will add powerful tools from algebra to their repertoire of counting techniques. We will explore the many kinds of generating functions, including ordinary power series and exponential generating functions, which we will use to derive closed forms, identities, and asymptotics for a range of classic sequences. Along the way, we will learn about combinatorial objects such as integer partitions, set partitions, and trees. The second half of the course will cover group actions and Pólya theory, a suite of methods for counting in problems with complex symmetries. In addition, each student will complete a research project examining an application of generating functions. | 677.169 | 1 |
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Using Casio FX9850 to Draw Real-Life Applications of Graphs of Functions
Brenda Lee blee@sun5.wfc.edu.tw
Wu Feng Inst. of Technology
Taiwan
Abstract
Modern hand-held technology has greatly aided the teaching and learning and doing of mathematics, especially when teaching the concept of function, with its ability to generate examples of graphs of functions. A motivational problem remains, however; students often do not see the necessity for learning how to graph functions, although with the help of hand-held technology we can identify the relationships between a function and the derivative of this function as well as determine the principal characteristics of this function. Many students still need convincing with real-life applications of graphing.
In this workshop, we will demonstrate how to use the graphs of functions to draw representations of real life situations. We will show how to have some fun with the graphs of functions by using them to draw some cartoons. We also demonstrate a couple of programs for drawing fractal diagrams, a fascinating area of modern mathematics. | 677.169 | 1 |
computational method Documents
Showing 1 to 30 of 50
Chapter 07.01
Primer on Integration
After reading this chapter, you should be able to:
1.
2.
3.
4.
define an integral,
use Riemanns sum to approximately calculate integrals,
use Riemanns sum and its limit to find the exact expression of integrals, and
fin
ESO 208A: Computational Methods in Engineering
Tutorial 12
Elliptic PDE
1. Determine steady state temperature in a square plate of length 3.0 cm for the following
conditions.
(a) All the edges are kept at 100C except the top edge, which is at 0C.
(b) Same
Chapter 07.05
Gauss Quadrature Rule of Integration
After reading this chapter, you should be able to:
1. derive the Gauss quadrature method for integration and be able to use it to solve
problems, and
2. use Gauss quadrature method to solve examples of ap
05.01
Definition of Interpolation
After reading this chapter, you should be able to:
1. Understand what Interpolation is.
What is Interpolation?
Many a times, a function y f x is given only at discrete points such as
x0 , y0 , x1, y1 ,., xn1, yn1 , xn , y
Chapter 08.07
Finite Difference Method for Ordinary Differential
Equations
After reading this chapter, you should be able to
1. Understand what the finite difference method is and how to use it to solve problems.
What is the finite difference method?
The | 677.169 | 1 |
Seventh SeventhThe Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
The goal of this text is to help students leam to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies. • The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelIed almost exactly on the exam pIes; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar ones.
Clearly written and well-illustrated, this text is geared toward undergraduate business and social science students. Topics include sets, functions, and graphs; limits and continuity; special functions; the derivative; the definite integral; and functions of several variables. Answers to more than half of the problems appear in the appendix. 1972 edition. Includes 142 figures.
This general review covers equations, functions, and graphs; limits, derivatives; integrals and antiderivatives; word problems; applications of integrals to geometry; and much more. Additional features make this volume especially helpful to students working on their own. They include worked-out examples, a summary of the main points of each chapter, exercises, and where needed, background material on algebra, geometry, and reading comprehension.
Eighth EighthCalculus: The Language of Change is an innovative new introductory text that blends traditional and reform approaches, and focuses on understanding calculus as its own language. With accessible writing and presentation, the text allows students to gradually understand the language – first by reviewing vocabulary, and then by quickly moving to present calculus conceptually, computationally, and theoretically. Within this framework, derivatives and integrals are developed side by side, coverage of theory is offered at various levels, and computing devices are incorporated generically. A full range of student and instructor resources make Calculus: The Language of Change an outstanding course package.
This text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. Over 1,200 problems, with hints and complete solutions. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, the differential, indefinite and definite integrals, more. 1963 edition. | 677.169 | 1 |
This section contains free e-books and guides on Combinatorics, some of the resources in this section can be viewed online and some of them can be downloaded.
This book covers the following topics: Fibonacci Numbers From a
Cominatorial Perspective, Functions,Sequences,Words,and Distributions, Subsets
with Prescribed Cardinality, Sequences of Two Sorts of Things with Prescribed
Frequency, Sequences of Integers with Prescribed Sum, Combinatorics and
Probability, Binary Relations, Factorial Polynomials, The Calculus of Finite
Differences, Principle of Inclusion and Exclusions.
The authors give full coverage of the underlying
mathematics and give a thorough treatment of both classical and modern
applications of the theory. The text is complemented with exercises, examples,
appendices and notes throughout the book to aid understanding. Major topics covered includes: Symbolic Methods, Complex
Asymptotics, Random Structures, Auxiliary Elementary Notions and Basic Complex Analysis. | 677.169 | 1 |
Holt Precalculus: A Graphing Approach: Student Edition 2004Earlier research funded by the NSF, such as "Project Follow Through," which reached very different conclusions about what works best in the classroom, would not be considered.62 Regardless of what cognitive psychology might say about teaching methodologies, only constructivist programs would be supported. Along with the Systemic Initiative awards, the NSF supported the creation and development of commercial mathematics curricula aligned to the NCTM Standards download. Or need help teaching mathematics to your English language learners? Participants will learn about core ESL principles and practices, the role that language and culture play in learning mathematics, planning and implementing instruction for ELLs, and assessment of mathematics knowledge Precalculus: A Graphing download online You'll also learn about the complexity of algorithms, how to use algorithmic thinking in problem solving, algorithmic applications of random processes, asymptotic analysis, finite calculus and partitions IB Mathematics Standard Level Online Course Book: Oxford IB Diploma Program download for free. John Wallis analyzed it in his Algebra in 1685. �, Martin, Knotted Doughnuts and Other Mathematical Entertainments (New York: W , source: IB Matematicas Nivel Medio Libro del Alumno: Programa del Diploma del IB Oxford (IB Diploma Program) download for free. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution ref.: Word Problems Interactive Years 1 & 2 Math 89S-01, "The Mathematics of the... read course details for Mathematics of the Universe » Mathematics 89S.02, "The Magic of Numbers" Fall 2016 Instructor: Lenny Ng This course explores some of the intriguing and beautiful mathematics that... read course details for The Magic of Numbers » Introduction to basic ideas of modern cryptography with emphasis on history and mathematics of encryption, applications in daily life, and... read course details for Cryptography and Society » Euclidean geometry, inverse and projective geometries, topology (Möbius strips, Klein bottle, projective space), and non-Euclidean geometries in two... read course details for Geometry » Advanced introduction to basic, non-measure theoretic proba- bility covering topics in more depth and with more rigor than MATH 230 , source: Nelson Probability and Statistics 1 for Cambridge International A Level
Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem 100 Addition Worksheets with download here Subbarayappa (New Delhi: Indian National Science Academy, 1971) Math Is Easy So Easy, download pdf travel.50thingstoknow.com. Please note that citing without naming the source can or will constitute plagiarism for which you can and will be held accountable. To request permission to reuse, reproduce, or translate IEA material, please fill out the permission request form » Please note that the website and its contents, together with all online and/or printed publications and released items ('works') by TIMSS, PIRLS, and IEA, were created with the utmost care , source: Step-by-Step Probability: Intermediate (Homework Booklet) He is also known as �the father of algebra,� from the title of his work, Hisab Al-Jabr wal Muqabalah, The Book of Calculations, Restoration and Reduction. He gave the name to that branch of mathematics. In fact, his algebra was a book of arithmetic featuring Hindi numerals -- a huge improvement over Roman numerals and other systems of dots, pictographs, and finger reckoning download.
In some institutions, this potential is already a reality.103 In an era of international competition, it is unlikely that the public will tolerate such trends indefinitely. It was the broad implementation of the NCTM reforms themselves that created the resistance to them 500 Addition Worksheets with download pdf On a personal note, when I was going through high school I relied heavily on a graphing calculator to do all of my work for me. Although I was very good at translating problems into a format the calculator can read, I ignored the underlying concepts which lead me to receive good grades, at least for the mean time Math For Everyone: 7th Grade download for free Students recognise angles in real situations. They interpret and compare data displays. They classify numbers as either odd or even. They recall addition and multiplication facts for single-digit numbers online. You are invited to present your research at the Math & Stat Research Day! We strive to advance the existing knowledge and application of mathematics and statistics Multiple Choice and read epub Students completing Stage 4 Honours will be considered for the award of honours according to the following scale based on performance in the thesis and the 5 courses taken as part of the honours year. The thesis contributes 37.5% to the overall honours mark, and the 5 courses each contribute 12.5% epub. That's fine, but when deciding what you do everyday, remember that this course is about $1000-important to whomever is paying for it. A majority of student learning occurs outside of class, which is largely spent doing homework 200 Addition Worksheets with download for free A-W. 1994. 0201558025 The serious student who wants to specialize in combinatorics should not specialize too much 365 Addition Worksheets with 4-Digit, 3-Digit Addends: Math Practice Workbook (365 Days Math Addition Series) (Volume 28) read for free. | 677.169 | 1 |
About this Category CBSE Class 9 NCERT Solutions for Mathematics, Science, Egnlish and other important subjects are available for download in PDF format completely free. NCERT Solutions contain the descriptive answers to all the exercise questions and will surely help the students preparing for the CBSE Examination 2016 - 2017 in a proficient manner.
CBSE Class 9 NCERT Solutions
Jagranjosh presents NCERT solution of CBSE Class 9 Mathematics in the form of eBook. These eBooks are completely free and can be downloaded in PDF format. One can access these eBooks anytime and anywhere with their mobile devices.
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Here you can find the Mathematics NCERT Solution for CBSE Class 9 Polynomials. It includes a detailed explanation of the NCERT Solution and the covers the various methods and Technique of solving the Questions assigned in the NCERT textbooks. | 677.169 | 1 |
IB Math SL Questions & Answers
IB Math SL Flashcards
IB Math SL Advice
IB Math SL Advice
Showing 1 to 3 of 3
The course is easy if you have a basic knowledge of calculus and common core. The class is very hands on and requires you to work well n groups. If you enjoy math and find a small classroom setting preferable, IB Math SL is a great course to take for your math credit.
Course highlights:
The course is very flexible, you will be able to help decide which area you study in order. Because the course is flexible, there is also a lot of time to study in depth topics such as in the category statistics. With stats, you will be able to learn the most difficult material but have time to really be able to understand different formulas and processes on the calculator.
Hours per week:
3-5 hours
Advice for students:
If you plan on taking this course, it is necessary for you to have a background with pre calc and calculus of either AB or AB BC. This course would be SL, which also might precipitate problems choosing which tests you would be taking at the HL level. If you take this course SL, you need to probably take HL history and HL science.
Course Term:Fall 2016
Professor:austin james
Course Required?Yes
Course Tags:Math-heavyBackground Knowledge ExpectedGroup Projects
Nov 18, 2016
| Would recommend.
This class was tough.
Course Overview:
I would recommend this class because it pushes you to work harder. For AP statistics you have to pay attention and really work on it at home. This will help prepare you for the future if youre up for a challlenge.
Course highlights:
The highlight of this course was being able to apply it to real life. You can use this class to come up with actual statistics.
Hours per week:
3-5 hours
Advice for students:
Stay on task and do your homework.
Course Term:Fall 2016
Professor:austin james
Course Tags:Math-heavyA Few Big AssignmentsRequires Presentations
Nov 16, 2016
| Would recommend.
This class was tough.
Course Overview:
I would recommend it because it is a good representation of how real math will be used in business' later in life.
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I gained general knowledge of statistics. I mastered probability and graphing. We graphed things like histograms, normal curves, box plots, tree graphs, scatter plots, and stem and leaf plots.
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3-5 hours
Advice for students:
Study Statistics before you start the class or else you are likely to be behind. | 677.169 | 1 |
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We bet at least once you dreamt about moving to a kind of wonderland where students don't have to do their homework every day. It may sound unreal, but such fairytale countries where kids spend only a couple of hours per week on homework exist. They had gone through a long way of trial and errors before realized all pros and cons of homework and introduced innovative changes into their system of education. However, why does the necessity of homework remain a topic for a debate, where the opposite side stands for giving as many home assignments to students as possible? It, therefore, sparks a question whether homework is actually harmful or helpful. Read the rest of this entry »
In this article, we will tell you about one more method of circuit analysis, which is called the Node Voltage Method. It also allows to reduce the calculations and save the time when we do the circuit analysis. Let's firstly consider the T-circuit scheme from our previous post and solve it with the help of the Node Voltage Method. Read the rest of this entry »
Is that possible to learn the language completely? Just any language, native or foreign in, say, 10, 25, 70 years? No. And to master English, it is of major importance to always refresh language structures, update your idiomatic base, modernize grammar, immerse into English speaking environment and add new lexical units to your word stock. Here we're going to speak about the latter and, believe, it doesn't come as a problem when you know some useful tips. Read the rest of this entry »
Practical exercises make an important part of a math course, which helps to deeper understand all theoretical information and further study more complex math sections. Quite often students do not even realize that they make mistakes and why that happens. In this article, we will analyze the most common student's problems met while solving calculus homework. Read the rest of this entry »
Previously, we've already introduced the Kirchhoff's Circuit Law as a separate topic. And today, we shall consider another method for the circuit analysis, which is called the Mesh Current Method. The advantage of this method is that we obtain less unknown variables and equations when we solve the circuit. Let's consider the T-circuit scheme from the mentioned post and solve it with the help of the Mesh Current Method. Read the rest of this entry »
There are a lot of different math tricks that can impress you with their beauty and compactness. In this post, you will find not only their examples but also will try to create your own. To begin with, a few "Mathematical Curiosities" are introduced by the famous Russian mathematician and physicist Yakov Perelman (1882-1942) in his popular book "Arithmetic for entertainment". Here are some of them: Read the rest of this entry »
Scientists all over the world usually argue about when it is better to learn a language. And we argue with them that there is no better time than NOW. No matter what county you live in, what age you are and what specialization you have, "I have no time" and "I was badly taught at school" are no longer the excuses. Actually, everybody has no time, but for many it's not a problem. Surprized? Let's then find out how to cope with this. Read the rest of this entry »
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Mathematics Grade 12
Description: The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Mathematics
Grade 12
Copyright c 2007 "Free High School Science Texts" Permission is granted to c...
The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Mathematics
Grade 12
Copyright c- Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License"
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The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Mathematics Grades 10 - 12
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Logarithms - Grade 12
In mathematics many ideas are related. We saw that addition and subtraction are related and that multiplication and division are related. Similarly, exponentials and logarithms are related. Logarithms, commonly referred to as logs, are the inverse of exponentials. The logarithm of a number x in the base a is defined as the number n such that an = x. So, if an = x, then: loga (x) = n (35.1)
Extension: Inverse Function When we say "inverse function" we mean that the answer becomes the question and the question becomes the answer. For example, in the equation ab = x the "question" is "what is a raised to the power b." The answer is "x." The inverse function would be loga x = b or "by what power must we raise a to obtain x." The answer is "b." The mathematical symbol for logarithm is loga (x) and it is read "log to the base a of x". For example, log10 (100) is "log to the base 10 of 100".
The logarithm of a number is the index to which the base must be raised to give that number. From the first example of the activity log2 (4) (read log to the base 2 of 4) means the power of 2 that will give 4. Therefore, log2 (4) = 2 (35.2) The index-form is then 22 = 4 and the logarithmic-form is log2 4 = 2. 445
Logarithms, like exponentials, also have a base and log2 (2) is not the same as log10 (2). We generally use the "common" base, 10, or the natural base, e. The number e is an irrational number between 2.71 and 2.72. It comes up surprisingly often in Mathematics, but for now suffice it to say that it is one of the two common bases.
Extension: Natural Logarithm The natural logarithm (symbol ln) is widely used in the sciences. The natural logarithm is to the base e which is approximately 2.71828183.... e is like π and is another example of an irrational number.
While the notation log10 (x) and loge (x) may be used, log10 (x) is often written log(x) in Science and loge (x) is normally written as ln(x) in both Science and Mathematics. So, if you see the log symbol without a base, it means log10 . It is often necessary or convenient to convert a log from one base to another. An engineer might need an approximate solution to a log in a base for which he does not have a table or calculator function, or it may be algebraically convenient to have two logs in the same base. Logarithms can be changed from one base to another, by using the change of base formula: loga x = logb x logb a (35.3)
where b is any base you find convenient. Normally a and b are known, therefore logb a is normally a known, if irrational, number. For example, change log2 12 in base 10 is: log2 12 = log10 12 log10 2
When the base is 10, we do not need to state it. From the work done up to now, it is also useful to summarise the following facts: 1. log 1 = 0 2. log 10 = 1 3. log 100 = 2 4. log 1000 = 3
35.6
Logarithm Law 3: loga (x · y) = loga (x) + loga (y)
The derivation of this law is a bit trickier than the first two. Firstly, we need to relate x and y to the base a. So, assume that x = am and y = an . Then from Equation 35.1, we have that: loga (x) and This means that we can write: loga (x · y) = = = = loga (am · an ) loga (am+n ) loga (y) = m = n (35.10) (35.11)
The derivation of this law is identical to the derivation of Logarithm Law 3 and is left as an exercise.
10 For example, show that log( 100 ) = log 10 − log 100. Start with calculating the left hand side:
In grade 10 you solved some exponential equations by trial and error, because you did not know the great power of logarithms yet. Now it is much easier to solve these equations by using logarithms. For example to solve x in 25x = 50 correct to two decimal places you simply apply the following reasoning. If the LHS = RHS then the logarithm of the LHS must be equal to the logarithm of the RHS. By applying Law 5, you will be able to use your calculator to solve for x.
In general, the exponential equation should be simplified as much as possible. Then the aim is to make the unknown quantity (i.e. x) the subject of the equation. For example, the equation 2(x+2) = 1 is solved by moving all terms with the unknown to one side of the equation and taking all constants to the other side of the equation 2x · 22 2
x
Worked Example 159: Exponential Equation Question: Solve for x in 7 · 5(3x+3) = 35 Answer Step 1 : Identify the base with x as an exponent There are two possible bases: 5 and 7. x is an exponent of 5. Step 2 : Eliminate the base with no x In order to eliminate 7, divide both sides of the equation by 7 to give: 5(3x+3) = 5 Step 3 : Take the logarithm of both sides log(5(3x+3) ) = log(5) Step 4 : Apply the log laws to make x the subject of the equation. 453
Logarithms are part of a number of formulae used in the Physical Sciences. There are formulae that deal with earthquakes, with sound, and pH-levels to mention a few. To work out time periods is growth or decay, logs are used to solve the particular equation.
Worked Example 160: Using the growth formula Question: A city grows 5% every 2 years. How long will it take for the city to triple its size? Answer Step 1 : Use the formula A = P (1 + i)n Assume P = x, then A = 3x. For this example n represents a period of 2 years, therefore the n is halved for this purpose. Step 2 : Substitute information given into formula 3 log 3 n n = = = = (1,05) 2 n × log 1.05 (usinglaw5) 2 2 log 3 ÷ log 1,05 45,034
n
Step 3 : Final answer So it will take approximately 45 years for the population to triple in size.
