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Solving Equations Graphic Organizer
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0.31 MB | 4 pages
PRODUCT DESCRIPTION
From my experience teaching general education and special education students how to solve equations (1-, 2- and multi-step), one of the most challenging parts for students is having them write their steps in a clear and organized manner.
So I made this graphic organizer out of necessity and for these reasons:
1. As a way to scaffold students' notes and example problems. The graphic organizer prompts students to solve equations step-by-step, in a vertical manner.
2. To encourage students to check their answers. Each problem box has a section for students to check their answer – substitute the value they think is the answer and make sure it is a true math statement. This simple step is sometimes overlooked, but it is vital in having students review prerequisite skills (PEMDAS and evaluating expression) and in having them self-monitor their work.
Included in this download is a PDF version (great for just making copies for students to take notes or show their work) and the Microsoft Word version in which you can type in your own problems and actually create assignments, quizzes, and tests using the graphic organizer format.
This tried and true graphic organizer has helped all levels of math students – from general education classes to resource classes | 677.169 | 1 |
Introductory Factoring Problems for Algebra 1 Spring 2014
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0.17 MB | 3 pages
PRODUCT DESCRIPTION
This worksheet is designed to replace a lecture on the topic of basic factoring, leading up to factoring trinomials. This worksheet emphasizes the area model for representing a product of polynomials. as in my experience, after 3-4 days of practice, every student can successfully factor a trinomial.
I start out class with a 15-minute "mini-lesson," giving my students some basic examples of what today's lesson will be about. Once the mini-lesson is over, I have them get to work within their groups on this worksheet | 677.169 | 1 |
Compare Linear, Exponential, and Quadratic Models SmartBoard Lesson
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0.07 MB
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The objective for this lesson is Choose a model that best fits a collection of data.
These lessons are done in Smart Notebook. Each lesson contains an Opening Activity (bell ringer); an objectives slide, which includes the common core standards the lesson is tied to; a definition slide; example slides; 'try' slides for the students; and a recap slide. Feel free to edit, change, add, or subtract from the lessons as you like. Make them | 677.169 | 1 |
A sequence with a constant difference between consecutive terms (e.g., 2, 5, 8, 11,... is an arithmetic sequence with a constant difference of 3).
Associative Property:
A property of addition or multiplication in which the regrouping of the addends or factors does not change the outcome of the operations [i.e., (a + b) + c = a + (b + c) and (ab)c = a(bc)].
Attribute:
A characteristic of an object, such as color, shape, or size.
Bar graph:
A graphical display representing data in different categories or groups. The length of a rectangle or bar is used to represent the numerical amount.
Box and whisker plot:
A graphical display which shows the median, quartiles, and extremes of a set of data but does not display any other specific data values.
Capacity:
The maximum amount that can be contained by an object. Often refers to measurement of liquid.
Cardinal numbers:
The counting numbers (1, 2, 3...).
Circle graph:
A graphical display that shows data as parts of a whole circle.
Circumference:
The distance around a circle; the formula for circumference of a circle is pi times the diameter (C = TTd).
Closed figure:
The boundary of a simple 2-dimensional region, including shapes with straight and curved sides.
Commutative Property:
A property of addition or multiplication in which the sum or product stays the same when the order of the addends or factors is changed (i.e., a + b = b + a and ab = ba).
Concrete:
Physical objects used to represent mathematical situations.
Congruency:
Geometric figures having the same size and shape; all corresponding parts of congruent figures have the same measure.
Conjecture:
A preliminary statement or hypothesis that something is true; a statement may later be confirmed or disproved through observation or testing.
Coordinate geometry:
The algebraic study of geometry through the use a coordinate plane or system.
Coordinate plane:
A 2-dimensional system in which the coordinates of a point are its distances from two intersecting perpendicular lines called axes. The formal name for this system is Cartesian coordinate system.
Counting techniques:
Methods to determine the number of possible outcomes of an event. Some of the methods are tree diagram, list, rules for multiplication, combinations, and permutations.
Curve fitting:
The sketching of a line or curve to best describe a relationship between two variables on a scatter plot.
Deductive reasoning:
The process of reasoning that starts from statements accepted as true and applied to a new situation to reach a conclusion (e.g., if 5+4 = 9, and 6+3 = 9, then 5+4 = 6+3).
Diagonal:
For a polygon in the plane, any line segment joining non-adjacent vertices. For a polyhedron in space, a line segment joining two vertices not in the same face.
Dilation:
A transformation which produces a figure similar to the original by proportionally shrinking or stretching the figure.
Dimensional analysis:
A method of converting units within a measurement system.
Direct proof:
The proof of a proposition by accepting the hypothesis of the proposition and arguing to the conclusion.
Distributive Property:
A property which establishes a relationship between multiplication and addition such that multiplication distributes across the addition [i.e., a(b+c) = ab + ac].
Divisibility (rules of) :
Special tests to determine if a particular integer is a factor of a given number, (e.g., a number is divisible by 10 if it ends in a 0).
Elapsed time:
The amount of time between a beginning time and an ending time.
Equally-likely outcomes:
Events in a sample space that have the same probability of occurring.
Equation:
A mathematical sentence of equality between two expressions (e.g., N + 50 = 75 or 75 = N + 50 means that N + 50 must have the same value as 75).
Equivalent:
Numbers or expressions that have the same value.
Estimation:
The process of finding a number close to an exact amount.
Euclidean geometry:
The geometry (plane and solid) based on Euclid's postulates.
Experimental probability:
A probability calculated from the results of an experiment.
Explicit relationship:
A sequence rule using the number of the term to define the function [e.g., in the sequence 3, 6, 9,..., the explicit rule is f(n) = 3n where n is the number of the term and f(n) is the value of the term].
Exponent:
A number which is placed to the right of and above another number (base). The value of the exponent determines how many times the base is used as a factor (e.g., 34 = 3 x 3 x 3 x3; {3 is the base and is used as a factor 4 times} the exponent is 4).