454
CHAPTER 35. LOGARITHMS - GRADE 12
35.12
35.11.1
Exercises
1. The population of a certain bacteria is expected to grow exponentially at a rate of 15 % every hour. If the initial population is 5 000, how long will it take for the population to reach 100 000 ? 2. Plus Bank is offering a savings account with an interest rate if 10 % per annum compounded monthly. You can afford to save R 300 per month. How long will it take you to save up R 20 000 ? (Answer to the nearest rand)
Worked Example 161: Logs in Compound Interest Question: I have R12 000 to invest. I need the money to grow to at least R30 000. If it is invested at a compound interest rate of 13% per annum, for how long (in full years) does my investment need to grow ? Answer Step 1 : The formula to use A = P (1 + i)n Step 2 : Substitute and solve for n 30 000 < 1,13n n n > > > 12 000(1 + 0,13)n 5 2 log 2,5 log 2,5 ÷ log 1,13 7,4972....
n log 1,13 >
Step 3 : Determine the final answer In this case we round up, because 7 years will not yet deliver the required R 30 000. The investment need to stay in the bank for at least 8 years.
17. √ Solve the following equation for x without the use of a calculator and using the fact that 10 ≈ 3,16 : 6 2 log(x + 1) = −1 log(x + 1) 18. Solve the following equation for x: 66x = 66 (Give answer correct to 2 decimal places.)
456
Chapter 36
Sequences and Series - Grade 12
36.1 Introduction
In this chapter we extend the arithmetic and quadratic sequences studied in earlier grades, to geometric sequences. We also look at series, which is the summing of the terms in a sequence.
36.2
Arithmetic Sequences
The simplest type of numerical sequence is an arithmetic sequence.
Definition: Arithmetic Sequence An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term
For example, 1,2,3,4,5,6, . . . is an arithmetic sequence because you add 1 to the current term to get the next term: first term: second term: third term: . . . nth term: 1 2=1+1 3=2+1 n = (n − 1) + 1
Given a1 and the common difference, d, the entire set of numbers belonging to an arithmetic sequence can be generated.
Definition: Arithmetic Sequence An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term: an = an−1 + d where • an represents the new term, the nth -term, that is calculated; • an−1 represents the previous term, the (n − 1)th -term; • d represents some constant. (36.2)
Important: Arithmetic Sequences
A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant: a2 − a1 = a3 − a2 = an − an−1 = d (36.3) This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condition does not hold, the sequence is not an arithmetic sequence.
Extension: Plotting a graph of terms in an arithmetic sequence Plotting a graph of the terms of sequence sometimes helps in determining the type of sequence involved. For an arithmetic sequence, plotting an vs. n results in: 458
Definition: Geometric Sequences A geometric sequence is a sequence in which every number in the sequence is equal to the previous number in the sequence, multiplied by a constant number.
This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example.
36.3.1
Example - A Flu Epidemic
Extension: What is influenza? Influenza (commonly called "the flu") is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severe illness that most of us get during winter time. The main way that the influenza virus is spread is from person to person in respiratory droplets of coughs and sneezes. (This is called "droplet spread".) This can happen when droplets from a cough or sneeze of an infected person are propelled (generally, up to a metre) through the air and deposited on the mouth or nose of people nearby. It is good practise to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu.
Assume that you have the flu virus, and you forgot to cover your mouth when two friends came to visit while you were sick in bed. They leave, and the next day they also have the flu. Let's assume that they in turn spread the virus to two of their friends by the same droplet spread the following day. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner: Again we can tabulate the events and formulate an equation for the general case: 459
The above table represents the number of newly-infected people after n days since you first infected your 2 friends. You sneeze and the virus is carried over to 2 people who start the chain (a1 = 2). The next day, each one then infects 2 of their friends. Now 4 people are newly-infected. Each of them infects 2 people the third day, and 8 people are infected, and so on. These events can be written as a geometric sequence: 2; 4; 8; 16; 32; . . . Note the common factor (2) between the events. Recall from the linear arithmetic sequence how the common difference between terms were established. In the geometric sequence we can determine the common ratio, r, by a3 a2 = =r a1 a2 Or, more general, an =r an−1 (36.4)
From the above example we know a1 = 2 and r = 2, and we have seen from the table that the nth -term is given by an = 2 × 2n−1 . Thus, in general, an = a1 · rn−1 where a1 is the first term and r is called the common ratio. So, if we want to know how many people are newly-infected after 10 days, we need to work out a10 : an a10 = = = = = a1 · rn−1 2 × 210−1 2 × 29 2 × 512 1024 (36.6)
That is, after 10 days, there are 1 024 newly-infected people. Or, how many days would pass before 16 384 people become newly infected with the flu virus? an = a1 · rn−1
1. What is the important characteristic of an arithmetic sequence? 461
36.4
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
2. Write down how you would go about finding the formula for the nth term of an arithmetic sequence? 3. A single square is made from 4 matchsticks. Two squares in a row needs 7 matchsticks and 3 squares in a row needs 10 matchsticks. Determine: A the first term B the common difference C the formula for the general term D how many matchsticks are in a row of 25 squares
4. 5; x; y is an arithmetic sequence and 81; x; y is a geometric sequence. All terms in the sequences are integers. Calculate the values of x and y.
36.4
Recursive Formulae for Sequences
When discussing arithmetic and quadratic sequences, we noticed that the difference between two consecutive terms in the sequence could be written in a general way. For an arithmetic sequence, where a new term is calculated by taking the previous term and adding a constant value, d: an = an−1 + d The above equation is an example of a recursive equation since we can calculate the nth -term only by considering the previous term in the sequence. Compare this with equation (36.1), an = a1 + d · (n − 1) (36.7)
where one can directly calculate the nth -term of an arithmetic sequence without knowing previous terms. For quadratic sequences, we noticed the difference between consecutive terms is given by (??): an − an−1 = D · (n − 2) + d Therefore, we re-write the equation as an = an−1 + D · (n − 2) + d Using (36.5), the recursive equation for a geometric sequence is: an = r · an−1 (36.9) (36.8)
which is then a recursive equation for a quadratic sequence with common second difference, D.
Recursive equations are extremely powerful: you can work out every term in the series just by knowing previous terms. As you can see from the examples above, working out an using the previous term an−1 can be a much simpler computation than working out an from scratch using a general formula. This means that using a recursive formula when using a computer to work out a sequence would mean the computer would finish its calculations significantly quicker.
The above sequence is called the Fibonacci sequence. Each new term is calculated by adding the previous two terms. Hence, we can write down the recursive equation: an = an−1 + an−2 (36.11)
36.5
Series
In this section we simply work on the concept of adding up the numbers belonging to arithmetic and geometric sequences. We call the sum of any sequence of numbers a series.
36.5.1
Some Basics
If we add up the terms of a sequence, we obtain what is called a series. If we only sum a finite amount of terms, we get a finite series. We use the symbol Sn to mean the sum of the first n terms of a sequence {a1 ; a2 ; a3 ; . . . ; an }: S n = a1 + a2 + a3 + . . . + an For example, if we have the following sequence of numbers 1; 4; 9; 25; 36; 49; . . . and we wish to find the sum of the first 4 terms, then we write S4 = 1 + 4 + 9 + 25 = 39 The above is an example of a finite series since we are only summing 4 terms. If we sum infinitely many terms of a sequence, we get an infinite series: S ∞ = a1 + a2 + a3 + . . . (36.13) (36.12)
In the case of an infinite series, the number of terms is unknown and simply increases to ∞.
36.5.2
Sigma Notation
In this section we introduce a notation that will make our lives a little easier. 463
36.5
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
A sum may be written out using the summation symbol . This symbol is sigma, which is the capital letter "S" in the Greek alphabet. It indicates that you must sum the expression to the right of it:
n
ai = am + am+1 + . . . + an−1 + an
i=m
(36.14)
where • i is the index of the sum; • m is the lower bound (or start index), shown below the summation symbol; • n is the upper bound (or end index), shown above the summation symbol; • ai are the terms of a sequence. The index i is increased from m to n in steps of 1. If we are summing from n = 1 (which implies summing from the first term in a sequence), then we can use either Sn - or -notation since they mean the same thing:
n
Sn =
i=1
ai = a1 + a2 + . . . + an
(36.15)
For example, in the following sum,
5
i
i=1
we have to add together all the terms in the sequence ai = i from i = 1 up until i = 5:
5
Remember that an arithmetic sequence is a set of numbers, such that the difference between any term and the previous term is a constant number, d, called the constant difference: an = a1 + d (n − 1) where • n is the index of the sequence; • an is the nth -term of the sequence; • a1 is the first term; • d is the common difference. When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series. The simplest arithmetic sequence is when a1 = 1 and d = 0 in the general form (36.18); in other words all the terms in the sequence are 1: ai = = a1 + d (i − 1) 1 + 0 · (i − 1) (36.18)
= {ai } =
n n
1 {1; 1; 1; 1; 1; . . .}
If we wish to sum this sequence from i = 1 to any positive integer n, we would write ai =
i=1 i=1
1 = 1 + 1 + 1 + ...+ 1 465
(n times)
36.6
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
Since all the terms are equal to 1, it means that if we sum to n we will be adding n-number of 1's together, which is simply equal to n:
n
If we wish to sum this sequence from i = 1 to any positive integer n, we would write i = 1 + 2 + 3 + ...+ n
i=1
(36.20)
This is an equation with a very important solution as it gives the answer to the sum of positive integers.
teresting Mathematician, Karl Friedrich Gauss, discovered this proof when he was only Interesting Fact Fact 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1 to 100. Young Karl realised how to do this almost instantaneously and shocked the teacher with the correct answer, 5050.
or, more sensibly, we could use equation (36.25) noting that a1 = 3, d = 7 and n = 20 so that
20
S20
= = =
i=1 20 2 [2
[3 + 7 (i − 1)] · 3 + 7 (20 − 1)]
1390
In this example, it is clear that using equation (36.25) is beneficial.
36.6.2
Exercises
n (7n + 15). 2
1. The sum to n terms of an arithmetic series is Sn =
A How many terms of the series must be added to give a sum of 425? B Determine the 6th term of the series. 2. The sum of an arithmetic series is 100 times its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero. 467
36.7
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
3. The common difference of an arithmetic series is 3. Calculate the values of n for which the nth term of the series is 93, and the sum of the first n terms is 975. 4. The sum of n terms of an arithmetic series is 5n2 − 11n for all values of n. Determine the common difference. 5. The sum of an arithmetic series is 100 times the value of its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero. 6. The third term of an arithmetic sequence is -7 and the 7t h term is 9. Determine the sum of the first 51 terms of the sequence. 7. Calculate the sum of the arithmetic series 4 + 7 + 10 + · · · + 901. 8. The common difference of an arithmetic series is 3. Calculate the values of n for which the nth term of the series is 93 and the sum of the first n terms is 975.
36.7
Finite Squared Series
When we sum a finite number of terms in a quadratic sequence, we get a finite quadratic series. The general form of a quadratic series is quite complicated, so we will only look at the simple case when D = 2 and d = (a2 − a1 ) = 3 in the general form (???). This is the sequence of squares of the integers: ai {ai } = = = i2 {12 ; 22 ; 32 ; 42 ; 52 ; 62 ; . . .} {1; 4; 9; 16; 25; 36; . . .}
n
If we wish to sum this sequence and create a series, then we write Sn =
i=1
i 2 = 1 + 4 + 9 + . . . + n2
which can be written, in general, as
n
Sn =
i=1
i2 =
n (2n + 1)(n + 1) 6
(36.26)
The proof for equation (36.26) can be found under the Advanced block that follows:
Extension: Derivation of the Finite Squared Series We will now prove the formula for the finite squared series:
n
When we sum a known number of terms in a geometric sequence, we get a finite geometric series. We know from (??) that we can write out each term of a geometric sequence in the general form: an = a1 · rn−1 (36.27) where • n is the index of the sequence; • an is the nth -term of the sequence; • a1 is the first term; • r is the common ratio (the ratio of any term to the previous term). By simply adding together the first n terms, we are actually writing out the series Sn = a1 + a1 r + a1 r2 + . . . + a1 rn−2 + a1 rn−1 We may multiply the above equation by r on both sides, giving us rSn = a1 r + a1 r2 + a1 r3 + . . . + a1 rn−1 + a1 rn 469 (36.29) (36.28)
36.8
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
You may notice that all the terms on the right side of (36.28) and (36.29) are the same, except the first and last terms. If we subtract (36.28) from (36.29), we are left with just rSn − Sn = a1 rn − a1 Sn (r − 1) = a1 (rn − 1) Dividing by (r − 1) on both sides, we arrive at the general form of a geometric series:
n
Sn =
i=1
a1 · ri−1 =
a1 (rn − 1) r−1
(36.30)
36.8.1
Exercises
a + ar + ar2 + ... + arn−1 = a (1 − rn ) (1 − r)
3 2
1. Prove that
2. Find the sum of the first 11 terms of the geometric series 6 + 3 + 3. Show that the sum of the first n terms of the geometric series
1 54 + 18 + 6 + ... + 5 ( 3 )n−1
+
3 4
+ ...
is given by 81 − 34−n . 4. The eighth term of a geometric sequence is 640. The third term is 20. Find the sum of the first 7 terms.
n
5. Solve for n:
t=1
3 1 8 ( 2 )t = 15 4 .
6. The ratio between the sum of the first three terms of a geometric series and the sum of the 4th -, 5th − and 6th -terms of the same series is 8 : 27. Determine the common ratio and the first 2 terms if the third term is 8. 7. Given the geometric series: 2 · (5)5 + 2 · (5)4 + 2 · (5)3 + . . . A Show that the series converges B Calculate the sum to infinity of the series C Calculate the sum of the first 8 terms of the series, correct to two decimal places. D Determine
∞ n=9
2 · 56−n
correct to two decimal places using previously calculated results. 8. Given the geometric sequence 1; −3; 9; . . . determine: A The 8th term of the sequence B The sum of the first 8 terms of the sequence. 9. Determine:
4 n=1
3 · 2n−1
470
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
36.9
36.9
Infinite Series
Thus far we have been working only with finite sums, meaning that whenever we determined the sum of a series, we only considered the sum of the first n terms. In this section, we consider what happens when we add infinitely many terms together. You might think that this is a silly question - surely the answer will be ∞ when one sums infinitely many numbers, no matter how small they are? The surprising answer is that in some cases one will reach ∞ (like when you try to add all the positive integers together), but in some cases one will get a finite answer. If you don't believe this, try doing the following sum, a geometric series, on your calculator or computer: 1 1 1 1 1 2 + 4 + 8 + 16 + 32 + . . . You might think that if you keep adding more and more terms you will eventually get larger and larger numbers, but in fact you won't even get past 1 - try it and see for yourself! We denote the sum of an infinite number of terms of a sequence by S∞ =
∞ i=1
ai
When we sum the terms of a series, and the answer we get after each summation gets closer and closer to some number, we say that the series converges. If a series does not converge, then we say that it diverges.
36.9.1
Infinite Geometric Series
There is a simple test for knowing instantly which geometric series converges and which diverges. When r, the common ratio, is strictly between -1 and 1, i.e. −1 < r < 1, the infinite series will converge, otherwise it will diverge. There is also a formula for working out what the series converges to. Let's start off with formula (36.30) for the finite geometric series:
n
calculate the smallest value of n for which the sum of the first n terms is greater than 80.99. 7. Determine the value of
∞ k=1 1 12( 5 )k−1 .
8. A new soccer competition requires each of 8 teams to play every other team once. A Calculate the total number of matches to be played in the competition. 472
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
36.10
B If each of n teams played each other once, determine a formula for the total number of matches in terms of n. 9. The midpoints of the sides of square with length equal to 4 units are joined to form a new square. The process is repeated indefinitely. Calculate the sum of the areas of all the squares so formed. 10. Thembi worked part-time to buy a Mathematics book which cost R29,50. On 1 February she saved R1,60, and saves everyday 30 cents more than she saved the previous day. (So, on the second day, she saved R1,90, and so on.) After how many days did she have enough money to buy the book? 11. Consider the geometric series: 5 + 21 + 11 + . . . 2 4 A If A is the sum to infinity and B is the sum of the first n terms, write down the value of: i. A ii. B in terms of n. B For which values of n is (A − B) <
1 24 ?
12. A certain plant reaches a height of 118 mm after one year under ideal conditions in a greenhouse. During the next year, the height increases by 12 mm. In each successive year, 5 the height increases by 8 of the previous year's growth. Show that the plant will never reach a height of more than 150 mm.
n
13. Calculate the value of n if
a=1
(20 − 4a) = −20.
14. Michael saved R400 during the first month of his working life. In each subsequent month, he saved 10% more than what he had saved in the previous month. A How much did he save in the 7th working month? B How much did he save all together in his first 12 working months? C In which month of his working life did he save more than R1,500 for the first time? 15. A man was injured in an accident at work. He receives a disability grant of R4,800 in the first year. This grant increases with a fixed amount each year. A What is the annual increase if, over 20 years, he would have received a total of R143,500? B His initial annual expenditure is R2,600 and increases at a rate of R400 per year. After how many years does his expenses exceed his income? 16. The Cape Town High School wants to build a school hall and is busy with fundraising. Mr. Manuel, an ex-learner of the school and a successful politician, offers to donate money to the school. Having enjoyed mathematics at school, he decides to donate an amount of money on the following basis. He sets a mathematical quiz with 20 questions. For the correct answer to the first question (any learner may answer), the school will receive 1 cent, for a correct answer to the second question, the school will receive 2 cents, and so on. The donations 1, 2, 4, ... form a geometric sequence. Calculate (Give your answer to the nearest Rand) A The amount of money that the school will receive for the correct answer to the 20th question. B The total amount of money that the school will receive if all 20 questions are answered correctly. 17. The first term of a geometric sequence is 9, and the ratio of the sum of the first eight terms to the sum of the first four terms is 97 : 81. Find the first three terms of the sequence, if it is given that all the terms are positive. 18. (k − 4); (k + 1); m; 5k is a set of numbers, the first three of which form an arithmetic sequence, and the last three a geometric sequence. Find k and m if both are positive. 473
C Determine the 10th term of this sequence correct to one decimal place. 20. The second and fourth terms of a convergent geometric series are 36 and 16, respectively. Find the sum to infinity of this series, if all its terms are positive.
5
A If the pattern continues, find the number of letters in the column containing M's. B If the total number of letters in the pattern is 361, which letter will the last column consist of. 31. The following question was asked in a test: Find the value of 22005 + 22005 . Here are some of the students' answers: A Megansaid the answer is 42005 . B Stefan wrote down 24010 . C Nina thinks it is 22006 . D Annatte gave the answer 22005×2005 . Who is correct? ("None of them" is also a possibility.) 32. Find the pattern and hence calculate: 1 − 2 + 3 − 4 + 5 − 6 . . . + 677 − 678 + . . . − 1000
∞
33. Determine A x=−
(x + 2)p , if it exists, when
p=1
5 2 B x = −5
∞ i=1
34. Calculate:
5 · 4−i 475
36.10
CHAPTER 36. SEQUENCES AND SERIES - GRADE 12
35. The sum of the first p terms of a sequence is p (p + 1). Find the 10th term. 36. The powers of 2 are removed from the set of positive integers 1; 2; 3; 4; 5; 6; . . . ; 1998; 1999; 2000 Find the sum of remaining integers. 37. A shrub of height 110 cm is planted. At the end of the first year, the shrub is 120 cm tall. Thereafter, the growth of the shrub each year is half of its growth in the previous year. Show that the height of the shrub will never exceed 130 cm.
476
Chapter 37
Finance - Grade 12
37.1 Introduction
In earlier grades simple interest and compound interest were studied, together with the concept of depreciation. Nominal and effective interest rates were also described. Since this chapter expands on earlier work, it would be best if you revised the work done in Chapters 8 and 21. If you master the techniques in this chapter, when you start working and earning you will be able to apply the techniques in this chapter to critically assess how to invest your money. And when you are looking at applying for a bond from a bank to buy a home, you will confidently be able to get out the calculator and work out with amazement how much you could actually save by making additional repayments. Indeed, this chapter will provide you with the fundamental concepts you will need to confidently manage your finances and with some successful investing, sit back on your yacht and enjoy the millionaire lifestyle.