A function whose general equation is a y=abx or y=abkx, where a, b, and k stand for constants.
Face:
A plane surface of a three-dimensional figure.
Factor:
The numbers or terms multiplied in an expression.
Formula:
An equation that states a fact or rule (e.g., A= w).
Frequency table:
A display to show how often items, numbers, or a range of numbers occur.
Function:
A relationship in which every value of x has a unique value of y (e.g., the relation y = 2x + 1 is a function because for every different x, there is one and only one y).
Function notation:
A notation that describes a function. For a function ƒ, when x is a member of the domain, the symbol ƒ(x) denotes the corresponding member of the range [e.g., an equation of a function might be ƒ(x) = x+3].
Geometric sequence:
A sequence with a constant ratio between two consecutive terms (e.g., 1, 2, 4, 8, 16,... is a geometric sequence with a ratio of 2).
Graph:
A pictorial representation of information or relationships between numbers.
Histogram:
A graphical display representing continuous data in different categories or groups.
Indirect measurement:
A measurement which is found by using a formula or other strategy and not actually measuring something (e.g., finding the height of a tree without actually holding a ruler next to it).
Indirect proof:
A proof that begins by assuming the denial of what is to be proved and then deducing a contradiction from this assumption.
Inductive reasoning:
A type of type of mathematical reasoning which involves observing patterns and using those observations to make generalizations.
Inequality:
A mathematical sentence in which the value of the expressions on either side of the relation symbol are unequal. Relation symbols include >, <, , , or (e.g., x < y, 7 > 3, n 4).
Two operations that "undo" each other (e.g., addition and subtraction).
Line graph:
A graphical representation using points connected by line segments to show how something changes over time.
Line of best fit:
A line drawn on a scatter plot to estimate the relationship between two sets of data.
Line plot:
A graph using marks (e.g., X, ·) above a number on a number line to show the frequency of data.
Linear function:
A function with no exponents other than one and with no products of the variables (e.g., y=x+4, y= -4, and 3x-4y = 1/2 are linear functions); in a rectangular coordinate system, the graph of a linear function is a line.
Manipulatives:
Tools, models, blocks, tiles, and other objects which are used to explore mathematical ideas and solve mathematical problems.
Matrices:
Rectangular arrays of numbers arranged in rows and columns.
Mean:
In a collection of data, the sum of all the data divided by the number of data.
Measures of central tendency:
Numbers which tend to cluster around the "middle" of a set of values. Three such numbers are mean, median, and mode.
Median:
The middle number (or the mean of the two middle numbers when necessary) in a collection of numbers that is arranged in order from least to greatest.
Mode:
The number(s) that occur(s) most often in a collection of data.
Model:
To represent or show mathematical ideas and relationships and real-world situations using objects, pictures, graphs, tables, functions, and other methods.
Multiple:
The product of a whole number and any other whole number.
Multiplicative inverse:
Two numbers are multiplicative inverses of each other if their product is 1 (e.g., since 4 x 1/4 = 1, 1/4 and 4 are multiplicative inverses).
Numerical perspective:
A mathematical idea expressed as a number or numbers.
One-dimensional:
A figure that has length but no width or height.
Ordinal numbers:
Numbers used to express order (e.g., 1st, 2nd, 3rd).
Outcome of an activity:
One of the possible events in a probability situation.
Parallel(ism):
Lines that lie in the same plane and never meet. Also, planes lying in space that never meet.
Patterns:
Regularities in situations such as those in nature, events, shapes, designs, and sets of numbers (e.g., spirals on pineapples, geometric designs in quilts, the number sequence 3, 6, 9, 12, . . . ).
Percent:
A special ratio that compares a number to 100 and uses the % sign (e.g., 1/2 = 50% and 2/3 = 66 2/3%).
Perimeter:
The distance around a geometric shape.
Perpendicular(ity):
Lines in the same plane which intersect to form a right angle.
Pictograph:
A graphical representation that shows numerical information by using picture symbols.
Place Value:
The value of a digit as determined by its position in a number (e.g., in the number "11" the one is worth either 10 or 1, depending on the position).
Population:
A group of people, objects, or events that fit a particular description.
Power:
A number expressed using an exponent (e.g., the number 53 is read five to the third power or five cubed).
Power function:
A function with a variable base and a constant exponent [e.g., f(x) = axb].
Precision:
The smallest place value to which an approximate number or measurement is expressed (e.g., if pi is represented as 3.14, then its precision is .01).
Prime number:
A whole number greater than 1 that can only be divided evenly by itself and 1 (e.g., 17).
Prism:
A three-dimensional figure with parallelogram faces and two parallel, congruent bases.
Probability:
The number of favorable outcomes compared to the number of possible outcomes of an experiment. A number from 0 to 1 which indicates how likely something is to happen.
A figure has symmetry if there exists some line or point through which all points of the figure can be reflected to generate another point on the figure.
System of linear equations:
Two or more equations that are conditions imposed simultaneously on all the variables, but that may or may not have common solutions (e.g., x+y=2, and 3x + 2y = 5).
Theoretical probability:
A probability calculated from mathematical counting techniques.
Three-dimensional:
An object that has length, width, and height.
Transformation:
A rule for moving every point in a plane figure to a new location.
Translation (slide):
A transformation that slides a figure a given distance in a given direction.
Trigonometric ratio:
A comparison of the measures of the lengths of two sides of a right triangle expressed in fractional or decimal form; there are six trigonometric ratios (sine, cosine, tangent, cotangent, secant, and cosecant) associated with any angle.
Trigonometry:
The study of right triangle measurements and ratios, useful for calculating indirect measurements.
Two-dimensional:
A figure that has length and width but not height (i.e., a plane figure such as a rectangle or circle).
Valid argument:
An explicit demonstration or proof that has been shown to be true.
Validate:
To give evidence that a solution or process is correct.
Variable:
A letter or symbol which represents one or more numbers.
Vertex (vertices):
The points where two line segments come together (corners).
Visual perspective:
A mathematical idea expressed as a picture, graph, or table.
Volume:
The amount of space enclosed in a three-dimensional figure, measured in cubic units. | 677.169 | 1 |
OSC Study Guides
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Mathematics Study & Revision Guides
Our International Baccalaureate (IB) Mathematics Revision Guides are designed to help students in their revision and cover the whole syllabus for all IBDP courses including separate guides for each HL option.
The guides contain numerous worked examples, questions and self-tests, and for HL and SL provide coverage of non-calculator as well as calculator techniques.
School/Bookseller?
Using the TI-Series Calculators in IB Mathematics 3rd Edition (SL/HL)
£16.50
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ISBN: 978-1-907374-68-5 | Edition: 3 | Publisher: OSC Publishing
Written By: Andy Kemp, Ian Lucas
This book addresses the important role, now obligatory in HL, SL and Maths Studies, that calculators play in International Baccalaureate (IBDP) Maths courses and exams. It is based on the widely used TI-Series calculators. It uses a step-by-step approach and includes an excellent section on programming your calculator.
About the Authors
Andy has taught Mathematics since 2004, and is currently Head of Mathematics and Director of Digital Strategy at Taunton School where he teaches the IB Higher Level course.
Ian is a Maths teacher with nearly 40 years' experience. He has been involved with IB teaching from its earliest days. He is a regular teacher at OSC Spring Revision Courses and Summer Schools.
Mathematical Studies 3rd Edition (SL)
£20.00
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ISBN: 978-1-907374-57-9 HL 3rd Edition
£23.00
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ISBN: 978-1-907374-55-5 Calculus 2nd Edition (HL)
£9.00
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ISBN: 978-1-907374-61-6 | Edition: 2 | Publisher: OSC Publishing
Written By: Wendy Stevens
This guide addresses the Mathematics HL Option Calculus. The Option is a separate part of the exam and generally it is the last part of the syllabus to be covered, and therefore often the part least revised and yet it is worth 20% of the final marks and should merit special attention. This dedicated guide will help students focus on it and achieve their best in the examination.
About the Author
Wendy has been teaching Maths for over 25 years and has taught regularly on OSC's Spring Revision courses and Summer Schools since 2001.
Mathematics Option: Discrete Mathematics 2nd Edition (HL)
£8.50
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ISBN: 978-1-907374-62 for Sets, Relations and Groups (HL)
£8.50
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ISBN: 978-1-907374-60-9 Statistics and Probability 2nd Edition (HL)
£8.50
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ISBN: 978-1-907374-59SL papers 1 and 2 both contain "extended response", or long answer, questions. Although the syllabus which these questions test is the same as for the short answer questions, the techniques required are very different.
This guide strips down the questions to their essential components, and shows students how to set about answering them. What do the keywords mean? How does the information in one part help with the answer to a different part? How do you cut through the words to find out exactly what is being asked? There are 32 example questions: some are analysed in detail, helping students build up a "toolbox" of techniques; others are set out as practice questions with comprehensive hints for those who get stuck. All the questions have full sets of answers.
The book should be seen as a companion guide to the SL Revision Guide. It will be invaluable for those students who are not yet confident answering the extended response questions, or where the student has not had extensive practice.
This guide is updated for the new syllabus introduced in Sep 2012, for exams beginning May 2014 SL 3rd Edition
£20.00
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ISBN: 978-1-907374-56-2The OSC Pre-IB Mathematics guide is designed to ensure that students do not only gain the minimum knowledge required but that they also develop the strong confidence and expertise in mathematics which will allow them to do well during the two years of the Diploma Programme.
Students who use this Pre-IB Mathematics guide will learn and refine their algebraic techniques so important while embarking on the future topics such as calculus. They will also study the basic and advanced concepts of functions, trigonometry, vectors, set theory, statistics, probability, matrices, sequences/series and logarithms.
About the Authors
Jacekis a teacher with 22 years of experience including 14 years with the IB programme and 9 years with OSC. He has worked at various international schools, currently at The British School in Warsaw as the Head of Mathematics Department.Accredited by the British Accreditation Council for Independent Further and Higher Education as a Short Course Provider | 677.169 | 1 |
Math 101 Advice
Math 101 Documents
Showing 1 to 30 of 100 and economics.
In Paul Krugmans article in theMath 101 Advice
Showing 1 to 1 of 1
I recommend it because this class will be able to help you understand complex formulas and patterns that you see in the world. It can also help you be able to pass the CRC and be able to apply for numerous jobs out there in the world.
Course highlights:
I've gained the ability to be able to undergo mathematical functions in my mind such as multiplying different variables to dividing complex formulas. It also helped me come up with different answers
Hours per week:
6-8 hours
Advice for students:
Take this course very seriously because this will be one of the very things that will help you out with life. Study as much math as possible for at least an hour a day to keep the knowledge fresh in your mind as well. | 677.169 | 1 |
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Show More fill the gap between the basic courses and the highly technical and specialized courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote at an early stage, a better appreciation and understanding of each other's discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. He existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics. Mathematical rigor combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics | 677.169 | 1 |
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Numeric information in graphic forms skills preTwenty-five students in physical science spring 2014 were given nine questions which focused on this particular outcome. The preassessment was done on the first day of class.
The students are not unfamiliar with mathematics. The last question asked the highest math class taken by the students. Fifteen of the students had completed college algebra, four had completed post-college algebra courses (two completed algebra and trigonometry, two completed statistics). The remaining six chose not to answer the question on the preassessment.
Despite 19 of the 25 students having completed college algebra, performance on the preassessment was abysmal. Although 14 students could plot xy coordinate pairs, only three students could determine the slope of a line from a graph, only six were able to determine that the y-intercept was zero (the first and third questions in the chart below). The number of students answering a question correctly is shown in the following chart.