37.2
Finding the Length of the Investment or Loan
In Grade 11, we used the formula A = P (1 + i)n to determine the term of the investment or loan, by trial and error. In other words, if we know what the starting sum of money is and what it grows to, and if we know what interest rate applies - then we can work out how long the money needs to be invested for all those other numbers to tie up. Now, that you have learnt about logarithms, you are ready to work out the proper algebraic solution. If you need to remind yourself how logarithms work, go to Chapter 35 (on page 445). The basic finance equation is: A = P · (1 + i)n If you don't know what A, P , i and n represent, then you should definitely revise the work from Chapters 8 and 21. Solving for n: A = (1 + i)n = log((1 + i)n ) = n log(1 + i) = n = P (1 + i)n (A/P ) log(A/P ) log(A/P ) log(A/P )/ log(1 + i)
Remember, you do not have to memorise this formula. It is very easy to derive any time you need it. It is simply a matter of writing down what you have, deciding what you need, and solving for that variable.
By this stage, you know how to do calculations such as "If I want R1 000 in 3 years' time, how much do I need to invest now at 10% ?" But what if we extend this as follows: If I want R1 000 next year and R1 000 the year after that and R1 000 after three years ... how much do I need to put into a bank account earning 10% p.a. right now to be able to afford that?" The obvious way of working that out is to work out how much you need now to afford the payments individually and sum them. We'll work out how much is needed now to afford the payment of R1 000 in a year (= R1 000 × (1,10)−1 = R909,0909), the amount needed now for the following year's R1 000 (= R1 000 × (1,10)−2 = R826,4463) and the amount needed now for the R1 000 after 3 years (= R1 000 × (1,10)−3 = R751,3148). Add these together gives you the amount needed to afford all three payments and you get R2486,85. So, if you put R2486,85 into a 10% bank account now, you will be able to draw out R1 000 in a year, R1 000 a year after that, and R1 000 a year after that - and your bank account will come down to R0. You would have had exactly the right amount of money to do that (obviously!). You can check this as follows: 478
CHAPTER 37. FINANCE - GRADE 12 Amount Amount Amount Amount Amount Amount Amount at Time 0 (i.e. Now) at Time 1 (i.e. a year later) after the R1 000 at Time 2 (i.e. a year later) after the R1 000 at Time 3 (i.e. a year later) after the R1 000 = = = = = = = R2486,85 R2735,54 R1735,54 R1909,09 R909,09 R1 000 R0
Perfect! Of course, for only three years, that was not too bad. But what if I asked you how much you needed to put into a bank account now, to be able to afford R100 a month for the next 15 years. If you used the above approach you would still get the right answer, but it would take you weeks! There is - I'm sure you guessed - an easier way! This section will focus on describing how to work with: • annuities - a fixed sum payable each year or each month either to provide a pre-determined sum at the end of a number of years or months (referred to as a future value annuity) or a fixed amount paid each year or each month to repay (amortise) a loan (referred to as a present value annuity). • bond repayments - a fixed sum payable at regular intervals to pay off a loan. This is an example of a present value annuity. • sinking funds - an accounting term for cash set aside for a particular purpose and invested so that the correct amount of money will be available when it is needed. This is an example of a future value annuity
37.3.1
Sequences and Series
Before we progress, you need to go back and read Chapter 36 (from page 457) to revise sequences and series. In summary, if you have a series of n terms in total which looks like this: a + ar + ar2 + ... + arn−1 = a[1 + r + r2 + ...rn−1 ] this can be simplified as: a(rn − 1) r−1 a(1 − rn ) 1−r useful when r > 1 useful when 0 ≤ r < 1
37.3.2
Present Values of a series of Payments
So having reviewed the mathematics of Sequences and Series, you might be wondering how this is meant to have any practical purpose! Given that we are in the finance section, you would be right to guess that there must be some financial use to all this Here is an example which happens in many people's lives - so you know you are learning something practical Let us say you would like to buy a property for R300 000, so you go to the bank to apply for a mortgage bond. The bank wants it to be repaid by annually payments for the next 20 years, starting at end of this year. They will charge you 15% per annum. At the end of the 20 years the bank would have received back the total amount you borrowed together with all the interest they have earned from lending you the money. You would obviously want to work out what the annual repayment is going to be! Let X be the annual repayment, i is the interest rate, and M is the amount of the mortgage bond you will be taking out. Time lines are particularly useful tools for visualizing the series of payments for calculations, and we can represent these payments on a time line as: 479
37.3
CHAPTER 37. FINANCE - GRADE 12
X 0 1
X 2
X 18
X 19
X 20
Cash Flows Time
Figure 37.1: Time Line for an annuity (in arrears) of X for n periods.
The present value of all the payments (which includes interest) must equate to the (present) value of the mortgage loan amount. Mathematically, you can write this as: M = X(1 + i)−1 + X(1 + i)−2 + X(1 + i)−3 + ... + X(1 + i)−20 The painful way of solving this problem would be to do the calculation for each of the terms above - which is 20 different calculations. Not only would you probably get bored along the way, but you are also likely to make a mistake. Naturally, there is a simpler way of doing this! You can rewrite the above equation as follows:
M = X(v 1 + v 2 + v 3 + ... + v 20 ) where v = (1 + i)−1 = 1/(1 + i) Of course, you do not have to use the method of substitution to solve this. We just find this a useful method because you can get rid of the negative exponents - which can be quite confusing! As an exercise - to show you are a real financial whizz - try to solve this without substitution. It is actually quite easy. Now, the item in square brackets is the sum of a geometric sequence, as discussion in section 36. This can be re-written as follows, using what we know from Chapter 36 of this text book: v 1 + v 2 + v 3 + ... + v n = v(1 + v + v 2 + ... + v n−1 ) 1 − vn ) = v( 1−v 1 − vn = 1/v − 1 1 − (1 + i)−n = i
Note that we took out a common factor of v before using the formula for the geometric sequence. So we can write: M = X[ This can be re-written: X= (1 − (1 + i)−n ) ] i M
−n [ (1−(1+i) ) ] i
So, this formula is useful if you know the amount of the mortgage bond you need and want to work out the repayment, or if you know how big a repayment you can afford and want to see what property you can buy. For example, if I want to buy a house for R300 000 over 20 years, and the bank is going to 480
= R47 928,44 This means, each year for the next 20 years, I need to pay the bank R47 928,44 per year before I have paid off the mortgage bond. On the other hand, if I know I will only have R30 000 a year to repay my bond, then how big a house can I buy? That is easy ....
So, for R30 000 a year for 20 years, I can afford to buy a house of R187 800 (rounded to the nearest hundred). The bad news is that R187 800 does not come close to the R300 000 you wanted to buy! The good news is that you do not have to memorise this formula. In fact , when you answer questions like this in an exam, you will be expected to start from the beginning - writing out the opening equation in full, showing that it is the sum of a geometric sequence, deriving the answer, and then coming up with the correct numerical answer.
Worked Example 163: Monthly mortgage repayments Question: Sam is looking to buy his first flat, and has R15 000 in cash savings which he will use as a deposit. He has viewed a flat which is on the market for R250 000, and he would like to work out how much the monthly repayments would be. He will be taking out a 30 year mortgage with monthly repayments. The annual interest rate is 11%. Answer Step 1 : Determine what is given and what is needed The following is given: • Deposit amount = R15 000 • Price of flat = R250 000 • interest rate, i = 11% We are required to find the monthly repayment for a 30-year mortgage. Step 2 : Determine how to approach the problem We know that: M X = (1−(1+i)−n ) ] [ i . In order to use this equation, we need to calculate M , the amount of the mortgage bond, which is the purchase price of property less the deposit which Sam pays upfront. M = = R250 000 − R15 000 R235 000 481
37.3
CHAPTER 37. FINANCE - GRADE 12 Now because we are considering monthly repayments, but we have been given an annual interest rate, we need to convert this to a monthly interest rate, i12. (If you are not clear on this, go back and revise section 21.8.) (1 + i12)12
12
=
(1 + i) 1,11 0,873459%
(1 + i12) = i12 =
We know that the mortgage bond is for 30 years, which equates to 360 months. Step 3 : Solve the problem Now it is easy, we can just plug the numbers in the formula, but do not forget that you can always deduce the formula from first principles as well! M
−n [ (1−(1+i) ) ] i
X
= = =
R235 000
−360 ) ] [ (1−(1.00876459) 0,008734594
R2 146,39
Step 4 : Write the final answer That means that to buy a house for R300 000, after Sam pays a R15 000 deposit, he will make repayments to the bank each month for the next 30 years equal to R2 146,39.
Worked Example 164: Monthly mortgage repayments Question: You are considering purchasing a flat for R200 000 and the bank's mortgage rate is currently 9% per annum payable monthly. You have savings of R10 000 which you intend to use for a deposit. How much would your monthly mortgage payment be if you were considering a mortgage over 20 years. Answer Step 1 : Determine what is given and what is required The following is given: • Deposit amount = R10 000 • Price of flat = R200 000 • interest rate, i = 9% We are required to find the monthly repayment for a 20-year mortgage. Step 2 : Determine how to approach the problem We are consider monthly mortgage repayments, so it makes sense to use months as our time period. The interest rate was quoted as 9% per annum payable monthly, which means that the monthly effective rate = 9%/12 = 0,75% per month. Once we have converted 20 years into 240 months, we are ready to do the calculations! First we need to calculate M , the amount of the mortgage bond, which is the purchase price of property less the deposit which Sam pays up-front. M = R200 000 − R10 000 = R190 000 482
But it is clearly much easier to use our formula that work out 240 factors and add them all up! Step 3 : Solve the problem 1 − (1 + 0,75%)−240 = 0,75% X × 111,14495 = X =
X×
R190 000 R190 000 R1 709,48
Step 4 : Write the final answer So to repay a R190 000 mortgage over 20 years, at 9% interest payable monthly, will cost you R1 709,48 per month for 240 months.
Show me the money Now that you've done the calculations for the worked example and know what the monthly repayments are, you can work out some surprising figures. For example, R1 709,48 per month for 240 month makes for a total of R410 275,20 (=R1 709,48 × 240). That is more than double the amount that you borrowed! This seems like a lot. However, now that you've studied the effects of time (and interest) on money, you should know that this amount is somewhat meaningless. The value of money is dependant on its timing. Nonetheless, you might not be particularly happy to sit back for 20 years making your R1 709,48 mortgage payment every month knowing that half the money you are paying are going toward interest. But there is a way to avoid those heavy interest charges. It can be done for less than R300 extra every month... So our payment is now R2 000. The interest rate is still 9% per annum payable monthly (0,75% per month), and our principal amount borrowed is R190 000. Making this higher repayment amount every month, how long will it take to pay off the mortgage? The present value of the stream of payments must be equal to R190 000 (the present value of the borrowed amount). So we need to solve for n in: R2 000 × [1 − (1 + 0,75%)−n ]/0,75% = 1 − (1 + 0,75%) log(1 + 0,75%)−n
−n
R190 000 (R190 000/2 000) × 0,75% log[(1 − (R190 000/R2 000) × 0,75%]
= =
−n × log(1 + 0,75%) = −n × 0,007472 = n = =
log[(1 − (R190 000/R2 000) × 0,75%] −1,2465 166,8 months 13,9 years
So the mortgage will be completely repaid in less than 14 years, and you would have made a total payment of 166,8× R2 000 = R333 600. Can you see what is happened? Making regular payments of R2 000 instead of the required R1,709,48, you will have saved R76 675,20 (= R410 275,20 - R333 600) in interest, and yet you have only paid an additional amount of R290,52 for 166,8 months, or R48 458,74. You surely 483
37.3
CHAPTER 37. FINANCE - GRADE 12
know by now that the difference between the additional R48 458,74 that you have paid and the R76 675,20 interest that you have saved is attributable to, yes, you have got it, compound interest!
37.3.3
Future Value of a series of Payments
In the same way that when we have a single payment, we can calculate a present value or a future value - we can also do that when we have a series of payments. In the above section, we had a few payments, and we wanted to know what they are worth now - so we calculated present values. But the other possible situation is that we want to look at the future value of a series of payments. Maybe you want to save up for a car, which will cost R45 000 - and you would like to buy it in 2 years time. You have a savings account which pays interest of 12% per annum. You need to work out how much to put into your bank account now, and then again each month for 2 years, until you are ready to buy the car. Can you see the difference between this example and the ones at the start of the chapter where we were only making a single payment into the bank account - whereas now we are making a series of payments into the same account? This is a sinking fund. So, using our usual notation, let us write out the answer. Make sure you agree how we come up with this. Because we are making monthly payments, everything needs to be in months. So let A be the closing balance you need to buy a car, P is how much you need to pay into the bank account each month, and i12 is the monthly interest rate. (Careful - because 12% is the annual interest rate, so we will need to work out later what the month interest rate is!) A = P (1 + i12)24 + P (1 + i12)23 + ... + P (1 + i12)1 Here are some important points to remember when deriving this formula: 1. We are calculating future values, so in this example we use (1 + i12)n and not (1 + i12)−n . Check back to the start of the chapter is this is not obvious to you by now. 2. If you draw a timeline you will see that the time between the first payment and when you buy the car is 24 months, which is why we use 24 in the first exponent. 3. Again, looking at the timeline, you can see that the 24th payment is being made one month before you buy the car - which is why the last exponent is a 1. 4. Always check that you have got the right number of payments in the equation. Check right now that you agree that there are 24 terms in the formula above. So, now that we have the right starting point, let us simplify this equation: A = = P [(1 + i12)24 + (1 + i12)23 + . . . + (1 + i12)1 ] P [X 24 + X 23 + . . . + X 1 ] using X = (1 + i12)
Note that this time X has a positive exponent not a negative exponent, because we are doing future values. This is not a rule you have to memorise - you can see from the equation what the obvious choice of X should be. Let us reorder the terms: A = P [X 1 + X 2 + . . . + X 24 ] = P · X[1 + X + X 2 + . . . + X 2 3] This is just another sum of a geometric sequence, which as you know can be simplified as: A = = P · X[X n − 1]/((1 + i12) − 1) P · X[X n − 1]/i12 484
CHAPTER 37. FINANCE - GRADE 12
37.4
So if we want to use our numbers, we know that A = R45 000, n=24 (because we are looking at monthly payments, so there are 24 months involved) and i = 12% per annum. BUT (and it is a big but) we need a monthly interest rate. Do not forget that the trick is to keep the time periods and the interest rates in the same units - so if we have monthly payments, make sure you use a monthly interest rate! Using the formula from Section 21.8, we know that (1 + i) = (1 + i12)12 . So we can show that i12 = 0,0094888 = 0,94888%. Therefore, 45 000 = P = P (1,0094888)[(1,0094888)24 − 1]/0,0094888 1662,67
This means you need to invest R1 662,67 each month into that bank account to be able to pay for your car in 2 years time. There is another way of looking at this too - in terms of present values. We know that we need an amount of R45 000 in 24 months time, and at a monthly interest rate of 0,94888%, the present value of this amount is R35 873,72449. Now the question is what monthly amount at 0,94888% interest over 24 month has a present value of R35 873,72449? We have seen this before - it is just like the mortgage questions! So let us go ahead and see if we get to the same answer
P
= R35 873,72449[(1 − (1,0094888)−24)/0,0094888] = R1 662,67
= M/[(1 − (1 + i)−n )/i]
37.3.4
Exercises - Present and Future Values
1. You have taken out a mortgage bond for R875 000 to buy a flat. The bond is for 30 years and the interest rate is 12% per annum payable monthly. A What is the monthly repayment on the bond? B How much interest will be paid in total over the 30 years? 2. How much money must be invested now to obtain regular annuity payments of R 5 500 per month for five years ? The money is invested at 11,1% p.a., compounded monthly. (Answer to the nearest hundred rand)
37.4
Investments and Loans
By now, you should be well equipped to perform calculations with compound interest. This section aims to allow you to use these valuable skills to critically analyse investment and load options that you will come across in your later life. This way, you will be able to make informed decisions on options presented to you. At this stage, you should understand the mathematical theory behind compound interest. However, the numerical implications of compound interest is often subtle and far from obvious. Recall the example in section ??FIXTHIS. For an extra payment of R290,52 a month, we could have paid off our loan in less than 14 years instead of 20 years. This provides a good illustration of the long term effect of compound interest that is often surprising. In the following section, we'll aim to explain the reason for drastic deduction in times it takes to repay the loan.
37.4.1
Loan Schedules
So far, we have been working out loan repayment amounts by taking all the payments and discounting them back to the present time. We are not considering the repayments individually. 485
37.4
CHAPTER 37. FINANCE - GRADE 12
Think about the time you make a repayment to the bank. There are numerous questions that could be raised: how much do you still owe them? Since you are paying off the loan, surely you must owe them less money, but how much less? We know that we'll be paying interest on the money we still owe the bank. When exactly do we pay interest? How much interest are we paying? The answer to these questions lie in something called the load schedule. We will continue to use the example from section ??FIXTHIS. There is a loan amount of R190 000. We are paying it off over 20 years at an interest of 9% per annum payable monthly. We worked out that the repayments should be R1 709,48. Consider the first payment of R1 709,48 one month into the loan. First, we can work out how much interest we owe the bank at this moment. We borrowed R190 000 a month ago, so we should owe:
I
= = =
M × i12 R190 000 × 0,75% R1 425
We are paying them R1 425 in interest. We calls this the interest component of the repayment. We are only paying off R1 709,48 - R1 425 = R284.48 of what we owe! This is called the capital component. That means we still owe R190 000 - R284,48 = R189 715,52. This is called the capital outstanding. Let's see what happens at end of the second month. The amount of interest we need to pay is the interest on the capital outstanding.
I
= = =
M × i12
R189 715,52 × 0,75% R1 422,87
Since we don't owe the bank as much as we did last time, we also owe a little less interest. The capital component of the repayment is now R1 709,48 - R1 422,87 = R286,61. The capital outstanding will be R189 715,52 - R286,61 = R189 428,91. This way, we can break each of our repayments down into an interest part and the part that goes towards paying off the loan. This is a simple and repetitive process. Table 37.1 is a table showing the breakdown of the first 12 payments. This is called a loan schedule. Now, let's see the same thing again, but with R2 000 being repaid each year. We expect the numbers to change. However, how much will they change by? As before, we owe R1 425 in interest in interest. After one month. However, we are paying R2 000 this time. That leaves R575 that goes towards paying off the capital outstanding, reducing it to R189 425. By the end of the second month, the interest owed is R1 420,69 (That's R189 425×i12). Our R2 000 pays for that interest, and reduces the capital amount owed by R2 000 - R1 420,69 = R579,31. This reduces the amount outstanding to R188 845,69. Doing the same calculations as before yields a new loan schedule shown in Table 37.2. The important numbers to notice is the "Capital Component" column. Note that when we are paying off R2 000 a month as compared to R1 709,48 a month, this column more than doubles? In the beginning of paying off a loan, very little of our money is used to pay off the captital outstanding. Therefore, even a small incread in repayment amounts can significantly increase the speed at which we are paying off the capital. Whatsmore, look at the amount we are still owing after one year (i.e. at time 12). When we were paying R1 709,48 a month, we still owe R186 441,84. However, if we increase the repayments to R2 000 a month, the amount outstanding decreases by over R3 000 to R182 808,14. This means we would have paid off over R7 000 in our first year instead of less than R4 000. This 486
increased speed at which we are paying off the capital portion of the loan is what allows us to pay off the whole load in around 14 years instead of the original 20. Note however, the effect of paying R2 000 instead of R1 709,48 is more significant in be beginning of the loan than near the end of the loan. It is noted that in this instance, by paying slightly more than what the bank would ask you to pay, you can pay off a loan a lot quicker. The natural question to ask here is: why are banks asking us to pay the lower amount for much longer then? Are they trying to cheat us out of our money? There is no simple answer to this. Banks provide a service to us in return for a fee, so they are out to make a profit. However, they need to be careful not to cheat their customers for fear that they'll simply use another bank. The central issue here is one of scale. For us, the changes involved appear big. We are paying off our loan 6 years earlier by paying just a bit more a month. To a bank, however, it doesn't matter much either way. In all likelihoxod, it doesn't affect their profit margins one bit! Remember that a bank calculates repayment amount using the same methods as we've been learning. Therefore, they are correct amounts for given interest rates and terms. As a result, which amount is repaid does generally make a bank more or less money. It's a simple matter of less money now or more money later. Banks generally use a 20 year repayment period by default. Learning about financial mathematics enables you to duplicate these calculations for yourself. This way, you can decide what's best for you. You can decide how much you want to repay each month and you'll know of its effects. A bank wouldn't care much either way, so you should pick something that suits you.