The last two questions were non-graphical questions. They presented the students with an equation in the format y = b + mx and asked the students to determine the slope and intercept. Only five identified the slope correctly, only four the intercept. Many students left this and other questions blank.
With nine questions, a perfect paper would have been a score of nine. The highest score was a single score of six. The average was 1.6 and the median was one. Seven students scored zero correct. The distribution of the student scores can be seen in the following box plot.
The score distribution is so low that the lower whisker (the minimum) is also the first quartile - seven zeroes out of 25 students.
The student performance was not just weak, the performance was weaker than the fall term 2013 performance on the same instrument. On five of the nine questions performance fell term-on-term. The sample sizes were nearly identical.
The red bars represent a drop from fall 2013 to spring 2014, the blue bars represent a gain (two questions) or no change (two questions) in the number of students answering correctly.
SC 130 Physical Science is intendedWhile these questions will be retested at term end, the question that remains unanswered is "downstream" retention. To what extent do students who have taken physical science retain some of these mathematical skills beyond the end of the | 677.169 | 1 |
Independent and Dependent Variable Notes- 6th Grade Mathematics
PDF (Acrobat) Document File
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1.05 MB
PRODUCT DESCRIPTION
These notes review independent and dependent variables. There is also a cut and paste activity where students match the independent and dependent variables.
The following activity is designed as an introduction to the following math TEKS:
6.(6) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to: (A) identify independent and dependent quantities from tables and graphs; (B) write an equation that represents the relationship between independent and dependent quantities from a table; and (C) represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b | 677.169 | 1 |
DESCRIPTION: This new edition of the Beta Mathematics Homework Book now contains even more of the questions and activities for which David Barton is famous: well graded, interesting, and linked throughout to real-world applications. The exercises in the Workbook are matched to corresponding exercises in Beta Mathematics 2E, (ISBN 9781741038842) making it easy for teachers to choose, and students to do, homework, extra practice and revision that matches what is done in class. As with Beta Mathematics 2E, the contents of this Workbook are organised into the three Strands of Level 5 (Years 9 and 10) Mathematics in the New Zealand Curriculum, and the examples and exercises follow the spirit of the Numeracy Project. The Beta Mathematics Workbook is organised so that each two page sheet can be removed from the book and handed in for assessment. More room has been provided within questions, for students to demonstrate their mathematical skills. Computers provide mathematicians, engineers, accountants and many other professionals with instant feedback in their explorations of number patterns and relationships. The Beta Mathematics CD in this Workbook features PowerPoint worked examples, data sets, spreadsheets and LiveMathTM computer algebra activities to give students practice with this vital aspect of maths | 677.169 | 1 |
Year 11
NCEA
NCEA Level 1
To pass Level 1 NCEA a student requires 10 Numeracy Credits, which are most easily gained in Maths. A few standards outside Maths give numeracy as well.
There is no difference between the credits from Internal and External Standards, though not every unit offers the same number of credits.
To enter Tertiary study or to take Level 2 Maths at St John's requires 14 Maths (not Numeracy) Credits. This should be the minimum aim of any student.
However, entry into the External Year 12 course requires passing the Algebra standard and a strong showing in the other Externals. Physics also has minimum Maths requirements. In general students cannot go on to do academic Maths courses without strong Algebra, and Year 13 Calculus and Algebra require taking the External courses.
The policy of the Mathematics department at St John's is that students will not be given a resit for internal assessments.
It is important that students practice assessments in the format that they will be given, and not just practice isolated skills.
Resources
NZQA
The exemplars for this year's units, showing what they expect at the various grade boundaries.
Homework
The Gamma Mathematics Workbook published by Pearson matches the textbook used at St John's for the
External course:
The Fast Track 3 from Eton Press is at a higher level, but covers the same
basic material and is recommended for more able students:
Nulake have booklets for each individual topic which are effectively textbooks with space for answers, which might be useful for some students. The EAS one is specifically for students doing all the external units.
There's some useful free material at mathscentre.co.nz but unfortunately still that annoying gap with Chance and Data.
Year Plan for Year 11 External Course
This is a plan only, and times will change somewhat with circumstances. | 677.169 | 1 |
Delivery: 10-20 Working Days
(31 reviews)
This brand-new hands-on workbook presents exercises, problems and quizzes with solutions and answers as it progresses through all math and science topics covered on the ACT college entrance test. Separate math chapters cover pre algebra, elementary algebra, intermediate algebra, plane geometry, coordinate geometry, and trigonometry. Separate science chapters cover data representation passages, research summary passages, and conflicting scientific viewpoints. The book's science sections emphasize the scientific method and focus on how to read scientific passages. A glossary of helpful science terms is provided. Strategies for success are included in every chapter. A full-length practice test reflecting the ACT science and math sections come with explained answers.
Similar Products
Specifications
Country
USA
Author
Roselyn Teukolsky M.S.
Binding
Paperback
EAN
9780764140341
Edition
Workbook
ISBN
0764140345
Label
Barron's Educational Series
Manufacturer
Barron's Educational Series
NumberOfItems
1
NumberOfPages
480
PublicationDate
2009-05-01
Publisher
Barron's Educational Series
Studio
Barron's Educational Series
Most Helpful Customer Reviews
From the student's perspective: I used this book to refresh myself on ACT math (my science score was already very high). While it's does a decent job of breaking down how to tackle problems in the lessons, it does not offer enough practice problems for the content learned to stick. After working through much of the math portion's lessons and practice problems, my math score went down 1 point when I tested again. I was disappointed to say the least. I have been able to raise my score with thorough and frequent practice using the Real ACT prep guide and old "preparing for the ACT" booklets that are available online for free: they consist of real ACT tests with the same type of problem you see on the real test. Additionally, the lack of practice tests is horrendous. I like to say that it includes half of a test: one math section and one science section. It might be a ploy to encourage customers to purchase other Barron's ACT books, but without additional practice tests,...
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This was a great book. it helped my Brother in law study for his exam. I would highly recommend ppl to buy this.
I would not recommend this book to anyone who scores less than 28 in these subjects. While I did find it helpful, I skipped over some of the material because the book doesn't really explain it very well and assumes that you know a lot (specifically in the math section | 677.169 | 1 |
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern... more...
Whether you're new to geometry or just looking for a refresher, this completely revised and updated third edition of Geometry Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with geometry essentials, Geometry Success... more...
This is an essentially self-contained book on the theory of convex functions and convex optimization in Banach spaces, with a special interest in Orlicz spaces. Approximate algorithms based on the stability principles and the solution of the corresponding nonlinear equations are developed in this text. A synopsis of the geometry of... more...
An innovative and appealing way for the layperson to develop math skills--while actually enjoying it Most people agree that math is important, but few would say it's fun. This book will show you that the subject you learned to hate in high school can be as entertaining as a witty remark, as engrossing as the mystery novel you can't put down--in... more...
Handy compilation of 100 practice problems, hints and solutions indispensable for students preparing for the William Lowell Putnam and other mathematical competitions. Preface to the First Edition. Sources. 1988 edition. more...
Any child can overcome the disadvantages of mediocre math teaching in school and parental math anxiety at home. Math Power offers easy-to-follow and concrete strategies for teaching math concepts. These lively techniques ? including games, questions, conversations, and specific math activities ? are suitable for children from preschool to age 10.... more...
Over 300 challenging problems in algebra, arithmetic, elementary number theory and trigonometry, selected from Mathematical Olympiads held at Moscow University. Only high school math needed. Includes complete solutions. Features 27 black-and-white illustrations. 1962 edition. more...
Students and others wishing to know more about the practical side of mathematics will find this volume a highly informative resource. Accessible explanations of important concepts feature worked examples and diagrams. 1963 edition. more... | 677.169 | 1 |
This textbook is written to develop thorough understanding the mastery behind the foundations of geometry that needed to solve geometric problems using vectors in three dimensional Euclidean space at a useful proficiency level, it introduces a comprehensive study on three dimensional Euclidean spaces' objects, namely; points, line segments, vectors, straight lines, planes, spheres, cylinders, quadratic surfaces in general analytically, and includes different comparisons between such studied objects among themselves, it focuses on the orthogonal projections of a point and directed line segment on line, on plane, and on sphere, and it is recommended world widely for the second level universities' students, its contents are written in a systematic, streaming, understandable and attractive way for student's interest, supported with adequate number of examples, exercises, problem solving and applications to meet the quality standard. Moreover; it is a prerequisite for some other fields of study, like Descriptive Geometry, Differential Geometry, Analytical Geometry, Algebraic geometry, Injective Geometry and some others. | 677.169 | 1 |
Learn or Review Trigonometry: Essential Skills (Step-by-Step
The Department has the degree programs and faculty from the institutions that form the new Kennesaw State University. The main topic of this book is the study of the representations of hypergroups via generalized permutations and hypermatrices. Well, if we had 200 apples increasing to 400 (double) that would be a 100% increase. For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. After two failures the student must have the approval of a committee formed by the department chair to review the case.
Multiple Choice and Free-Response Questions in Preparation for the Ap Statistics Examination
Here is EducationWorld's state certification listing and USC's Certification Map. To teach at the community college level, you should get a Master's degree in mathematics or a Master of Arts in Teaching; to teach at the college level, you should get a Ph download. Q: Sir My question is A) 5 raised to 3 divided by 5 raised to 6 , source: 60 Addition Worksheets with 5-Digit, 2-Digit Addends: Math Practice Workbook (60 Days Math Addition Series) (Volume 27) Continued enrolment at the Advanced level depends on meeting the standards set for the degree. Conversely, students who miss out on a place initially may qualify for admission part-way through their degree 500 Addition Worksheets with download epub theleadershiplink.org. Help With Math: Offers MathXpert software to help learn precalculus and calculus. Precalculus Notes: Extensive collection of nice explanations of precalculus topics by Ken Kuniyuki of San Diego Mesa College Creative Problem Solving, Grade 8: Multiple Solutions for the Same Answer read epub. Calculating the Volume of a Vase: A very nicely presented project in which the volume of a real vase is computed. From the Duke University Mathematics Department. Karl's Calculus Tutor: Lots of nice explanations of calculus concepts. Mudd Math Fun Facts: Nice explanations of many calculus (and other mathematical) concepts , source: Mathematics - Mechanics and download for free travel.50thingstoknow.com. Through a problem-solving approach, this aspect of mathematics can be developed. Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context. Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989) download. For students using this pattern, all grades awarded by the school are averaged in the GPA calculation. One yearlong course (1.0 unit value) must be either a course in geometry or part of an integrated sequence that includes sufficient geometry. Other rigorous courses that use mathematical concepts, include a mathematics prerequisite, and are intended for 11th and 12th grade levels, may also satisfy the requirement Solutions Manual for Ap Prep Book for Bc Calculus read online.