Worked Example 165: Monthly Payments Question: Stefan and Marna want to buy a house that costs R 1 200 000. Their parents offer to put down a 20% payment towards the cost of the house. They need to get a moratage for the balance. What are their monthly repayments if the term of the home loan is 30 years and the interest is 7,5%, compounded monthly ? Answer Step 1 : Determine how much money they need to borrow R1 200 00 − R240 000 = R960 000 Step 2 : Determine how to approach the problem Use the formula: P = Where P = 960 000 n = 30 × 12 = 360months i = 0,075 ÷ 12 = 0,00625 Step 3 : Solve the problem x[1 − (1 + 0,00625)−360] 0,00625 x(143,017 627 3) R6 712,46 x[1 − (1 + i)−n ] i
R960 000 = = x =
Step 4 : Write the final answer The monthly repayments = R6 712,46
488
CHAPTER 37. FINANCE - GRADE 12
37.5
37.4.2
Exercises - Investments and Loans
1. A property costs R1 800 000. Calculate the monthly repayments if the interest rate is 14% p.a. compounded monthly and the loan must be paid of in 20 years time. 2. A loan of R 4 200 is to be returned in two equal annual instalments. If the rate of interest os 10% per annum, compounded annually, calculate the amount of each instalment.
37.4.3
Calculating Capital Outstanding
As defined in Section 37.4.1, Capital outstanding is the amount we still owe the people we borrowed money from at a given moment in time. We also saw how we can calculate this using loan schedules. However, there is a significant disadvantage to this method: it is very time consuming. For example, in order to calculate how much capital is still outstanding at time 12 using the loan schedule, we'll have to first calculate how much capital is outstanding at time 1 through to 11 as well. This is already quite a bit more work than we'd like to do. Can you imagine calculating the amount outstanding after 10 years (time 120)? Fortunately, there is an easier method. However, it is not immediately why this works, so let's take some time to examine the concept. Prospective method for Capital Outstanding Let's say that after a certain number of years, just after we made a repayment, we still owe amount Y . What do we know about Y ? We know that using the loan schedule, we can calculate what it equals to, but that is a lot of repetitive work. We also know that Y is the amount that we are still going to pay off. In other words, all the repayments we are still going to make in the future will exactly pay off Y . This is true because in the end, after all the repayments, we won't be owing anything. Therefore, the present value of all outstanding future payments equal the present amount outstanding. This is the prospective method for calculating capital outstanding. Let's return to a previous example. Recall the case where we were trying to repay a loan of R200 000 over 20 years. At an interested rate of 9% compounded monthly, the monthly repayment is R1 709,48. In table 37.1, we can see that after 12 month, the amount outstanding is R186 441,84. Let's try to work this out using the the prospective method. After time 12, there is still 19 × 12 = 228 repayments left of R1 709,48 each. The present value is:
n = i = Y = =
228 0,75% R1 709,48 × R186 441,92 1 − 1,0075−228 0,0075
Oops! This seems to be almost right, but not quite. We should have got R186 441,84. We are 8 cents out. However, this is in fact not a mistake. Remember that when we worked out the monthly repayments, we rounded to the nearest cents and arrived at R1 709,48. This was because one cannot make a payment for a fraction of a cent. Therefore, the rounding off error was carried through. That's why the two figures don't match exactly. In financial mathematics, this is largely unavoidable.
37.5
Formulae Sheet
As an easy reference, here are the key formulae that we derived and used during this chapter. While memorising them is nice (there are not many), it is the application that is useful. Financial 489
37.6
CHAPTER 37. FINANCE - GRADE 12
experts are not paid a salary in order to recite formulae, they are paid a salary to use the right methods to solve financial problems.
37.5.1
P i n iT
Definitions
Principal (the amount of money at the starting point of the calculation) interest rate, normally the effective rate per annum period for which the investment is made the interest rate paid T times per annum, i.e. iT = Nominal Interest Rate T
Important: Always keep the interest and the time period in the same units of time (e.g. both in years, or both in months etc.).
37.6
End of Chapter Exercises
1. Thabo is about to invest his R8 500 bonus in a special banking product which will pay 1% per annum for 1 month, then 2% per annum for the next 2 months, then 3% per annum for the next 3 months, 4% per annum for the next 4 months, and 0% for the rest of the year. The are going to charge him R100 to set up the account. How much can he expect to get back at the end of the period? 2. A special bank account pays simple interest of 8% per annum. Calculate the opening balance required to generate a closing balance of R5 000 after 2 years. 3. A different bank account pays compound interest of 8% per annum. Calculate the opening balance required to generate a closing balance of R5 000 after 2 years. 4. Which of the two answers above is lower, and why? 5. After 7 months after an initial deposit, the value of a bank account which pays compound interest of 7,5% per annum is R3 650,81. What was the value of the initial deposit? 6. Suppose you invest R500 this year compounded at interest rate i for a year in Bank T. In the following year you invest the accumulation that you received for another year at the same interest rate and on the third year, you invested the accumulation you received at the same interest rate too. If P represents the present value (R500), find a pattern for this investment. [Hint: find a formula] 7. Thabani and Lungelo are both using UKZN Bank for their saving. Suppose Lungelo makes a deposit of X today at interest rate of i for six years. Thabani makes a deposit of 3X at an interest rate of 0.05. Thabani made his deposit 3 years after Lungelo made his first deposit. If after 6 years, their investments are equal, calculate the value of i and find X. if the sum of their investment is R20 000, use X you got to find out how much Thabani got in 6 years. 490
CHAPTER 37. FINANCE - GRADE 12
37.6
8. Sipho invests R500 at an interest rate of log(1,12) for 5 years. Themba, Sipho's sister invested R200 at interest rate i for 10 years on the same date that her brother made his first deposit. If after 5 years, Themba's accumulation equals Sipho's, find the interest rate i and find out whether Themba will be able to buy her favorite cell phone after 10 years which costs R2 000. 9. Moira deposits R20 000 in her saving account for 2 years at an interest rate of 0.05. After 2 years, she invested her accumulation for another 2 years, at the same interest rate. After 4 years, she invested her accumulation for which she got for another 2 years at an interest rate of 5 %. After 6 years she choose to buy a car which costs R26 000. Her husband, Robert invested the same amount at interest rate of 5 % for 6 years. A Without using any numbers, find a pattern for Moira's investment? B How Moira's investment differ from Robert's? 10. Calculate the real cost of a loan of R10 000 for 5 years at 5% capitalised monthly and half yearly. 11. Determine how long, in years, it will take for the value of a motor vehicle to decrease to 25% of its original value if the rate of depreciation, based on the reducing-balance method, is 21% per annum. 12. Andr´ and Thoko, decided to invest their winnings (amounting to R10 000) from their e science project. They decided to divide their winnings according to the following: Because Andr was the head of the project and he spent more time on it, Andr´ got 65,2 % of the e winnings and Thoko got 34,8%. So, Thoko decided to invest only 0,5 % of the share of her sum and Andr´decided to invest 1,5 % of the share of his sum. When they calculated how e much each contributed in the investment, Thoko had 25 % and Andr´ had 75 % share. e They planned to invest their money for 20 years , but, as a result of Thoko finding a job in Australia 7 years after their initial investment. They both decided to take whatever value was there and split it according to their initial investment(in terms of percentages). Find how much each will get after 7 years, if the interest rate is equal to the percentage that Thoko invested (NOT the money but the percentage).
491
37.6
CHAPTER 37. FINANCE - GRADE 12
492
Chapter 38
Factorising Cubic Polynomials Grade 12
38.1 Introduction
In grades 10 and 11, you learnt how to solve different types of equations. Most of the solutions, relied on being able to factorise some expression and the factorisation of quadratics was studied in detail. This chapter focusses on the factorisation of cubic polynomials, that is expressions with the highest power equal to 3.
38.2
The Factor Theorem
The factor theorem describes the relationship between the root of a polynomial and a factor of the polynomial. Definition: Factor Theorem For any polynomial, f (x), for all values of a which satisfy f (a) = 0, (x − a) is a factor of f (x). Or, more concisely: f (x) = q(x) x−a is a polynomial. In other words: If the remainder when dividing f (x) by (x − a) is zero, then (ax + b) is a factor of f (x). b So if f (− a ) = 0, then (ax + b) is a factor of f (x).
Cubic expressions have a highest power of 3 on the unknown variable. This means that there should be at least 3 factors. We have seen in Grade 10 that the sum and difference of cubes is factorised as follows.: (x + y)(x2 − xy + y 2 ) = x3 + y 3 and (x − y)(x2 + xy + y 2 ) = x3 − y 3 We also saw that the quadratic terms do not have rational roots. There are many methods of factorising a cubic polynomial. The general method is similar to that used to factorise quadratic equations. If you have a cubic polynomial of the form: f (x) = ax3 + bx2 + cx + d then you should expect factors of the form: (Ax + B)(Cx + D)(Ex + F ). (38.1)
We will deal with simplest case first. When a = 1, then A = C = E = 1, and you only have to determine B, D and F . For example, find the factors of: x3 − 2x2 − 5x + 6. In this case we have a b = 1 = −2
This is a set of three equations in three unknowns. However, we know that B, D and F are factors of 6 because BDF = 6. Therefore we can use a trial and error method to find B, D and F . This can become a very tedious method, therefore the Factor Theorem can be used to find the factors of cubic polynomials.
In general, to factorise a cubic polynomial, you find one factor by trial and error. Use the factor theorem to confirm that the guess is a root. Then divide the cubic polynomial by the factor to obtain a quadratic. Once you have the quadratic, you can apply the standard methods to factorise the quadratic. For example the factors of x3 − 2x2 − 5x + 6 can be found as follows: There are three factors which we can write as (x − a)(x − b)(x − c). 495
In grades 10 and 11 you have learnt about linear functions and quadratic functions as well as the hyperbolic functions and exponential functions and many more. In grade 12 you are expected to demonstrate the ability to work with various types of functions and relations including the inverses of some functions and generate graphs of the inverse relations of functions, in particular the inverses of: y = ax + q y = ax2 y = ax; a > 0 .
39.2
Definition of a Function
A function is a relation for which there is only one value of y corresponding to any value of x. We sometimes write y = f (x), which is notation meaning 'y is a function of x'. This definition makes complete sense when compared to our real world examples — each person has only one height, so height is a function of people; on each day, in a specific town, there is only one average temperature. However, some very common mathematical constructions are not functions. For example, consider the relation x2 + y 2 = 4. This relation describes a circle of radius 2 centred at the origin, as in figure 39.1. If we let x = 0, we see that y 2 = 4 and thus either y = 2 or y = −2. Since there are two y values which are possible for the same x value, the relation x2 + y 2 = 4 is not a function. There is a simple test to check if a relation is a function, by looking at its graph. This test is called the vertical line test. If it is possible to draw any vertical line (a line of constant x) which crosses the relation more than once, then the relation is not a function. If more than one intersection point exists, then the intersections correspond to multiple values of y for a single value of x. We can see this with our previous example of the circle by looking at its graph again in Figure 39.1. We see that we can draw a vertical line, for example the dotted line in the drawing, which cuts the circle more than once. Therefore this is not a function.
39.2.1
Exercises
1. State whether each of the following equations are functions or not: A x+y =4 501
39.3
CHAPTER 39. FUNCTIONS AND GRAPHS - GRADE 12
2 1
−2
−1
1 −1 −2
2
Figure 39.1: Graph of y 2 + x2 = 4
B y=x 4 C y = 2x D x2 + y 2 = 4 2. The table gives the average per capita income, d, in a region of the country as a function of the percent unemployed, u. Write down the equation to show that income is a function of the persent unemployed. u d 1 22500 2 22000 3 21500 4 21000
39.3
Notation used for Functions
In grade 10 you were introduced to the notation used to "name" a function. In a function y = f (x), y is called the dependent variable, because the value of y depends on what you choose as x. We say x is the independent variable, since we can choose x to be any number. Similarly if g(t) = 2t + 1, then t is the independent variable and g is the function name. If f (x) = 3x − 5 and you are ask to determine f (3), then you have to work out the value for f (x) when x = 3. For example, f (x) = f (3) = = 3x − 5 3(3) − 5
4
39.4
Graphs of Inverse Functions
In earlier grades, you studied various types of functions and understood the effect of various parameters in the general equation. In this section, we will consider inverse functions. An inverse function is a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f −1 is its inverse function if and only if for every x ∈ X we have: f −1 (f (x)) = f (f −1 (x)) = x (39.1) For example, if the function x → 3x + 2 is given, then its inverse function is x → is usually written as: f f −1 : : x → 3x + 2 (x − 2) x→ 3 502 (x − 2) . This 3 (39.2) (39.3)
CHAPTER 39. FUNCTIONS AND GRAPHS - GRADE 12 The superscript "-1" is not an exponent. If a function f has an inverse then f is said to be invertible.
39.4
If f is a real-valued function, then for f to have a valid inverse, it must pass the horizontal line test, that is a horizontal line y = k placed anywhere on the graph of f must pass through f exactly once for all real k. It is possible to work around this condition, by defining a multi-valued function as an inverse. If one represents the function f graphically in a xy-coordinate system, then the graph of f −1 is the reflection of the graph of f across the line y = x. Algebraically, one computes the inverse function of f by solving the equation y = f (x) for x, and then exchanging y and x to get y = f −1 (x)
The inverse function of a straight line is also a straight line. For example, the straight line equation given by y = 2x − 3 has as inverse the function, y = 1 3 2 x + 2 . The graphs of these functions are shown in Figure 39.2. It can be seen that the two graphs are reflections of each other across the line y = x.
We have seen that the domain of a function of the form y = ax + q is {x : x ∈ R} and the range is {y : y ∈ R}. Since the inverse function of a straight line is also a straight line, the inverse function will have the same domain and range as the original function.
Intercepts
q 1 The general form of the inverse function of the form y = ax + q is y = a x − a .
q By setting x = 0 we have that the y-intercept is yint = − a . Similarly, by setting y = 0 we have that the x-intercept is xint = q. q 1 It is interesting to note that if f (x) = ax + q, then f −1 (x) = a x − a and the y-intercept of −1 f (x) is the x-intercept of f (x) and the x-intercept of f (x) is the y-intercept of f −1 (x).
39.4.2
Exercises
1. Given f (x) = 2x − 3, find f −1 (x) 2. Consider the function f (x) = 3x − 7. A Is the relation a function? B Identify the domain and range. 3. Sketch the graph of the function f (x) = 3x − 1 and its inverse on the same set of axes. 4. The inverse of a function is f −1 (x) = 2x − 4, what is the function f (x)?
We see that the inverse function of y = ax2 is not a function because it fails the vertical line √ test. If we draw a vertical line through the graph of f −1 (x) = ± x, the line intersects the graph more than once. There has to be a restriction on the domain of a parabola for the inverse to also be a function. Consider the function f (x) = −x2 + 9. The inverse of f can be found by witing f (y) = x. Then x = y2 = y = −y 2 + 9 9−x √ ± 9−x
√ √ If x ≥ 0, then 9 − x is a function. If the restriction on the domain of f is x ≤ 0 then − 9 − x would be a function.
39.4.4
Exercises
1. The graph of f −1 is shown. Find the equation of f , given that the graph of f is a parabola. (Do not simplify your answer) 504
2. f (x) = 2x2 . A Draw the graph of f and state its domain and range. B Find f −1 and state the domain and range. C What must the domain of f be, so that f −1 is a function ? 3. Sketch the graph of x = − 10 − y 2 . Label a point on the graph other than the intercepts with the axes. 4. A Sketch the graph of y = x2 labelling a point other than the origin on your graph. B Find the equation of the inverse of the above graph in the form y = . . .. √ C Now sketch the y = x. √ D The tangent to the graph of y = x at the point A(9;3) intersects the x-axis at B. Find the equation of this tangent and hence or otherwise prove that the y-axis bisects the straight line AB. 5. Given: g(x) = −1 + √ x, find the inverse of g(x) in the form g −1 (x). 505
Figure 39.4: The function f (x) = 10x and its inverse f −1 (x) = log(x). The line y = x is shown as a dashed line. The exponential function and the logarithmic function are inverses of each other; the graph of the one is the graph of the other, reflected in the line y = x. The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.
39.4.6
Exercises
1 1. Given that f (x) = [ 5 ]x , sketch the graphs of f and f −1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is f and which is f −1 .
2. Given that f (x) = 4−x , A Sketch the graphs of f and f −1 on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is f and which is f −1 . B Write f −1 in the form y = . . .. √ 3. Given g(x) = −1 + x, find the inverse of g(x) in the form g −1 (x) = . . . 4. A B C D
Sketch the graph of y = x2 , labeling a point other than the origin on your graph. Find the equation of the inverse of the above graph in the form y = . . . √ Now, sketch y = x. √ 506
6. Given the equation h(x) = 3x A Write down the inverse in the form h−1 (x) = ... B Sketch the graphs of h(x) and h−1 (x) on teh same set of axes, labelling the intercepts with the axes. C For which values of x is h−1 (x) undefined ? 7. A Sketch the graph of y = x2 , labelling a point other than the origin on your graph. B Find the equation of the inverse of the above graph in the form y = . . . √ C Now, sketch y = x. √ D
508
Chapter 40
Differential Calculus - Grade 12
40.1 Why do I have to learn this stuff?
Calculus is one of the central branches of mathematics and was developed from algebra and geometry. Calculus is built on the concept of limits, which will be discussed in this chapter. Calculus consists of two complementary ideas: differential calculus and integral calculus. Only differential calculus will be studied. Differential calculus is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behaviour of functions. This can be illustrated by the slope of a function's graph. Examples of typical differential calculus problems include: finding the acceleration and velocity of a free-falling body at a particular moment and finding the optimal number of units a company should produce to maximize its profit. Calculus is fundamentally different from the mathematics that you have studied previously. Calculus is more dynamic and less static. It is concerned with change and motion. It deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Calculus is a tool to understand many natural phenomena like how the wind blows, how water flows, how light travels, how sound travels and how the planets move. However, other human activities such as economics are also made easier with calculus. In this section we give a glimpse of some of the main ideas of calculus by showing how limits arise when we attempt to solve a variety of problems. Extension: Integral Calculus Integral calculus is concerned with the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. Differential and integral calculus act inversely to each other. Examples of typical integral calculus problems include finding areas and volumes, finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure and finding the amount of rain that fell in a certain area if the rain fell at a specific rate.
teresting Both Isaac Newton (4 January 1643 – 31 March 1727) and Gottfried Liebnitz Interesting Fact Fact (1 July 1646 – 14 November 1716 (Hanover, Germany)) are credited with the 'invention' of calculus. Newton was the first to apply calculus to general physics, while Liebnitz developed most of the notation that is still in use today.