This list is just a sample of the organisations who employ graduates with mathematical skills. Biosciences […] You may not have heard about many famous female mathematicians. This is because until relatively recently it wasn't easy for women to go to university, let alone have a career in science or mathematics , e.g. Pre-algebra: Arkansas Edition Successful applicants will receive a stipend upon completing and finalizing the report pdf. The surcharge is subject to review and may change. Information about payment methods and the surcharge is set out at: AQA GCSE Statistics theleadershiplink.org. D. and pursue a career as an academic mathematician. D. in mathematics, you should first try your best to talk yourself out of it. It's a little like aspiring to be a pro athlete. Even under the best of circumstances, the chances are too high that you'll end up in a not-very-well-paying job in a not-very-attractive geographic location , cited: 60 Subtraction Worksheets with download pdf 60 Subtraction Worksheets with 3-Digit. I learned the entire S1 course from this book! I purchased this book a few weeks before the final exam, and it only took me several hours of solid work to learn everything that I needed to know for the exam. Some of the text can seem daunting at times as the authors do tend to write quite a lot of content, but i see this as a positive; the explanations are clear and easy to follow, so when you actually get round to reading it all, it doesn't seem so bad online. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems. Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC pdf. MathGL3d Interactive 3d visualiztion system for Mathematica. MathGrapher Mathematical graphing tool for 2D and 3D functions and data. Includes nonlinear curve fitting and integration of coupled ordinary differential equations. MathPoint a software company producing mathematical libraries inActiveX and Java 60 Subtraction Worksheets with read for free
When asked how long a mathematics major should ideally spend studying a proof outside of the classroom, their average response was between 30 and 37 minutes, with 81% giving a response of greater than 15 minutes. Again, mathematics majors see things differently. Their average response to the same question was 17 to 20 minutes, with only 41% giving a response of greater than 15 minutes. (Weber & Mejia-Ramos, 2014) , cited: 100 Subtraction Worksheets with 4-Digit Minuends, 2-Digit Subtrahends: Math Practice Workbook (100 Days Math Subtraction Series) (Volume 8) online. Note that the evaluation of the expression requires matrix multiplication. You can use the Cholesky Decomposition to solve systems of equations.. 30 Addition Worksheets with read for free They construct histograms and back-to-back stem-and-leaf plots. By the end of Year 10, students recognise the connection between simple and compound interest Holt Precalculus: A Graphing Approach: Student Edition 2004 Her office is Room 425 of the Physical Sciences Center. Our junior and senior years are spent at Western Kentucky University taking college classes in math, science, and lots of other subjects. While many students are looking for their locker combination, we're searching for alternative fuels and a cure for cancer Solutions Manual for Ap Prep download online Solutions Manual for Ap Prep Book for Bc. I've read all the material available on the EMF website, but I still have questions that I need answered before enrolling my child , source: Holt Precalculus: A Graphing Approach: Student Edition 2004 Holt Precalculus: A Graphing Approach:. Students use metric units for length, mass and capacity. Students make models of three-dimensional objects. Students conduct chance experiments and list possible outcomes. They conduct simple data investigations for categorical variables. By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division Pre-Calculus Essentials read for free We strive to help our students to understand mathematical concepts at a very high level. To do this, we ask even our youngest students to become thinkers and analysts. Students are routinely asked to compare, contrast and explain epub. I trust math would inspire neither of these in an average person. Trying to make the best of it, I'll seek refuge in a third quote from Kant, "The sublime must always be great; the beautiful can also be small." talking about experimental sciences, has the following to say about proofs: "Notice also that scientists generally avoid the use of the word proof 30 Addition Worksheets with download online download online. Zeta and L-functions; Dedekind zeta functions; Artin L-functions; the class-number formula and generalizations; density theorems online. Family Prize for the best SPUR paper was awarded to Lingfu Zhang and his mentor Hong Wang for the paper, "Refinements of the 2-Dimensional Strichartz Estimate using the Maximum Wave Packet" (project provided by Larry Guth ). Instructor Roger Casals has been awarded the José Luis Rubio de Francia Prize from the Royal Spanish Mathematical Society. This award recognizes and encourages the work of young researchers in mathematics , cited: 500 Addition Worksheets with download for free 500 Addition Worksheets with 4-Digit,. | 677.169 | 1 |
The Complete Idiot's Guide to Algebra Word Problems (Complete Idiot)
You are here: Home Online resources Problem solving, find here an annotated list of problem solving websites and books, and a list of math contests. There are many fine resources for word problems on the net!
If you're a seller, Fulfillment by Amazon can help you. The book does a great job of providing a structure by classifying algebra word problems into age.
/teaching/p, a Click, a Click features a graduated set of over 4700 challenging problems for students in grades one through twelve, starting from the very simple to the extremely difficult. No fees, no ads, no calculators, and no sign in.
Math Circle Presentations Math circle presentations for grades 6-12 from University of Waterloo and their related student exercises, available as PDF files. These can be used as enrichment, as challenging word problems or as review of certain topics.
The problems are non-routine problem solving questions that are adapted to many math competitions. Price is about 20 for a semester. M/gifted/index. Jsp Archive for the Handley Math Page Problems of the Week From 19lots of good problems with solutions.
An easy way to start understanding variables is to replace them with question marks in algebra problems. When converting a word problem into algebra.
Open-Ended Math Problems, collection of problems that lend themselves to more than one way of.edu/school/math2/. Thinking Blocks, learn to model or make visual representations of word problems with this interactive m.
Word, problems for Kids, a great selection of word problems for grades 5-12. A hint and a complete solution available for each, problem Solving Decks from North Carolina Public Schools. Includes a deck of problem cards for grades 1-8, student sheets, and solutions. | 677.169 | 1 |
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Calculus with Vectors grew out of a strong need for a beginning calculus textbook for undergraduates who intend to pursue careers in STEM fields. The approach introduces vector-valued functions from the start, emphasizing the connections between one-variable and multi-variable calculus.
Calculus with Applications, Tenth Edition by Lial, Greenwell, and Ritchey, is our most applied text to date, making the math relevant and accessible for students of business, life science, and social sciences. | 677.169 | 1 |
This volume teaches calculus in the biology context without compromising the level of regular calculus. The material is organized in the standard way and explains how the different concepts are logically related. Each new concept is typically introduced with a biological example; the concept is then developed without the biological context and then the concept is tied into additional biological examples. This allows readers to first see why a certain concept is important, then lets them focus on how to use the concepts without getting distracted by applications, and then, once readers feel more comfortable with the concepts, it revisits the biological applications to make sure that they can apply the concepts. The book features exceptionally detailed, step-by-step, worked-out examples and a variety of problems, including an unusually large number of word problems. The volume begins with a preview and review and moves into discrete time models, sequences, and difference equations, limits and continuity, differentiation, applications of differentiation, integration techniques and computational methods, differential equations, linear algebra and analytic geometry, multivariable calculus, systems of differential equations and probability and statistics. For faculty and postdocs in biology departments.