When Newton and Leibniz first published their results, there was some controversy over whether Leibniz's work was independent of Newton's. While Newton derived his results years before Leibniz, it was only some time after Leibniz published in 1684 that Newton published. Later, Newton would claim that Leibniz 509
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12 got the idea from Newton's notes on the subject; however examination of the papers of Leibniz and Newton show they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Today, both Newton and Leibniz are given credit for independently developing calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluxions".
40.2
40.2.1
Limits
A Tale of Achilles and the Tortoise
teresting Zeno (circa 490 BC - circa 430 BC) was a pre-Socratic Greek philosopher of Interesting Fact Fact southern Italy who is famous for his paradoxes.
One of Zeno's paradoxes can be summarised by:
Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles is very fast. So, the tortoise asks Achilles for a head-start. Achilles agrees to give the tortoise a 1 000 m head start. Does Achilles overtake the tortoise?
However, Zeno (the Greek philosopher who thought up this problem) looked at it as follows: Achilles takes 1000 = 500 s t= 2 to travel the 1 000 m head start that the tortoise had. However, in this 500 s, the tortoise has travelled a further x = (500)(0,25) = 125 m. 125 = 62,5 s 2 to travel the 125 m. In this 62,5 s, the tortoise travels a further t= x = (62,5)(0,25) = 15,625 m. Zeno saw that Achilles would always get closer but wouldn't actually overtake the tortoise. Achilles then takes another
40.2.2
Sequences, Series and Functions
So what does Zeno, Achilles and the tortoise have to do with calculus? Well, in Grades 10 and 11 you studied sequences. For the sequence 1 2 3 4 0, , , , , . . . 2 3 4 5 which is defined by the expression 1 n the terms get closer to 1 as n gets larger. Similarly, for the sequence an = 1 − 1 1 1 1 1, , , , , . . . 2 3 4 5 which is defined by the expression 1 n the terms get closer to 0 as n gets larger. We have also seen that the infinite geometric series has a finite total. The infinite geometric series is an = S∞ =
∞ i=1
a1 .ri−1 =
a1 1−r
for
−1
where a1 is the first term of the series and r is the common ratio. 511
40.2
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
We see that there are some functions where the value of the function gets close to or approaches a certain value. x2 + 4x − 12 x+6 The numerator of the function can be factorised as: y= y= (x + 6)(x − 2) . x+6 Similarly, for the function:
Then we can cancel the x − 6 from numerator and denominator and we are left with: y = x − 2. However, we are only able to cancel the x + 6 term if x = −6. If x = −6, then the denominator becomes 0 and the function is not defined. This means that the domain of the function does not include x = −6. But we can examine what happens to the values for y as x gets close to -6. These values are listed in Table 40.1 which shows that as x gets closer to -6, y gets close to 8. (x + 6)(x − 2) as x gets close to -6. x+6
The graph of this function is shown in Figure 40.1. The graph is a straight line with slope 1 and intercept -2, but with a missing section at x = −6. Extension: Continuity We say that a function is continuous if there are no values of the independent variable for which the function is undefined.
Worked Example 172: Limits Notation Question: Summarise the following situation by using limit notation: As x gets close to 1, the value of the function y =x+2 gets close to 3. 513
40.2 Answer This is written as:
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
x→1
lim x + 2 = 3
in limit notation.
We can also have the situation where a function has a different value depending on whether x approaches from the left or the right. An example of this is shown in Figure 40.2.
4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 1 2 3 4 5 6 7
1 Figure 40.2: Graph of y = x .
1 As x → 0 from the left, y = x approaches −∞. As x → 0 from the right, y = +∞. This is written in limits notation as:
1 x
approaches
x→0−
lim
1 = −∞ x
for x approaching zero from the left and lim 1 =∞ x
x→0+
for x approaching zero from the right. You can calculate the limit of many different functions using a set method. Method: Limits If you are required to calculate a limit like limx→a then: 1. Simplify the expression completely. 2. If it is possible, cancel all common terms. 3. Let x approach the a.
In Grade 10 you learnt about average gradients on a curve. The average gradient between any two points on a curve is given by the gradient of the straight line that passes through both points. In Grade 11 you were introduced to the idea of a gradient at a single point on a curve. We saw that this was the gradient of the tangent to the curve at the given point, but we did not learn how to determine the gradient of the tangent. Now let us consider the problem of trying to find the gradient of a tangent t to a curve with equation y = f (x) at a given point P . tangent P f (x) We know how to calculate the average gradient between two points on a curve, but we need two points. The problem now is that we only have one point, namely P . To get around the problem we first consider a secant to the curve that passes through point P and another point on the curve Q. We can now find the average gradient of the curve between points P and Q. secant f (a) f (a − h) P Q f (x)
a−h
a
If the x-coordinate of P is a, then the y-coordinate is f (a). Similarly, if the x-coordinate of Q is a − h, then the y-coordinate is f (a − h). If we choose a as x2 and a − h as x1 , then: y1 = f (a − h) y2 = f (a). We can now calculate the average gradient as: y2 − y1 x2 − x1 = = f (a) − f (a − h) a − (a − h) f (a) − f (a − h) h (40.12) (40.13)
Now imagine that Q moves along the curve toward P . The secant line approaches the tangent line as its limiting position. This means that the average gradient of the secant approaches the gradient of the tangent to the curve at P . In (40.13) we see that as point Q approaches point P , h gets closer to 0. When h = 0, points P and Q are equal. We can now use our knowledge of limits to write this as: gradient at P = lim
h→0
f (a) − f (a − h) . h
(40.14)
and we say that the gradient at point P is the limit of the average gradient as Q approaches P along the curve. 516
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.2
Activity :: Investigation : Limits The gradient at a point x on a curve defined by f (x) can also be written as: lim f (x + h) − f (x) h (40.15)
h→0
Show that this is equivalent to (40.14).
Worked Example 176: Limits Question: For the function f (x) = 2x2 − 5x, determine the gradient of the tangent to the curve at the point x = 2. Answer Step 1 : Calculating the gradient at a point We know that the gradient at a point x is given by: lim f (x + h) − f (x) h
The tangent problem has given rise to the branch of calculus called differential calculus and the equation: f (x + h) − f (x) lim h→0 h defines the derivative of the function f (x). Using (40.15) to calculate the derivative is called finding the derivative from first principles.
There are a few different notations used to refer to derivatives. If we use the traditional notation y = f (x) to indicate that the dependent variable is y and the independent variable is x, then some common alternative notations for the derivative are as follows: f ′ (x) = y ′ = dy df d = = f (x) = Df (x) = Dx f (x) dx dx dx
d The symbols D and dx are called differential operators because they indicate the operation of differentiation, which is the process of calculating a derivative. It is very important that you learn to identify these different ways of denoting the derivative, and that you are consistent in your usage of them when answering questions.
dy Important: Though we choose to use a fractional form of representation, dx is a limit and dy dy is not a fraction, i.e. dx does not mean dy ÷ dx. dx means y differentiated with respect to dp d x. Thus, dx means p differentiated with respect to x. The ' dx ' is the "operator", operating on some function of x.
Worked Example 178: Derivatives - First Principles Question: Calculate the derivative of g(x) = x − 1 from first principles. Answer Step 1 : Calculating the gradient at a point We know that the gradient at a point x is given by: g ′ (x) = lim g(x + h) − g(x) h→0 h 519
Thus far we have learnt about how to differentiate various functions, but I am sure that you are beginning to ask, What is the point of learning about derivatives? Well, we know one important fact about a derivative: it is a gradient. So, any problems involving the calculations of gradients or rates of change can use derivatives. One simple application is to draw graphs of functions by firstly determine the gradients of straight lines and secondly to determine the turning points of the graph.
40.5.1
Finding Equations of Tangents to Curves
In section 40.2.4 we saw that finding the gradient of a tangent to a curve is the same as finding the slope of the same curve at the point of the tangent. We also saw that the gradient of a function at a point is just its derivative. Since we have the gradient of the tangent and the point on the curve through which the tangent passes, we can find the equation of the tangent.
Worked Example 181: Finding the Equation of a Tangent to a Curve Question: Find the equation of the tangent to the curve y = x2 at the point (1,1) and draw both functions. Answer Step 1 : Determine what is required We are required to determine the equation of the tangent to the curve defined by y = x2 at the point (1,1). The tangent is a straight line and we can find the equation by using derivatives to find the gradient of the straight line. Then we will have the gradient and one point on the line, so we can find the equation using: y − y1 = m(x − x1 ) from grade 11 Coordinate Geometry. Step 2 : Differentiate the function Using our rules of differentiation we get: y ′ = 2x Step 3 : Find the gradient at the point (1,1) In order to determine the gradient at the point (1,1), we substitute the x-value into the equation for the derivative. So, y ′ at x = 1 is: 2(1) = 2 Step 4 : Find equation of tangent y − y1 y−1 y y = = m(x − x1 ) (2)(x − 1)
Differentiation can be used to sketch the graphs of functions, by helping determine the turning points. We know that if a graph is increasing on an interval and reaches a turning point, then the graph will start decreasing after the turning point. The turning point is also known as a stationary point because the gradient at a turning point is 0. We can then use this information to calculate turning points, by calculating the points at which the derivative of a function is 0. Important: If x = a is a turning point of f (x), then: f ′ (a) = 0 This means that the derivative is 0 at a turning point. Take the graph of y = x2 as an example. We know that the graph of this function has a turning point at (0,0), but we can use the derivative of the function: y ′ = 2x and set it equal to 0 to find the x-value for which the graph has a turning point. 2x = 0 x = 0 We then substitute this into the equation of the graph (i.e. y = x2 ) to determine the y-coordinate of the turning point: f (0) = (0)2 = 0 This corresponds to the point that we have previously calculated. 524
We are now ready to sketch graphs of functions. Method: Sketching GraphsSuppose we are given that f (x) = ax3 + bx2 + cx + d, then there are five steps to be followed to sketch the graph of the function:
1. If a > 0, then the graph is increasing from left to right, and has a maximum and then a minimum. As x increases, so does f (x). If a < 0, then the graph decreasing is from left to right, and has first a minimum and then a maximum. f (x) decreases as x increases. 2. Determine the value of the y-intercept by substituting x = 0 into f (x) 3. Determine the x-intercepts by factorising ax3 + bx2 + cx + d = 0 and solving for x. First try to eliminate constant common factors, and to group like terms together so that the expression is expressed as economically as possible. Use the factor theorem if necessary. 525
40.5
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
df dx
4. Find the turning points of the function by working out the derivative zero, and solving for x.
and setting it to
5. Determine the y-coordinates of the turning points by substituting the x values obtained in the previous step, into the expression for f (x). 6. Draw a neat sketch.
which does not have real roots. Therefore, the graph of g(x) does not have any x-intercepts. Step 3 : Find the turning points of the function dg Work out the derivative dx and set it to zero to for the x coordinate of the turning point. dg = 2x − 1 dx dg = dx 2x − 1 = 2x = x =
0 0 1 1 2
Step 4 : Determine the y-coordinates of the turning points by substituting the x values obtained in the previous step, into the expression for f (x). 1 y coordinate of turning point is given by calculating g( 2 ). 1 g( ) = 2 = =
7 The turning point is at ( 1 , 4 ) 2 Step 5 : Draw a neat sketch
Exercise: Sketching Graphs 1. Given f (x) = x3 + x2 − 5x + 3: A Show that (x − 1) is a factor of f (x) and hence fatorise f (x) fully. B Find the coordinates of the intercepts with the axes and the turning points and sketch the graph
dy If the derivative ( dx ) is zero at a point, the gradient of the tangent at that point is zero. It means that a turning point occurs as seen in the previous example.
y 9 8 7 6 5 (3;4) 4 3 2 1 −1 −1 (1;0) 1 2 3 4 (4;0) x
From the drawing the point (1;0) represents a local minimum and the point (3;4) the local maximum. A graph has a horizontal point of inflexion where the derivative is zero but the sign of the sign of the gradient does not change. That means the graph always increases or always decreases. 529
40.6 y
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
(3;1) x
From this drawing, the point (3;1) is a horizontal point of inflexion, because the sign of the derivative stays positive.
40.6
Using Differential Calculus to Solve Problems
We have seen that differential calculus can be used to determine the stationary points of functions, in order to sketch their graphs. However, determining stationary points also lends itself to the solution of problems that require some variable to be optimised. For example, if fuel used by a car is defined by: f (v) = 3 2 v − 6v + 245 80 (40.20)
where v is the travelling speed, what is the most economical speed (that means the speed that uses the least fuel)? If we draw the graph of this function we find that the graph has a minimum. The speed at the minimum would then give the most economical speed.
We have seen that the coordinates of the turning point can be calculated by differentiating the function and finding the x-coordinate (speed in the case of the example) for which the derivative is 0. Differentiating (40.20), we get: 3 v−6 40 If we set f ′ (v) = 0 we can calculate the speed that corresponds to the turning point. 530 f ′ (v) =
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.6
f ′ (v) 0 v
3 v−6 40 3 v−6 = 40 6 × 40 = 3 = 80 =
This means that the most economical speed is 80 km·hr−1 .
Worked Example 185: Optimisation Problems Question: The sum of two positive numbers is 10. One of the numbers is multiplied by the square of the other. If each number is greater than 0, find the numbers that make this product a maximum. Answer Step 1 : Examine the problem and formulate the equations that are required Let the two numbers be a and b. Then we have: a + b = 10 (40.21)
Step 4 : Write the final answer The product is maximised if a and b are both equal to 5.
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CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
Worked Example 186: Optimisation Problems Question: Michael wants to start a vegetable garden, which he decides to fence off in the shape of a rectangle from the rest of the garden. Michael only has 160 m of fencing, so he decides to use a wall as one border of the vegetable garden. Calculate the width and length of the garden that corresponds to largest possible area that Michael can fence off. wall length, l
garden
width, w Answer Step 1 : Examine the problem and formulate the equations that are required The important pieces of information given are related to the area and modified perimeter of the garden. We know that the area of the garden is: A= w·l (40.25)
We are also told that the fence covers only 3 sides and the three sides should add up to 160 m. This can be written as: 160 = w + l + l However, we can use (40.26) to write w in terms of l: w = 160 − 2l Substitute (40.27) into (40.25) to get: A = (160 − 2l)l = 160l − 2l2 (40.28) (40.27) (40.26)
Exercise: Solving Optimisation Problems using Differential Calculus 1. The sum of two positive numbers is 20. One of the numbers is multiplied by the square of the other. Find the numbers that make this products a maximum. 2. A wooden block is made as shown in the diagram. The ends are right-angled triangles having sides 3x, 4x and 5x. The length of the block is y. The total surface area of the block is 3 600 cm2 .
3x
4x
y
300 − x2 . x B Find the value of x for which the block will have a maximum volume. (Volume = area of base × height.) A Show that y = 3. The diagram shows the plan for a verandah which is to be built on the corner of a cottage. A railing ABCDE is to be constructed around the four edges of the verandah. y C D
x
verandah F B A E cottage If AB = DE = x and BC = CD = y, and the length of the railing must be 30 metres, find the values of x and y for which the verandah will have a maximum area.
533
40.6
CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
40.6.1
Rate of Change problems
f (b)−f (a) b−a
Two concepts were discussed in this chapter: Average rate of change = limh→0 f (x+h)−f (x) . h
and Instan-
taneous rate of change = When we mention rate of change, the latter is implied. Instantaneous rate of change is the derivative. When Average rate of change is required, it will be specifically refer to as average rate of change. Velocity is one of the most common forms of rate of change. Again, average velocity = average rate of change and instantaneous velocity = instantaneous rate of change = derivative. Velocity refers to the increase of distance(s) for a corresponding increade in time (t). The notation commonly used for this is: v(t) = ds = s′ (t) dt
Acceleration is the change in velocity for a corersponding increase in time. Therefore, acceleration is the derivative of velocity a(t) = v ′ (t) This implies that acceleration is the second derivative of the distance(s).
Worked Example 187: Rate of Change Question: The height (in metres) of a golf ball that is hit into the air after t seconds, is given by h(t) = 20t = 5t2 . Determine 1. the average velocity of the ball during the first two seconds 2. the velocity of the ball after 1,5 seconds 3. when the velocity is zero 4. the velocity at which the ball hits the ground 5. the acceleration of the ball Answer Step 1 : Average velocity h(2) − h(0) 2−0 [20(2) − 5(2)2 ] − [20(0) − 5(0)2 ] 2 40 − 20 2 10 ms−1
A Determine the co-ordinates of the turning points of f . B Draw a neat sketch graph of f . Clearly indicate the co-ordinates of the intercepts with the axes, as well as the co-ordinates of the turning points. C For which values of k will the equation f (x) = k , have exactly two real roots? D Determine the equation of the tangent to the graph of f (x) = 2x3 − 5x2 − 4x + 3 at the point where x = 1. 6. A Sketch the graph of f (x) = x3 − 9x2 + 24x − 20, showing all intercepts with the axes and turning points. B Find the equation of the tangent to f (x) at x = 4. 7. Calculate: 1 − x3 x→1 1 − x lim f (x) = 2x2 − x A Use the definition of the derivative to calculate f ′ (x). B Hence, calculate the co-ordinates of the point at which the gradient of the tangent to the graph of f is 7. √ 9. If xy − 5 = x3 , determine dx dy 10. Given: g(x) = (x−2 + x2 )2 . Calculate g ′ (2). 11. Given: A Find: B Solve: f (x) = 2x − 3 f −1 (x) f −1 (x) = 3f ′ (x)
8. Given:
12. Find f ′ (x) for each of the following: √ 5 x3 + 10 A f (x) = 3 (2x2 − 5)(3x + 2) B f (x) = x2 13. Determine the minimum value of the sum of a positive number and its reciprocal. 14. If the displacement s (in metres) of a particle at time t (in seconds) is governed by the 1 equation s = 2 t3 − 2t, find its acceleration after 2 seconds. (Acceleration is the rate of change of velocity, and velocity is the rate of change of displacement.) 15. A After doing some research, a transport company has determined that the rate at which petrol is consumed by one of its large carriers, travelling at an average speed of x km per hour, is given by: P (x) = 55 x + 2x 200 litres per kilometre
i. Assume that the petrol costs R4,00 per litre and the driver earns R18,00 per hour (travelling time). Now deduce that the total cost, C, in Rands, for a 2 000 km trip is given by: 256000 + 40x C(x) = x ii. Hence determine the average speed to be maintained to effect a minimum cost for a 2 000 km trip. 536
i. Determine an expression for the rate of change of temperature with time. ii. During which time interval was the temperature dropping? 16. The depth, d, of water in a kettle t minutes after it starts to boil, is given by d = 1 1 86 − 8 t − 4 t3 , where d is measured in millimetres. A How many millimetres of water are there in the kettle just before it starts to boil? B As the water boils, the level in the kettle drops. Find the rate at which the water level is decreasing when t = 2 minutes. C How many minutes after the kettle starts boiling will the water level be dropping at 1 a rate of 12 8 mm/minute?
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CHAPTER 40. DIFFERENTIAL CALCULUS - GRADE 12
538
Chapter 41
Linear Programming - Grade 12
41.1 Introduction
In Grade 11 you were introduced to linear programming and solved problems by looking at points on the edges of the feasible region. In Grade 12 you will look at how to solve linear programming problems in a more general manner.
41.2
Terminology
Here is a recap of some of the important concepts in linear programming.
41.2.1
Feasible Region and Points
Constraints mean that we cannot just take any x and y when looking for the x and y that optimise our objective function. If we think of the variables x and y as a point (x,y) in the xyplane then we call the set of all points in the xy-plane that satisfy our constraints the feasible region. Any point in the feasible region is called a feasible point. For example, the constraints x≥0 y≥0 mean that every (x,y) we can consider must lie in the first quadrant of the xy plane. The constraint x≥y means that every (x,y) must lie on or below the line y = x and the constraint x ≤ 20 means that x must lie on or to the left of the line x = 20. We can use these constraints to draw the feasible region as shown by the shaded region in Figure 41.1.
Important: ax + by = c If b = 0, feasible points must lie on the line c y = −ax + b b If b = 0, feasible points must lie on the line x = c/a If b = 0, feasible points must lie on or below the line y = − a x + c . b b If b = 0, feasible points must lie on or to the left of the line x = c/a.
ax + by ≤ c
When a constraint is linear, it means that it requires that any feasible point (x,y) lies on one side of or on a line. Interpreting constraints as graphs in the xy plane is very important since it allows us to construct the feasible region such as in Figure 41.1.
41.3
Linear Programming and the Feasible Region
If the objective function and all of the constraints are linear then we call the problem of optimising the objective function subject to these constraints a linear program. All optimisation problems we will look at will be linear programs. The major consequence of the constraints being linear is that the feasible region is always a polygon. This is evident since the constraints that define the feasible region all contribute a line segment to its boundary (see Figure 41.1). It is also always true that the feasible region is a convex polygon. The objective function being linear means that the feasible point(s) that gives the solution of a linear program always lies on one of the vertices of the feasible region. This is very important since, as we will soon see, it gives us a way of solving linear programs. We will now see why the solutions of a linear program always lie on the vertices of the feasible region. Firstly, note that if we think of f (x,y) as lying on the z axis, then the function f (x,y) = ax + by (where a and b are real numbers) is the definition of a plane. If we solve for y in the equation defining the objective function then
=
f (x,y) −a x+ b b
x
(41.1)
What this means is that if we find all the points where f (x,y) = c for any real number c (i.e. f (x,y) is constant with a value of c), then we have the equation of a line. This line we call a level line of the objective function. 540
have also been drawn in. It is very important to realise that these are not the only level lines; in fact, there are infinitely many of them and they are all parallel to each other. Remember that if we look at any one level line f (x,y) has the same value for every point (x,y) that lies on that line. Also, f (x,y) will always have different values on different level lines. y 20 15 10 5 5 10 15 20 f (x,y) = −20 f (x,y) = −10 f (x,y) = 0 f (x,y) = 10 f (x,y) = 20 x
If a ruler is placed on the level line corresponding to f (x,y) = −20 in Figure 41.2 and moved down the page parallel to this line then it is clear that the ruler will be moving over level lines which correspond to larger values of f (x,y). So if we wanted to maximise f (x,y) then we simply move the ruler down the page until we reach the "lowest" point in the feasible region. This point will then be the feasible point that maximises f (x,y). Similarly, if we wanted to minimise f (x,y) then the "highest" feasible point will give the minimum value of f (x,y). Since our feasible region is a polygon, these points will always lie on vertices in the feasible region. The fact that the value of our objective function along the line of the ruler increases as we move it down and decreases as we move it up depends on this particular example. Some other examples might have that the function increases as we move the ruler up and decreases as we move it down. It is a general property, though, of linear objective functions that they will consistently increase or decrease as we move the ruler up or down. Knowing which direction to move the ruler in order to maximise/minimise f (x,y) = ax + by is as simple as looking at the sign of b (i.e. "is b negative, positive or zero?"). If b is positive, then f (x,y) increases as we move the ruler up and f (x,y) decreases as we move the ruler down. The opposite happens for the case when b is negative: f (x,y) decreases as we move the ruler up and f (x,y) increases as we move the ruler down. If b = 0 then we need to look at the sign of a. 541
41.3
CHAPTER 41. LINEAR PROGRAMMING - GRADE 12
If a is positive then f (x,y) increases as we move the ruler to the right and decreases if we move the ruler to the left. Once again, the opposite happens for a negative. If we look again at the objective function mentioned earlier, f (x,y) = x − 2y with a = 1 and b = −2, then we should find that f (x,y) increases as we move the ruler down the page since b = −2 < 0. This is exactly what we found happening in Figure 41.2. The main points about linear programming we have encountered so far are • The feasible region is always a polygon. • Solutions occur at vertices of the feasible region. • Moving a ruler parallel to the level lines of the objective function up/down to the top/bottom of the feasible region shows us which of the vertices is the solution. • The direction in which to move the ruler is determined by the sign of b and also possibly by the sign of a. These points are sufficient to determine a method for solving any linear program. Method: Linear Programming If we wish to maximise the objective function f (x,y) then: 1. Find the gradient of the level lines of f (x,y) (this is always going to be − a as we saw in b Equation ??) 2. Place your ruler on the xy plane, making a line with gradient − a (i.e. b units on the b x-axis and −a units on the y-axis) 3. The solution of the linear program is given by appropriately moving the ruler. Firstly we need to check whether b is negative, positive or zero. A If b > 0, move the ruler up the page, keeping the ruler parallel to the level lines all the time, until it touches the "highest" point in the feasible region. This point is then the solution. B If b < 0, move the ruler in the opposite direction to get the solution at the "lowest" point in the feasible region. C If b = 0, check the sign of a i. If a < 0 move the ruler to the "leftmost" feasible point. This point is then the solution. ii. If a > 0 move the ruler to the "rightmost" feasible point. This point is then the solution.
Worked Example 188: Prizes! Question: As part of their opening specials, a furniture store has promised to give away at least 40 prizes with a total value of at least R2 000. The prizes are kettles and toasters. 1. If the company decides that there will be at least 10 of each prize, write down two more inequalities from these constraints. 2. If the cost of manufacturing a kettle is R60 and a toaster is R50, write down an objective function C which can be used to determine the cost to the company of both kettles and toasters. 542
CHAPTER 41. LINEAR PROGRAMMING - GRADE 12 3. Sketch the graph of the feasibility region that can be used to determine all the possible combinations of kettles and toasters that honour the promises of the company. 4. How many of each prize will represent the cheapest option for the company? 5. How much will this combination of kettles and toasters cost? Answer Step 1 : Identify the decision variables Let the number of kettles be xk and the number of toasters be yt and write down two constraints apart from xk ≥ 0 and yt ≥ 0 that must be adhered to. Step 2 : Write constraint equations Since there will be at least 10 of each prize we can write: xk ≥ 10 and yt ≥ 10 Also the store has promised to give away at least 40 prizes in total. Therefore: xk + yt ≥ 40 Step 3 : Write the objective function The cost of manufacturing a kettle is R60 and a toaster is R50. Therefore the cost the total cost C is: C = 60xk + 50yt Step 4 : Sketch the graph of the feasible region yt 100 90 80 70 60 50 40 30 20 10 10 20 A xk 30 40 50 60 70 80 90 100 B
41.3
Step 5 : Determine vertices of feasible region From the graph, the coordinates of vertex A is (3,1) and the coordinates of vertex B are (1,3). Step 6 : Draw in the search line The seach line is the gradient of the objective function. That is, if the equation C = 60x + 50y is now written in the standard form y = ..., then the gradient is: 6 m=− , 5 which is shown with the broken line on the graph. 543
Step 8 : Write the final answer The cheapest combination of prizes is 10 kettles and 30 toasters, costing the company R2 100.
Worked Example 189: Search Line Method Question: As a production planner at a factory manufacturing lawn cutters your job will be to advise the management on how many of each model should be produced per week in order to maximise the profit on the local production. The factory is producing two types of lawn cutters: Quadrant and Pentagon. Two of the production processes that the lawn cutters must go through are: bodywork and engine work. • The factory cannot operate for less than 360 hours on engine work for the lawn cutters. • The factory has a maximum capacity of 480 hours for bodywork for the lawn cutters. 544
CHAPTER 41. LINEAR PROGRAMMING - GRADE 12 • Half an hour of engine work and half an hour of bodywork is required to produce one Quadrant. • One third of an hour of engine work andone fifth of an hour of bodywork is required to produce one Pentagon. • The ratio of Pentagon lawn cutters to Quadrant lawn cutters produced per week must be at least 3:2. Let the number of Quadrant lawn cutters manufactured in a week be x. Let the number of Pentagon lawn cutters manufactured in a week be y. Two of the constraints are: x ≥ 200 3x + 2y ≥ 2 160 1. Write down the remaining constraints in terms of x and y to represent the abovementioned information. 2. Use graph paper to represent the constraints graphically. 3. Clearly indicate the feasible region by shading it. 4. If the profit on one Quadrant lawn cutter is R1 200 and the profit on one Pentagon lawn cutter is R400, write down an equation that will represent the profit on the lawn cutters. 5. Using a search line and your graph, determine the number of Quadrant and Pentagon lawn cutters that will yield a maximum profit. 6. Determine the maximum profit per week. Answer Step 1 : Remaining constraints: 1 1 x + ≤ 480 2 5 3 y ≥ x 2 Step 2 : Graphical representation y 2400 • A minimum of 200 Quadrant lawn cutters must be produced per week.
1. Polkadots is a small company that makes two types of cards, type X and type Y. With the available labour and material, the company can make not more than 150 cards of type X and not more than 120 cards of type Y per week. Altogether they cannot make more than 200 cards per week. There is an order for at least 40 type X cards and 10 type Y cards per week. Polkadots makes a profit of R5 for each type X card sold and R10 for each type Y card. Let the number of type X cards be x and the number of type Y cards be y, manufactured per week. A One of the constraint inequalities which represents the restrictions above is x ≤ 150. Write the other constraint inequalities. B Represent the constraints graphically and shade the feasible region. C Write the equation that represents the profit P (the objective function), in terms of x and y. D On your graph, draw a straight line which will help you to determine how many of each type must be made weekly to produce the maximum P E Calculate the maximum weekly profit. 2. A brickworks produces "face bricks" and "clinkers". Both types of bricks are produced and sold in batches of a thousand. Face bricks are sold at R150 per thousand, and clinkers at R100 per thousand, where an income of at least R9,000 per month is required to cover costs. The brickworks is able to produce at most 40,000 face bricks and 90,000 clinkers per month, and has transport facilities to deliver at most 100,000 bricks per month. The number of clinkers produced must be at least the same number of face bricks produced. Let the number of face bricks in thousands be x, and the number of clinkers in thousands be y. A List all the constraints. B Graph the feasible region. C If the sale of face bricks yields a profit of R25 per thousand and clinkers R45 per thousand, use your graph to determine the maximum profit. D If the profit margins on face bricks and clinkers are interchanged, use your graph to determine the maximum profit. 3. A small cell phone company makes two types of cell phones: Easyhear and Longtalk. Production figures are checked weekly. At most, 42 Easyhear and 60 Longtalk phones can be manufactured each week. At least 30 cell phones must be produced each week to cover costs. In order not to flood the market, the number of Easyhear phones cannot be more than twice the number of Longtalk phones. It takes 2 hour to assemble an Easyhear 3 phone and 1 hour to put together a Longtalk phone. The trade unions only allow for a 2 50-hour week. Let x be the number of Easyhear phones and y be the number of Longtalk phones manufactured each week. A Two of the constraints are: 0 ≤ x ≤ 42 and 0 ≤ y ≤ 60
C If the profit on an Easyhear phone is R225 and the profit on a Longtalk is R75, determine the maximum profit per week. 4. Hair for Africa is a firm that specialises in making two kinds of up-market shampoo, Glowhair and Longcurls. They must produce at least two cases of Glowhair and one case of Longcurls per day to stay in the market. Due to a limited supply of chemicals, they cannot produce more than 8 cases of Glowhair and 6 cases of Longcurls per day. It takes half-an-hour to produce one case of Glowhair and one hour to produce a case of Longcurls, and due to restrictions by the unions, the plant may operate for at most 7 hours per day. The workforce at Hair for Africa, which is still in training, can only produce a maximum of 10 cases of shampoo per day. Let x be the number of cases of Glowhair and y the number of cases of Longcurls produced per day. A Write down the inequalities that represent all the constraints. B Sketch the feasible region. C If the profit on a case of Glowhair is R400 and the profit on a case of Longcurls is R300, determine the maximum profit that Hair for Africa can make per day. 5. A transport contracter has 6 5-ton trucks and 8 3-ton trucks. He must deliver at least 120 tons of sand per day to a construction site, but he may not deliver more than 180 tons per day. The 5-ton trucks can each make three trips per day at a cost of R30 per trip, and the 3-ton trucks can each make four trips per day at a cost of R120 per trip. How must the contracter utilise his trucks so that he has minimum expense ?
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41.4
CHAPTER 41. LINEAR PROGRAMMING - GRADE 12
548
Chapter 42
Geometry - Grade 12
42.1 Introduction
Activity :: Discussion : Discuss these Research Topics Research one of the following geometrical ideas and describe it to your group: 1. taxicab geometry, 2. sperical geometry, 3. fractals, 4. the Koch snowflake.
42.2
42.2.1
Circle Geometry
Terminology
The following is a recap of terms that are regularly used when referring to circles. arc An arc is a part of the circumference of a circle. chord A chord is defined as a straight line joining the ends of an arc. radius The radius, r, is the distance from the centre of the circle to any point on the circumference. diameter The diameter, , is a special chord that passes through the centre of the circle. The diameter is the straight line from a point on the circumference to another point on the circumference, that passes through the centre of the circle. segment A segment is the part of the circle that is cut off by a chord. A chord divides a circle into two segments. tangent A tangent is a line that makes contact with a circle at one point on the circumference. (AB is a tangent to the circle at point P . 549
42.2
CHAPTER 42. GEOMETRY - GRADE 12
segment chord
rad
i us
O diameter
A
a rc
P tangent B C O P 550 B
Figure 42.1: Parts of a Circle
42.2.2
Axioms
An axiom is an established or accepted principle. For this section, the following are accepted as axioms. 1. The Theorem of Pythagoras, which states that the square on the hypotenuse of a rightangled triangle is equal to the sum of the squares on the other two sides. In △ABC, this means that AB 2 + BC 2 = AC 2 A
B
Figure 42.2: A right-angled triangle 2. A tangent is perpendicular to the radius, drawn at the point of contact with the circle.
42.2.3
Theorems of the Geometry of Circles
A theorem is a general proposition that is not self-evident but is proved by reasoning (these proofs need not be learned for examination purposes). Theorem 6. The line drawn from the centre of a circle, perpendicular to a chord, bisects the chord. Proof:
A
CHAPTER 42. GEOMETRY - GRADE 12
42.2
Consider a circle, with centre O. Draw a chord AB and draw a perpendicular line from the centre of the circle to intersect the chord at point P . The aim is to prove that AP = BP 1. △OAP and △OBP are right-angled triangles. 2. OA = OB as both of these are radii and OP is common to both triangles. Apply the Theorem of Pythagoras to each triangle, to get: OA2 OB 2 However, OA = OB. So, OP 2 + AP 2 ∴ AP 2 and AP This means that OP bisects AB. Theorem 7. The line drawn from the centre of a circle, that bisects a chord, is perpendicular to the chord. Proof: = = = OP 2 + BP 2 BP 2 BP = OP 2 + AP 2 = OP 2 + BP 2
Theorem 8. The perpendicular bisector of a chord passes through the centre of the circle.
Proof:
Q
A
P
B
Consider a circle. Draw a chord AB. Draw a line P Q perpendicular to AB such that P Q bisects AB at point P . Draw lines AQ and BQ. The aim is to prove that Q is the centre of the circle, by showing that AQ = BQ. In △OAP and △OBP ,
1. AP = P B (given)
2. ∠QP A = ∠QP B (QP ⊥ AB)
3. QP is common to both triangles.
∴ △QAP ≡ △QBP (SAS). From this, QA = QB. Since the centre of a circle is the only point inside a circle that has points on the circumference at an equal distance from it, Q must be the centre of the circle.
Exercise: Circles I
1. Find the value of x: 552
CHAPTER 42. GEOMETRY - GRADE 12
42.2
a)
b)
O x 5 Q P PR=8 x R P
O 4 Q R
PR=6
c)
10 O x R P Q PR=8
d)
Q 2 6 P x O S 6
R
e)
S 5 P 8 10 O x T U R Q
f)
24 T P 5 x 25 S
Q
R O
R
Theorem 9. The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circumference of the circle. Proof: P
Theorem 10. The angles subtended by a chord at the circumference of a circle on the same side of the chord are equal. Proof: Q P
O A
B Consider a circle, with centre O. Draw a chord AB. Select any points P and Q on the circumference of the circle, such that both P and Q are on the same side of the chord. Draw lines P A, P B, QA and QB. 554
Theorem 11. (Converse of Theorem 10) If a line segment subtends equal angles at two other points on the same side of the line, then these four points lie on a circle. Proof: Q P R
A
B
Consider a line segment AB, that subtends equal angles at points P and Q on the same side of AB.
∴ the assumption that the circle does not pass through P , must be false, and A, B, P and Q lie on the circumference of a circle.
Exercise: Circles III 1. Find the values of the unknown letters. 555
42.2 1. A
a
CHAPTER 42. GEOMETRY - GRADE 12 2. E
B
21◦
F
15◦
D G C 3. J 4. N
b
I
H
O K
c 17◦
M Q
24◦ d
L
P
5.
T
S R
45◦ 35◦
6.
35◦
W
X O
12◦ e
f
Y V
Z
U
Cyclic Quadrilaterals
Cyclic quadrilaterals are quadrilaterals with all four vertices lying on the circumference of a circle. The vertices of a cyclic quadrilateral are said to be concyclic. Theorem 12. The opposite angles of a cyclic quadrilateral are supplementary.
Theorem 13. (Converse of Theorem 12) If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. Proof: Q R P
A
B ˆ ˆ ˆ ˆ Consider a quadrilateral ABP Q, such that ABP + AQP = 180◦ and QAB + QP B = 180◦ . Draw BR.
Theorem 14. Two tangents drawn to a circle from the same point outside the circle are equal in length. Proof:
A
O
P B Consider a circle, with centre O. Choose a point P outside the circle. Draw two tangents to the circle from point P , that meet the circle at A and B. Draw lines OA, OB and OP . The aim is to prove that AP = BP . In △OAP and △OBP , 558
Theorem 16. (Converse of 15) If the angle formed between a line, that is drawn through the end point of a chord, and the chord, is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle. Proof:
Q A O
Y S X B 561 R
42.2
CHAPTER 42. GEOMETRY - GRADE 12
Consider a circle, with centre O and chord AB. Let line SR pass through point B. Chord AB ˆ ˆ subtends an angle at point Q such that ABS = AQB. The aim is to prove that SBR is a tangent to the circle. By contradiction. Assume that SBR is not a tangent to the circle and draw XBY such that XBY is a tangent to the circle.
However,
ˆ ABX ˆ ABS
= = = = =
ˆ AQB ˆ AQB
(tan-chord theorem) (given) (42.2)
ˆ ∴ ABX ˆ But since, ABX ˆ (42.2) can only be true if, XBS
ˆ ABS ˆ ˆ ABS + XBS 0
ˆ If XBS is zero, then both XBY and SBR coincide and SBR is a tangent to the circle.