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Written so that biology students can immediately grasp how the material is relevant to their training, this textbook differs from traditional calculus texts in several ways: concepts are motivated with biological examples, differential equations are introduced early, and multivariable calculus is taught the first year<-->recognizing that most life science students don't go on to take the second year of calculus. Annotation c. Book News, Inc., Portland, OR (booknews.com) | 677.169 | 1 |
Related coursesThe purpose of this course is to review the material covered in the Fundamentals of Engineering (FE) exam to enable the student to pass it. It will be presented in modules corresponding to the FE topics, particularly those in Civil and Mechanical Engineering.
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MTH206: GeometryStudents move at their own pace and then are assessed by computer-scored unit tests for a grade. Teacher-graded assignments are available as optional or for review only. Students review core geometric concepts as they develop sound ideas of inductive and deductive reasoning, logic, concepts, and techniques and applications of Euclidean plane and solid geometry. Students use visualizations, spatial reasoning, and geometric modeling to solve problems. Topics include points, lines, and angles; triangles, polygons, and circles; coordinate geometry; three-dimensional solids; geometric constructions; symmetry; and the use of transformations. | 677.169 | 1 |
The Year 10 Mathematics course follows Levels 4 to 6 of the New Zealand Mathematics Curriculum. Students are banded into separate classes based on their mathematical ability. The three bands: 10MATA, 10MATB and 10MATC, work at different levels of difficulty and pace, but all courses are considered the second step in the three year programme leading to success in NCEA Level 1. | 677.169 | 1 |
Multiple Choice and Free-Response Questions in Preparation
Clearly there must be some starting point for explaining concepts in terms of simpler concepts. The main topic of this book is the study of the representations of hypergroups via generalized permutations and hypermatrices. For discrete methods, in addition to Math 340 and 341, 450, 570 (previously 473), and 580 (previously 440), students should consider Math 524-525 (previously 470) and 581 (previously 441). The documents feature standards organized in units with key concepts and skills identified, and a suggested pacing guide for the unit.
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Nelson Probability and Statistics 1 for Cambridge International A Level
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There are also resources to help implement CCSS. This book is an excellent resource for teachers and administrators who work with gifted and advanced learners. Advanced Studies in Pure Mathematics contains survey articles as well as original papers of lasting interest Catholic High School Entrance Exams (TACHS /HSPT) 2016-2017 Test Prep (Argo Brothers®) download here. The alternative mechanism to horizontal projection is the even more elaborate three-dimensional path integration, which the researchers plan to investigate. One way suggested would be by training ants to find food at the end of a ramp, then testing them on terrain with a totally vertical channel, where the horizontal projection is zero, at the end of a completely horizontal channel online. In the preface Simmons states that the goal is to illuminate these words' meaning and their relation to each other, which is exactly what he does in the remaining pages. Self-study students will find Simmons to be a phenomenal communicator and will have no problem at all going through chapter after chapter of his writings , e.g. Holt Precalculus: Student Edition 2006 Using calculus, we understand and explain how water flows, the sun shines, the wind blows, and the planets cycle through the heavens. Differential calculus deals with the rate of change of one quantity with respect to another, for example the rate at which an object's speed changes with respect to time IB Matematicas Nivel Medio download epub IB Matematicas Nivel Medio Libro del. Also useful as a comprehensive math reference. It does no good to have a text without practice of the concepts. Students read the content found in the main text "Understanding Mathematics," then complete the corresponding section in the solutions (companion) guide. The first half of the guide offers hints, examples, explanations, exercises, and problems for practice , source: 365 Addition Worksheets with read online read online.
Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions 200 Addition Worksheets with 4-Digit, 3-Digit Addends: Math Practice Workbook (200 Days Math Addition Series) (Volume 28) 200 Addition Worksheets with 4-Digit,. Eigenvalue and singular value computations. Calculation of roots of polynomials and nonlinear equations. Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Prerequisites: Math 20D or 21D and Math 170B, or consent of instructor Complete Mathematics for Cambridge IGCSERG Student Book (Extended) travel.50thingstoknow.com. Together with a translation of Euclid�s Elements, they became the two foundations of subsequent mathematical developments in Al-Andalus download. You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick online. Finally, learning mathematics is believed to be one of the many ways to improve human brain reflection. Mathematics has been widely taught for thousands of years in all nations. As can be seen, children in countries where mathematics is highly considered tend to perform better in problem solving than those in countries with less attention in mathematics , source: International Mathematics for Cambridge IGCSERG
If you REALLY want to learn Advanced mathematics, get this book ref.: 30 Addition Worksheets with download online 30 Addition Worksheets with 3-Digit,! Students should receive extensive practice in doing derivations. For example, students should know techniques for expressing sin(a+b) as a function of sin(a), sin(b), cos(a), and cos(b). Engineers usually consult tables of identities for such relations, but learning how to do such derivations is an important intellectual skill. This skill is required in courses on electromagnetic field theory, signal processing, semiconductor physics, etc ref.: Core Maths Advanced Level 3rd Edition Tagged with: #solutions to advanced engineering mathematics.#advanced engineering mathematics stroud pdf.#advanced engineering mathematics 5th edition.#advanced engineering mathematics 4th edition.#advanced engineering mathematics 10th edition.#advanced engineering mathematics 2nd edition.#advanced math.#advanced engineering mathematics solutions 10.#advanced engineering mathematics figure 13.3.1 Developing Skills in Estimation (Blackline Masters, Book B, Grades 8-9) The honours year in science is a widely recognised and highly regarded additional year of undergraduate study available to you after you complete your undergraduate course. It's a unique opportunity for you to explore your research potential and put the theory from your undergraduate studies into practice , source: Holt Precalculus: A Graphing Approach: Student Edition 2004 theleadershiplink.org. Provides a deeper and more extensive treatment of the topics in MA 0520 and can be substituted for MA 0520 in fulfilling requirements. The standard requirements for all 100-level mathematics courses except Mathematics 1010 and 1260 are MA 0180, MA 0200, or MA 0350; and MA 0520 or MA 0540 epub. These types are all the possible pairs of equations of the following seven forms: x + y = a, x - y = b, xy = c, x2 + y2 = d, x2 - y2 = e, x3 + y3 = f, and x3 - y3 = g. For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. Now we have presented the latter part of the history of Indian mathematics in an unlikely way 100 Subtraction Worksheets read pdf 100 Subtraction Worksheets with 4-Digit.