Exercise: Applying Theorem 9
1. Show that Theorem 9 also applies to the following two cases:
A
P
O
O R P
R
B A B
562
CHAPTER 42. GEOMETRY - GRADE 12
42.2
Worked Example 190: Circle Geometry I BD is a tangent to the circle with centre O. BO ⊥ AD. Prove that: 1. CF OE is a cyclic quadrilateral
O A E D
Question:
F C
2. F B = BC 3. △COE///△CBF 4. CD2 = ED.AD 5.
OE BC
=
CD CO
B
Answer 1. Step 1 : To show a quadrilateral is cyclic, we need a pair of opposite angles to be supplementary, so lets look for that. ˆ F OE ˆ F CE
2. Step 1 : Since these two sides are part of a triangle, we are proving that triangle to be isosceles. The easiest way is to show the angles opposite to those sides to be equal. ˆ Let OEC = x. ∴ ∴ ∴ ˆ F CB = x (∠ between tangent BD and chord CE) ˆ BF C = x (exterior ∠ to cyclic quadrilateral CF OE) BF = BC (sides opposite equal ∠'s in isosceles △BF C)
3. Step 1 : To show these two triangles similar, we will need 3 equal angles. We already have 3 of the 6 needed angles from the previous question. We need only find the missing 3 angles. ˆ CBF OC
Step 3 : The third equal angle is an angle both triangles have in common. ˆ ˆ Lastly, ADC = EDC since they are the same ∠. Step 4 : Now we know that the triangles are similar and can use the proportionality relation accordingly.
∴ △ADC///△CDE (3 ∠'s equal) ED CD ∴ = CD AD ∴ CD2 = ED.AD 5. Step 1 : This looks like another proportionality relation with a little twist, since not all sides are contained in 2 triangles. There is a quick observation we can make about the odd side out, OE.
OE
=
CD (△OEC is isosceles)
Step 2 : With this observation we can limit ourselves to proving triangles BOC and ODC similar. Start in one of the triangles. In △BCO ˆ OCB ˆ CBO = = 90◦ (radius OC on tangent BD) 180◦ − 2x (sum of ∠'s in △BF C)
Step 3 : Then we move on to the other one. In △OCD ˆ OCD ˆ COD = = 90◦ (radius OC on tangent BD) 180◦ − 2x (sum of ∠'s in △OCE)
Step 4 : Again we have a common element. Lastly, OC is a common side to both △'s. Step 5 : Then, once we've shown similarity, we use the proportionality relation , as well as our first observation, appropriately. 564
Step 2 : We have already proved 1 pair of angles equal in the previous question.
∠BCD
= 565
∠F AE (above)
42.3
CHAPTER 42. GEOMETRY - GRADE 12 Step 3 : Proving the last set of angles equal is simply a matter of adding up the angles in the triangles. Then we have proved similarity.
∠AF E ∠CBD
= = ∴
△AF E///△CBD (3 ∠'s equal)
180◦ − x − y (∠'s in △AF E) 180◦ − x − y (∠'s in △CBD)
3. Step 1 : This equation looks like it has to do with proportionality relation of similar triangles. We already showed triangles AF E and CBD similar in the previous question. So lets start there. DC BD FA FE DC.F E = FA BD
= ∴
Step 2 : Now we need to look for a hint about side F A. Looking at triangle CAH we see that there is a line F G intersecting it parallel to base CH. This gives us another proportionality relation. AG GH FA (F G CH splits up lines AH and AC proportionally) FC F C.AG FA = GH
We know that every point on the circumference of a circle is the same distance away from the centre of the circle. Consider a point (x1 ,y1 ) on the circumference of a circle of radius r with centre at (x0 ,y0 ).
P (x1 ,y1 ) (x0 ,y0 ) O Q
Figure 42.3: Circle h with centre (x0 ,y0 ) has a tangent, g passing through point P at (x1 ,y1 ). Line f passes through the centre and point P . 566
Worked Example 192: Equation of a Circle I Question: Find the equation of a circle (centre O) with a diameter between two points, P at (−5,5) and Q at (5, − 5).
Answer
Step 1 : Draw a picture Draw a picture of the situation to help you figure out what needs to be done.
P
5 O
−5
5 −5 Q
Step 2 : Find the centre of the circle We know that the centre of a circle lies on the midpoint of a diameter. Therefore the co-ordinates of the centre of the circle is found by finding the midpoint of the line between P and Q. Let the co-ordinates of the centre of the circle be (x0 ,y0 ), let the co-ordinates of P be (x1 ,y1 ) and let the co-ordinates of Q be (x2 ,y2 ). Then, 567
We are given that a tangent to a circle is drawn through a point P with co-ordinates (x1 ,y1 ). In this section, we find out how to determine the equation of that tangent.
g
h f (x0 ,y0 )
P (x1 ,y1 )
Figure 42.4: Circle h with centre (x0 ,y0 ) has a tangent, g passing through point P at (x1 ,y1 ). Line f passes through the centre and point P .
We start by making a list of what we know: 1. We know that the equation of the circle with centre (x0 ,y0 ) is (x − x0 )2 + (y − y0 )2 = r2 . 2. We know that a tangent is perpendicular to the radius, drawn at the point of contact with the circle. As we have seen in earlier grades, there are two steps to determining the equation of a straight line: Step 1: Calculate the gradient of the line, m. Step 2: Calculate the y-intercept of the line, c. The same method is used to determine the equation of the tangent. First we need to find the gradient of the tangent. We do this by finding the gradient of the line that passes through the centre of the circle and point P (line f in Figure 42.4), because this line is a radius line and the tangent is perpendicular to it. mf = y1 − y0 x1 − x0 (42.4)
For example, find the equation of the tangent to the circle at point (1,1). The centre of the circle is at (0,0). The equation of the circle is x2 + y 2 = 2. Use y − y1 = − with (x0 ,y0 ) = (0,0) and (x1 ,y1 ) = (1,1). x1 − x0 (x − x1 ) y1 − y0
Exercise: Rotations Any line OP is drawn (not necessarily in the first quadrant), making an angle of θ degrees with the x-axis. Using the co-ordinates of P and the angle α, calculate the co-ordinates of P ′ , if the line OP is rotated about the origin through α degrees. 1. 2. 3. 4. 5. 6. P (2, 6) (4, 2) (5, -1) (-3, 2) (-4, -1) (2, 5) α 60◦ 30◦ 45◦ 120◦ 225◦ -150◦ O
P
θ
42.4.2
Characteristics of Transformations
Rigid transformations like translations, reflections, rotations and glide reflections preserve shape and size, and that enlargement preserves shape but not size.
42.4.3
Characteristics of Transformations
Rigid transformations like translations, reflections, rotations and glide reflections preserve shape and size, and that enlargement preserves shape but not size.
Activity :: : Geometric Transformations
15
10
Draw a large 15×15 grid and plot △ABC with A(2; 6), B(5; 6) and C(5; 1). Fill in the lines y = x and y = −x. Complete the table below , by drawing the images of △ABC under the given transformations. The first one has been done for you.
−15 −10 −5
We have, for any angles α and β, that sin(α + β) = sin α cos β + sin β cos α How do we derive this identity? It is tricky, so follow closely. Suppose we have the unit circle shown below. The two points L(a,b) and K(x,y) are on the circle.
y
K(x; y) L(a; b) 1 (α − β) α β
a
1
b
O
M (x; y)
x
We can get the coordinates of L and K in terms of the angles α and β. For the triangle LOK, we have that b 1 a cos β = 1 sin β = =⇒ =⇒ 577 b = sin β a = cos β
The most important thing to remember when asked to prove identities is:
Important: Trigonometric Identities
Never assume that the left hand side is equal to the right hand side. You need to show that both sides are equal. A suggestion for proving identities: It is usually much easier simplifying the more complex side of an identity to get the simpler side than the other way round.
sin θ + sin 2θ = tan θ 1 + cos θ + cos 2θ For which values is the identity not valid? Answer Step 1 : Identify a strategy The right-hand side (RHS) of the identity cannot be simplified. Thus we should try simplify the left-hand side (LHS). We can also notice that the trig function on the RHS does not have a 2θ dependance. Thus we will need to use the doubleangle formulas to simplify the sin 2θ and cos 2θ on the LHS. We know that tan θ is undefined for some angles θ. Thus the identity is also undefined for these θ, and hence is not valid for these angles. Also, for some θ, we might have division by zero in the LHS, which is not allowed. Thus the identity won't hold for these angles also. Step 2 : Execute the strategy sin θ + 2 sin θ cos θ 1 + cos θ + (2 cos2 θ − 1) sin θ(1 + 2 cos θ) cos θ(1 + 2 cos θ) sin θ cos θ tan θ RHS
Worked Example 199: Height of tower Question: D is the top of a tower of height h. Its base is at C. The triangle ABC lies on ˆ ˆ the ground (a horizontal plane). If we have that BC = b, DBA = α, DBC = x ˆ and DCB = θ, show that b sin α sin x h= sin(x + θ) 584
CHAPTER 43. TRIGONOMETRY - GRADE 12
D
43.2
h
C θ
A
b x B
α
Answer Step 1 : Identify a strategy We have that the triangle ABD is right-angled. Thus we can relate the height h with the angle α and either the length BA or BD (using sines or cosines). But we have two angles and a length for △BCD, and thus can work out all the remaining lengths and angles of this triangle. We can thus work out BD. Step 2 : Execute the strategy We have that h BD =⇒ h = = sin α BD sin α
Exercise: 1. The line BC represents a tall tower, with C at its foot. Its angle of elevation from D is θ. We are also given that BA = AD = x. 585
43.3 C
CHAPTER 43. TRIGONOMETRY - GRADE 12
B
θ α x A x
D
A Find the height of the tower BC in terms of x, tan θ and cos 2α. B Find BC if we are given that k = 140m, α = 21◦ and θ = 9◦ .
43.3
43.3.1
Other Geometries
Taxicab Geometry
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates.
43.3.2
Manhattan distance
The metric in taxi-cab geometry, is known as the Manhattan distance, between two points in an Euclidean space with fixed Cartesian coordinate system as the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. For example, in the plane, the Manhattan distance between the point P1 with coordinates (x1 , y1 ) and the point P2 at (x2 , y2 ) is |x1 − x2 | + |y1 − y2 | (43.1)
Figure 43.1: Manhattan Distance (dotted and solid) compared to Euclidean Distance (dashed). √ In each case the Manhattan distance is 12 units, while the Euclidean distance is 36 586
CHAPTER 43. TRIGONOMETRY - GRADE 12
43.3
The Manhattan distance depends on the choice on the rotation of the coordinate system, but does not depend on the translation of the coordinate system or its reflection with respect to a coordinate axis. Manhattan distance is also known as city block distance or taxi-cab distance. It is given these names because it is the shortest distance a car would drive in a city laid out in square blocks. Taxicab geometry satisfies all of Euclid's axioms except for the side-angle-side axiom, as one can generate two triangles with two sides and the angle between them the same and have them not be congruent. In particular, the parallel postulate holds. A circle in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These circles are squares whose sides make a 45◦ angle with the coordinate axes.
43.3.3
Spherical Geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. In plane geometry the basic concepts are points and line. On the sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of "straight line" but in the sense of "the shortest paths between points" which is called a geodesic. On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. Thus, in spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects (for example, the sum of the interior angles of a triangle exceeds 180◦). Spherical geometry is the simplest model of elliptic geometry, in which a line has no parallels through a given point. Contrast this with hyperbolic geometry, in which a line has two parallels, and an infinite number of ultra-parallels, through a given point. Spherical geometry has important practical uses in celestial navigation and astronomy.
Extension: Distance on a Sphere The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form. The distance between two points in Euclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In non-Euclidean geometry, straight lines are replaced with geodesics. Geodesics on the sphere are the great circles (circles on the sphere whose centers are coincident with the center of the sphere). Between any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. Between two points which are directly opposite each other (called antipodal points) there infinitely many great circles, but all have the same length, equal to half the circumference of the circle, or πr, where r is the radius of the sphere. Because the Earth is approximately spherical (see spherical Earth), the equations for great-circle distance are important for finding the shortest distance between points on the surface of the Earth, and so have important applications in navigation. Let φ1 ,λ1 ; φ2 ,λ2 , be the latitude and longitude of two points, respectively. Let ∆λ be the longitude difference. Then, if r is the great-circle radius of the sphere, the great-circle distance is r∆σ, where ∆σ is the angular difference/distance and can be determined from the spherical law of cosines as: ∆σ = arccos {sin φ1 sin φ2 + cos φ1 cos φ2 cos ∆λ}
587
43.3
CHAPTER 43. TRIGONOMETRY - GRADE 12 Extension: Spherical Distance on the Earth The shape of the Earth more closely resembles a flattened spheroid with extreme values for the radius of curvature, or arcradius, of 6335.437 km at the equator (vertically) and 6399.592 km at the poles, and having an average great-circle radius of 6372.795 km. Using a sphere with a radius of 6372.795 km thus results in an error of up to about 0.5%.
43.3.4
Fractal Geometry
The word "fractal" has two related meanings. In colloquial usage, it denotes a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex." In mathematics a fractal is a geometric object that satisfies a specific technical condition, namely having a Hausdorff dimension greater than its topological dimension. The term fractal was coined in 1975 by Benot Mandelbrot, from the Latin fractus, meaning "broken" or "fractured." Three common techniques for generating fractals are:
• Escape-time fractals - Fractals defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, the Burning Ship fractal and the Lyapunov fractal.
Fractals in nature Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, mountains, river networks, and systems of blood vessels. Trees and ferns are fractal in nature and can be modeled on a computer using a recursive algorithm. This recursive nature is clear in these examples - a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The surface of a mountain can be modeled on a computer using a fractal: Start with a triangle in 3D space and connect the central points of each side by line segments, resulting in 4 triangles. The central points are then randomly moved up or down, within a defined range. The procedure is repeated, decreasing at each iteration the range by half. The recursive nature of the algorithm guarantees that the whole is statistically similar to each detail. 588
In this chapter, you will use the mean, median, mode and standard deviation of a set of data to identify whether the data is normally distributed or whether it is skewed. You will learn more about populations and selecting different kinds of samples in order to avoid bias. You will work with lines of best fit, and learn how to find a regression equation and a correlation coefficient. You will analyse these measures in order to draw conclusions and make predictions.
1. Calculate the mean, median, mode and standard deviation of the data. 2. What percentage of the data is within one standard deviation of the mean? 3. Draw a histogram of the data using intervals 60 ≤ x < 64, 64 ≤ x < 68, etc. 4. Join the midpoints of the bars to form a frequency polygon.
If large numbers of data are collected from a population, the graph will often have a bell shape. If the data was, say, examination results, a few learners usually get very high marks, a few very low marks and most get a mark in the middle range. We say a distribution is normal if • the mean, median and mode are equal. • it is symmetric around the mean. • ±68% of the sample lies within one standard deviation of the mean, 95% within two standard deviations and 99% within three standard deviations of the mean. 591
44.2
CHAPTER 44. STATISTICS - GRADE 12
68% 95% 99%
¯ x − 3σ x − 2σ x − σ ¯ ¯ x ¯ x + σ x + 2σ x + 3σ ¯ ¯ ¯
What happens if the test was very easy or very difficult? Then the distribution may not be symmetrical. If extremely high or extremely low scores are added to a distribution, then the mean tends to shift towards these scores and the curve becomes skewed.
If the test was very difficult, the mean score is shifted to the left. In this case, we say the distribution is positively skewed, or skewed right. Skewed right If it was very easy, then many learners would get high scores, and the mean of the distribution would be shifted to the right. We say the distribution is negatively skewed, or skewed left. Skewed left
Draw the histogram of the results. Join the midpoints of each bar and draw a frequency polygon. What mark must one obtain in order to be in the top 2% of the class? Approximately 84% of the pupils passed the test. What was the pass mark? Is the distribution normal or skewed? 592
CHAPTER 44. STATISTICS - GRADE 12 3. In a road safety study, the speed of 175 cars was monitored along a specific stretch of highway in order to find out whether there existed any link between high speed and the large number of accidents along the route. A frequency table of the results is drawn up below. Speed (km.h−1 ) 50 60 70 80 90 100 110 120 Number of cars (Frequency) 19 28 23 56 20 16 8 5
44.3
The mean speed was determined to be around 82 km.h−1 while the median speed was worked out to be around 84,5 km.h−1 . A Draw a frequency polygon to visualise the data in the table above. B Is this distribution symmetrical or skewed left or right? Give a reason fro your answer.
44.3
Extracting a Sample Population
Suppose you are trying to find out what percentage of South Africa's population owns a car. One way of doing this might be to send questionnaires to peoples homes, asking them whether they own a car. However, you quickly run into a problem: you cannot hope to send every person in the country a questionnaire, it would be far to expensive. Also, not everyone would reply. The best you can do is send it to a few people, see what percentage of these own a car, and then use this to estimate what percentage of the entire country own cars. This smaller group of people is called the sample population. The sample population must be carefully chosen, in order to avoid biased results. How do we do this? First, it must be representative. If all of our sample population comes from a very rich area, then almost all will have cars. But we obviously cannot conclude from this that almost everyone in the country has a car! We need to send the questionnaire to rich as well as poor people. Secondly, the size of the sample population must be large enough. It is no good having a sample population consisting of only two people, for example. Both may very well not have cars. But we obviously cannot conclude that no one in the country has a car! The larger the sample population size, the more likely it is that the statistics of our sample population corresponds to the statistics of the entire population. So how does one ensure that ones sample is representative? There are a variety of methods available, which we will look at now.
Random Sampling. Every person in the country has an equal chance of being selected. It is unbiased and also independant, which means that the selection of one person has no effect on the selection on another. One way of doing this would be to give each person in the country a number, and then ask a computer to give us a list of random numbers. We could then send the questionnaire to the people corresponding to the random numbers. Systematic Sampling. Again give every person in the country a number, and then, for example, select every hundredth person on the list. So person with number 1 would be selected, person with number 100 would be selected, person with number 200 would be selected, etc. 593
44.4
CHAPTER 44. STATISTICS - GRADE 12 Stratified Sampling. We consider different subgroups of the population, and take random samples from these. For example, we can divide the population into male and female, different ages, or into different income ranges. Cluster Sampling. Here the sample is concentrated in one area. For example, we consider all the people living in one urban area.
Exercise: Sampling 1. Discuss the advantages, disadvantages and possible bias when using A systematic sampling B random sampling C cluster sampling 2. Suggest a suitable sampling method that could be used to obtain information on: A passengers views on availability of a local taxi service. B views of learners on school meals. C defects in an item made in a factory. D medical costs of employees in a large company. 3. 5% of a certain magazines' subscribers is randomly selected. The random number 16 out of 50, is selected. Then subscribers with numbers 16, 66, 116, 166, . . . are chosen as a sample. What kind of sampling is this?
44.4
Function Fitting and Regression Analysis
In Grade 11 we recorded two sets of data (bivariate data) on a scatter plot and then we drew a line of best fit as close to as many of the data items as possible. Regression analysis is a method of finding out exactly which function best fits a given set of data. We can find out the equation of the regression line by drawing and estimating, or by using an algebraic method called "the least squared method", or we can use a calculator. The linear regression equation is written y = a + bx (we say y-hat) or y = A + Bx. Of course these are both variations of a more familiar ˆ equation y = mx + c. Suppose you are doing an experiment with washing dishes. You count how many dishes you begin with, and then find out how long it takes to finish washing them. So you plot the data on a graph of time taken versus number of dishes. This is plotted below.
t 200 180
Time taken (seconds)
160 140 120 100 80 60 40 20 0 0 1 2 3 4 5 6 Number of dishes d
594
CHAPTER 44. STATISTICS - GRADE 12
44.4
If t is the time taken, and d the number of dishes, then it looks as though t is proportional to d, ie. t = m · d, where m is the constant of proportionality. There are two questions that interest us now. 1. How do we find m? One way you have already learnt, is to draw a line of best-fit through the data points, and then measure the gradient of the line. But this is not terribly precise. Is there a better way of doing it? 2. How well does our line of best fit really fit our data? If the points on our plot don't all lie close to the line of best fit, but are scattered everywhere, then the fit is not 'good', and our assumption that t = m · d might be incorrect. Can we find a quantitative measure of how well our line really fits the data? In this chapter, we answer both of these questions, using the techniques of regression analysis.
Worked Example 200: Fitting by hand Question: Use the data given to draw a scatter plot and line of best fit. Now write down the equation of the line that best seems to fit the data. x y 1,0 2,5 2,4 2,8 3,1 3,0 4,9 4,8 5,6 5,1 6,2 5,3
Step 2 : Calculating the equation of the line The equation of the line is y = mx + c From the graph we have drawn, we estimate the y-intercept to be 1,5. We estimate that y = 3,5 when x = 3. So we have that points (3; 3,5) and (0; 1,6) lie on the line. The gradient of the line, m, is given by m = = = y2 − y1 x2 − x1 3,5 − 1,5 3−0 2 3
So we finally have that the equation of the line of best fit is y= 2 x + 1,5 3 595
44.4
CHAPTER 44. STATISTICS - GRADE 12
44.4.1
The Method of Least Squares
We now come to a more accurate method of finding the line of best-fit. The method is very simple. Suppose we guess a line of best-fit. Then at at every data point, we find the distance between the data point and the line. If the line fitted the data perfectly, this distance should be zero for all the data points. The worse the fit, the larger the differences. We then square each of these distances, and add them all together. y
y
The best-fit line is then the line that minimises the sum of the squared distances. Suppose we have a data set of n points {(x1 ; y1 ), (x2 ; y2 ), . . . , (xn ,yn )}. We also have a line f (x) = mx + c that we are trying to fit to the data. The distance between the first data point and the line, for example, is distance = y1 − f (x) = y1 − (mx + c) We now square each of these distances and add them together. Lets call this sum S(m,c). Then we have that S(m,c) = =
i=1
Worked Example 201: Method of Least Squares Question: In the table below, we have the records of the maintenance costs in Rands, compared with the age of the appliance in months. We have data for 5 appliances.
Now press [2] for linear regression. Your screen should look something like this: x 1 2 3 Step 2 : Entering the data Press [52] and then [=] to enter the first mark under x. Then enter the other values, in the same way, for the x-variable (the Chemistry marks) in the order in which they are given in the data set. Then move the cursor across and up and enter 48 under y opposite 52 in the x-column. Continue to enter the other y-values (the Accounting marks) in order so that they pair off correctly with the corresponding x-values. 1 2 3 x 52 55 y y
Step 3 : Getting regression results from the calculator a) Press [1] and [=] to get the value of the y-intercept, a = −5,065.. = −5,07(to 2 d.p.) Finally, to get the slope, use the following key sequence: [SHIFT][1][7][2][=]. The calculator gives b = 1,169.. = 1,17(to 2 d.p.) The equation of the line of regression is thus: y = −5,07 + 1,17x ˆ 598
A Use the formulae to find the regression equation coefficients a and b. B Draw a scatter plot of the data on graph paper. C Now use algebra to find a more accurate equation. 2. Footlengths and heights of 7 students are given in the table below. Height (cm) Footlength (cm) 170 27 163 23 131 20 181 28 146 22 134 20 166 24
A Draw a scatter plot of the data on graph paper. B Indentify and describe any trends shown in the scatter plot. C Find the equation of the least squares line by using algebraic methods and draw the line on your graph. D Use your equation to predict the height of a student with footlength 21,6 cm. E Use your equation to predict the footlength of a student 176 cm tall. 3. Repeat the data in question 2 and find the regression line using a calculator
44.4.3
Correlation coefficients
Once we have applied regression analysis to a set of data, we would like to have a number that tells us exactly how well the data fits the function. A correlation coefficient, r, is a tool that tells us to what degree there is a relationship between two sets of data. The correlation coefficient r ∈ [−1; 1] when r = −1, there is a perfect negative relationship, when r = 0, there is no relationship and r = 1 is a perfect positive correlation.
y y y y y
x
x
x
x
x
Positive, strong r ≈ 0,9
Positive, fairly strong r ≈ 0,7
Positive, weak r ≈ 0,4
No association r=0
Negative, fairly strong r ≈ −0,7
We often use the correlation coefficient r2 in order to work with the strength of the correlation only (no whether it is positive or negative). In this case: 599
• where n is the number of data points, • sx is the standard deviation of the x-values and • sy is the standard deviation of the y-values. This is known as the Pearson's product moment correlation coefficient. It is a long calculation and much easier to do on the calculator where you simply follow the procedure for the regression equation, and go on to find r.
Draw a scatter plot of the data set and your estimate of a line of best fit. Calculate equation of the line of regression using the method of least squares. Draw the regression line equation onto the graph. Calculate the correlation coefficient r. What conclusion can you reach, regarding the relationship between CO2 emission and GDP per capita for the countries in the data set?
2. A collection of data on the peculiar investigation into a foot size and height of students was recorded in the table below. Answer the questions to follow. Length of right foot (cm) 25,5 26,1 23,7 26,4 27,5 24 22,6 27,1 600 Height (cm) 163,3 164,9 165,5 173,7 174,4 156 155,3 169,3
CHAPTER 44. STATISTICS - GRADE 12 A Draw a scatter plot of the data set and your estimate of a line of best fit.
44.5
B Calculate equation of the line of regression using the method of least squares or your calculator. C Draw the regression line equation onto the graph. D Calculate the correlation coefficient r. E What conclusion can you reach, regarding the relationship between the length of the right foot and height of the students in the data set? 3. A class wrote two tests, and the marks for each were recorded in the table below. Full marks in the first test was 50, and the second test was out of 30. A Is there a strong association between the marks for the first and second test? Show why or why not. B One of the learners (in row 18) did not write the second test. Given their mark for the first test, calculate an expected mark for the second test. Learner 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Test 1 (Full marks: 50) 42 32 31 42 35 23 43 23 24 15 19 13 36 29 29 25 29 17 30 28 Test 2 (Full marks: 30) 25 19 20 26 23 14 24 12 14 10 11 10 22 17 17 16 18 19 17
4. A fast food company produces hamburgers. The number of hamburgers made, and the costs are recorded over a week. Hamburgers made Costs 495 R2382 550 R2442 515 R2484 500 R2400 480 R2370 530 R2448 585 R2805 A Find the linear regression function that best fits the data. B If the total cost in a day is R2500, estimate the number of hamburgers produced. C What is the cost of 490 hamburgers? 5. The profits of a new shop are recorded over the first 6 months. The owner wants to predict his future sales. The profits so far have been R90 000 , R93 000, R99 500, R102 000, R101 300, R109 000. A For the profit data, calculate the linear regression function. 601
44.5
CHAPTER 44. STATISTICS - GRADE 12 B Give an estimate of the profits for the next two months. C The owner wants a profit of R130 000. Estimate how many months this will take.
6. A company produces sweets using a machine which runs for a few hours per day. The number of hours running the machine and the number of sweets produced are recorded. Machine hours 3,80 4,23 4,37 4,10 4,17 Sweets produced 275 287 291 281 286
Find the linear regression equation for the data, and estimate the machine hours needed to make 300 sweets.
602
Chapter 45
Combinations and Permutations Grade 12
45.1 Introduction
Mathematics education began with counting. At the beginning, fingers, beans, buttons, and pencils were used to help with counting, but these are only practical for small numbers. What happens when a large number of items must be counted? This chapter focuses on how to use mathematical techniques to count combinations of items.
45.2
Counting
An important aspect of probability theory is the ability to determine the total number of possible outcomes when multiple events are considered. For example, what is the total number of possible outcomes when a die is rolled and then a coin is tossed? The roll of a die has six possible outcomes (1, 2, 3, 4, 5 or 6) and the toss of a coin, 2 outcomes (head or tails). Counting the possible outcomes can be tedious.
The use of lists, tables and tree diagrams is only feasible for events with a few outcomes. When the number of outcomes grows, it is not practical to list the different possibilities and the fundamental counting principle is used. The fundamental counting principle describes how to determine the total number of outcomes of a series of events. Suppose that two experiments take place. The first experiment has n1 possible outcomes, and the second has n2 possible outcomes. Therefore, the first experiment, followed by the second experiment, will have a total of n1 × n2 possible outcomes. This idea can be generalised to m experiments as the total number of outcomes for m experiments is:
m
n1 × n2 × n3 × . . . × nm = is the multiplication equivalent of .
ni
i=1
Note: the order in which the experiments are done does not affect the total number of possible outcomes.
Worked Example 204: Lunch Special Question: A take-away has a 4-piece lunch special which consists of a sandwich, soup, dessert and drink for R25.00. They offer the following choices for : Sandwich: chicken mayonnaise, cheese and tomato, tuna, and ham and lettuce Soup: tomato, chicken noodle, vegetable Dessert: ice-cream, piece of cake Drink: tea, coffee, coke, Fanta and Sprite. How many possible meals are there? 604
CHAPTER 45. COMBINATIONS AND PERMUTATIONS - GRADE 12 Answer Step 1 : Determine how many parts to the meal there are There are 4 parts: sandwich, soup, dessert and drink. Step 2 : Identify how many choices there are for each part Meal Component Number of choices Sandwich 4 Soup 3 Dessert 2 Drink 5
The fundamental counting principle describes how to calculate the total number of outcomes when multiple independent events are performed together. A more complex problem is determining how many combinations there are of selecting a group of objects from a set. Mathematically, a combination is defined as an un-ordered collection of unique elements, or more formally, a subset of a set. For example, suppose you have fifty-two playing cards, and select five cards. The five cards would form a combination and would be a subset of the set of 52 cards. In a set, the order of the elements in the set does not matter. These are represented usually with curly braces, for example {2, 4, 6} is a subset of the set {1,2,3,4,5,6}. Since the order of the elements does not matter, only the specific elements are of interest. Therefore, {2, 4, 6} = {6, 4, 2} and {1, 1, 1} is the same as {1} because a set is defined by its elements; they don't usually appear more than once
45.5.1
Counting Combinations
Calculating the number of ways that certain patterns can be formed is the beginning of combinatorics, the study of combinations. Let S be a set with n objects. Combinations of k objects from this set S are subsets of S having k elements each (where the order of listing the elements does not distinguish two subsets). Combination without Repetition When the order does not matter, but each object can be chosen only once, the number of combinations is: n! n = r!(n − r)! r
where n is the number of objects from which you can choose and r is the number to be chosen. For example, if you have 10 numbers and wish to choose 5 you would have 10!/(5!(10 - 5)!) = 252 ways to choose. For example how many possible 5 card hands are there in a deck of cards with 52 cards? 52! / (5!(52-5)!) = 2 598 960 combinations 605
45.6
CHAPTER 45. COMBINATIONS AND PERMUTATIONS - GRADE 12
Combination with Repetition When the order does not matter and an object can be chosen more than once, then the number of combinations is: (n + r − 1)! n+r−1 n+r−1 = = r!(n − 1)! r n−1
where n is the number of objects from which you can choose and r is the number to be chosen. For example, if you have ten types of donuts to choose from and you want three donuts there are (10 + 3 - 1)! / 3!(10 - 1)! = 220 ways to choose.
45.5.2
Combinatorics and Probability
Combinatorics is quite useful in the computation of probabilities of events, as it can be used to determine exactly how many outcomes are possible in a given event.
Worked Example 205: Probability Question: At a school, learners each play 2 sports. They can choose from netball, basketball, soccer, athletics, swimming, or tennis. What is the probability that a learner plays soccer and either netball, basketball or tennis? Answer Step 1 : Identify what events we are counting We count the events: soccer and netball, soccer and basketball, soccer and tennis. This gives three choices. Step 2 : Calculate the total number of choices There are 6 sports to choose from and we choose 2 sports. There are 6 2 = 6!/(2!(6 − 2)!) = 15 choices. Step 3 : Calculate the probability The probability is the number of events we are counting, divided by the total number of choices. 3 Probability = 15 = 1 = 0,2 5
45.6
Permutations
The concept of a combination did not consider the order of the elements of the subset to be important. A permutation is a combination with the order of a selection from a group being important. For example, for the set {1,2,3,4,5,6}, the combination {1,2,3} would be identical to the combination {3,2,1}, but these two combinations are permutations, because the elements in the set are ordered differently. More formally, a permutation is an ordered list without repetitions, perhaps missing some elements. This means that {1, 2, 2, 3, 4, 5, 6} and {1, 2, 4, 5, 5, 6} are not permutations of the set {1, 2, 3, 4, 5, 6}. Now suppose you have these objects: 1, 2, 3 Here is a list of all permutations of those: 1 2 3; 1 3 2; 2 1 3; 2 3 1; 3 1 2; 3 2 1; 606
CHAPTER 45. COMBINATIONS AND PERMUTATIONS - GRADE 12
45.6
45.6.1
Counting Permutations
Let S be a set with n objects. Permutations of k objects from this set S refer to sequences of k different elements of S (where two sequences are considered different if they contain the same elements but in a different order, or if they have a different length). Formulas for the number of permutations and combinations are readily available and important throughout combinatorics. It is easy to count the number of permutations of size r when chosen from a set of size n (with r ≤ n). 1. Select the first member of all permutations out of n choices because there are n distinct elements in the set. 2. Next, since one of the n elements has already been used, the second member of the permutation has (n − 1) elements to choose from the remaining set. 3. The third member of the permutation can be filled in (n − 2) ways since 2 have been used already. 4. This pattern continues until there are r members on the permutation. This means that the last member can be filled in (n − (r − 1)) = (n − r + 1) ways. 5. Summarizing, we find that there is a total of n(n − 1)(n − 2)...(n − r + 1) different permutations of r objects, taken from a pool of n objects. This number is denoted by P (n, r) and can be written in factorial notation as: P (n,r) = n! . (n − r)!
For example, if we have a total of 5 elements, the integers {1, 2, 3,4,5}, how many ways are there for a permutation of three elements to be selected from this set? In this case, n = 10 and r = 3. Then, P (10,3) = 10!/7! = 720.
Worked Example 206: Permutations Question: Show that a collection of n objects has n! permutations. Answer Proof: Constructing an ordered sequence of n objects is equivalent to choosing the position occupied by the first object, then choosing the position of the second object, and so on, until we have chosen the position of each of our n objects. There are n ways to choose a position for the first object. Once its position is fixed, we can choose from (n-1) possible positions for the second object. With the first two placed, there are (n-2) remaining possible positions for the third object; and so on. There are only two positions to choose from for the penultimate object, and the nth object will occupy the last remaining position. Therefore, according to the multiplicative principle, there are n(n − 1)(n − 2)...2 × 1 = n! ways of constructing an ordered sequence of n objects.
607
45.7
CHAPTER 45. COMBINATIONS AND PERMUTATIONS - GRADE 12
Permutation with Repetition When order matters and an object can be chosen more than once then the number of permutations is: nr where n is the number of objects from which you can choose and r is the number to be chosen. For example, if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three letter patterns (trigrams) you find that there are 43 or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total. Permutation without Repetition When the order matters and each object can be chosen only once, then the number of permutations is: n! (n − r)!
where n is the number of objects from which you can choose and r is the number to be chosen.
For example, if you have five people and are going to choose three out of these, you will have 5!/(5-3)! = 60 permutations. Note that if n = r (meaning number of chosen elements is equal to number of elements to choose from) then the formula becomes n! n! = = n! (n − n)! 0! For example, if you have three people and you want to find out how many ways you may arrange them it would be 3! or 3 × 2 × 1 = 6 ways. The reason for this is because you can choose from three for the initial slot, then you are left with only two to choose from for the second slot, and that leaves only one for the final slot. Multiplying them together gives the total.
45.7
Applications
Extension: The Binomial Theorem In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads
n
The coefficients form a triangle, where each number is the sum of the two numbers above it:
1
1 1 1 4 3 6 2
1 1 3 4 1 1
This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayym in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.
Worked Example 207: Number Plates Question: The number plate on a car consists of any 3 letters of the alphabet (excluding the vowels and 'Q'), followed by any 3 digits (0 to 9). For a car chosen at random, what is the probability that the number plate starts with a 'Y' and ends with an odd digit? Answer Step 1 : Identify what events are counted The number plate starts with a 'Y', so there is only 1 choice for the first letter, and ends with an even digit, so there are 5 choices for the last digit (1,3,5,7,9). Step 2 : Find the number of events Use the counting principle. For each of the other letters, there are 20 possible choices (26 in the alphabet, minus 5 vowels and 'Q') and 10 possible choices for each of the other digits. Number of events = 1 × 20 × 20 × 10 × 10 × 5 = 200 000 Step 3 : Find the number of total possible number plates Use the counting principle. This time, the first letter and last digit can be anything. Total number of choices = 20 × 20 × 20 × 10 × 10 × 10 = 8 000 000 Step 4 : Calculate the probability The probability is the number of events we are counting, divided by the total number of choices. 1 Probability = 8200 000 = 40 = 0,025 000 000
1. Tshepo and Sally go to a restaurant, where the menu is: Starter Main Course Dessert Chicken wings Beef burger Chocolate ice cream Mushroom soup Chicken burger Strawberry ice cream Greek salad Chicken curry Apple crumble Lamb curry Chocolate mousse Vegetable lasagne A How many different combinations (of starter, main meal, and dessert) can Tshepo have? B Sally doesn't like chicken. How many different combinations can she have? 2. Four coins are thrown, and the outcomes recorded. How many different ways are there of getting three heads? First write out the possibilites, and then use the formula for combinations. 3. The answers in a multiple choice test can be A, B, C, D, or E. In a test of 12 questions, how many different ways are there of answering the test? 4. A girl has 4 dresses, 2 necklaces, and 3 handbags. A How many different choices of outfit (dress, necklace and handbag) does she have? B She now buys two pairs of shoes. How many choices of outfit (dress, necklace, handbag and shoes) does she now have? 5. In a soccer tournament of 9 teams, every team plays every other team. A How many matches are there in the tournament? B If there are 5 boys' teams and 4 girls' teams, what is the probability that the first match will be played between 2 girls' teams? 6. The letters of the word 'BLUE' are rearranged randomly. How many new words (a word is any combination of letters) can be made? 7. The letters of the word 'CHEMISTRY' are arranged randomly to form a new word. What is the probability that the word will start and end with a vowel? 610
CHAPTER 45. COMBINATIONS AND PERMUTATIONS - GRADE 12
45.8
8. There are 2 History classes, 5 Accounting classes, and 4 Mathematics classes at school. Luke wants to do all three subjects. How many possible combinations of classes are there? 9. A school netball team has 8 members. How many ways are there to choose a captain, vice-captain, and reserve? 10. A class has 15 boys and 10 girls. A debating team of 4 boys and 6 girls must be chosen. How many ways can this be done? 11. A secret pin number is 3 characters long, and can use any digit (0 to 9) or any letter of the alphabet. Repeated characters are allowed. How many possible combinations are there?
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A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section A is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.
TRANSLATION
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FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See 623
APPENDIX A. GNU FREE DOCUMENTATION LICENSE Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.
ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright c YEAR YOUR NAME. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. | 677.169 | 1 |
Algebra 1 and 2: Systems of Equations Task Cards
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this file type before downloading and/or purchasing.
918 KB|27 pages
Product Description
This set of task cards is made up of 6 sets of 5 cards. Each set emphasizes
a different method for solving systems of equations:
Set 1: Solving by Graphing
Set 2: Solving by Substitution
Set 3: Solving by Addition or Subtraction
Set 4: Solving by Multiplication First
Set 5: Special Types (No solution or infinite solutions)
Set 6: Mixture of Systems
The last problem in each set is a real life problem where students
must first write the system before solving it.
There are many ways to incorporate these
cards in your classroom. | 677.169 | 1 |
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