Notetaking in college classes: Student patterns and instructional strategies. The Journal of General Education, 51(3), 173-199 download. May be taken for credit two times when topics change. Prerequisites: Math 20D and either Math 20F or Math 31AH, and Math 109, or consent of instructor ref.: Creative Problem Solving, read for free Here is also an actuarial job recruiting site and a European actuarial job search site and an actuarial info and jobs site Complete Solutions Manual to Accompany Preparing for the AP Calculus (AB) Examination download online. Topics covered include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization; series, including geometric and Taylor series; ordinary differential equations; matrix algebra; eigenvalues; and (possibly) dynamic optimization and multivariable integration. Prerequisites: MATH-UA 212 Mathematics for Economics II epub. Semester II provides a rigorous mathematical foundation to probability theory and covers conditional probabilities and expectations, limit theorems for sums of random variables. martingales, ergodic theory, Brownian motion and an introduction to stochastic process theory , source: IB Mathematics Higher Level read epub read epub. The course will be rigorous, but the emphasis will be on application 200 Subtraction Worksheets download here download here. I think this shows that with enought motivation you can do anything. Perhaps you'll have a decreased memory capacity compared to your fellow students, but only perhaps. I think, you never know, if you don't try. Are you looking to learn the different rules of mathematics? For those who find it hard to understand, mathematics can be quite challenging ref.: 200 Subtraction Worksheets with 4-Digit Minuends, 2-Digit Subtrahends: Math Practice Workbook (200 Days Math Subtraction Series) (Volume 8) download for free. A team of educators examines all students and their educational performance in order to evaluate the best instructional fit for students based upon their readiness, potential, and achievement. This process helps ensure that access to Advanced Academics is provided to all students based upon a team assessment of multiple data points , e.g. Nelson Probability and Statistics 1 for Cambridge International A Level The content demands of mathematics itself have limited the direct influence of some pedagogical fashions on high school math teachers , e.g. 100 Addition Worksheets with download online The Internet was a powerful organizing tool for parents of school children during the 1990s Word Problems Interactive download epub download epub. That fits pretty well. (Note that (2,10,0) was just an initial guess for the amplitude, period, and phase parameters. As long as your initial guess isn't too crazy, its precise value isn't important and Meta. Numerics will find the best-fit point.) So, what's the best-fit oscillation period? Let's work with one more kind of experimental data. Suppose we randomly assign patients to receive or not receive an experimental treatment and obtain the following results Complete Mathematics for read for free A Concise Course in Advanced Level Statistics with worked examples travel.50thingstoknow.com. The unfortunate thing is that, at least at my school, you don't get to do much mathematics until your junior year, so you may be in for a long slog of general education classes you are not particularly interested in. (I actually think this is a big problem... but that's a whole different issue I guess.) Also, you say "I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction." | 677.169 | 1 |
Book Review: Pressley's Elementary Differential Geometry, 2nd Ed
It is time to return to the book reviews! Our next book is Elementary Differential Geometry, 2nd Ed by Andrew Pressley. This is a pretty recent text. The first edition is from 2002, with the update published in 2010. The book has an attractive price point from Springer, and you can get it from Amazon.com for even cheaper.
Pressley's desired approach is to make the subject as accessible as possible. In the preface, he writes:
Thus, for virtually all of the book, the only prerequisites are a good working knowledge of Calculus (including partial differentiation), Vectors and Linear Algebra (including matrices and determinants).
The tone of the writing bears this out, as does the author's care to explain basic material. This text is definitely aimed at the modern student, and it conforms to the standard expectations for what a recent textbook on an advanced subject should look like.
The Chapter Headings
Pressley has organized the material as follows:
Curves in the plane and in space
How much does a curve curve?
Global properties of curves
Surfaces in three dimensions
Examples of surfaces
The first fundamental form
Curvature of Surfaces
Gaussian, mean and principal curvatures
Geodesics
Gauss' Theorema Egregium
Hyperbolic Geometry
Minimal surfaces
The Gauss-Bonnet theorem
(As you can see, Pressley doesn't use a serial comma. We are already at odds.) There are also three appendices, enumerated computer-science-style:
A0. Inner product spaces and self-adjoint linear maps
A1. Isometries of Euclidean Spaces
A2. Möbius Transformations
Clearly, these are chosen to support some of the prerequisite material for describing curvature, understanding congruence, and dealing with hyperbolic geometry.
Keeping to the plan, I read chapter 4 Surfaces in three dimensions.
Notable Features
As you can see from the chapter headings, Pressley wants to lay out the basics very carefully. There are whole chapters devoted to the basic process of defining curves (chapter 1) and defining smooth surfaces (chapter 4). The writing is direct but relatively friendly. You don't get the "marble temple of mysteries" feel that some advanced textbooks can fall into.
There are over 200 exercises. (I am quoting the preface for this. I didn't count.) There is a selection of hints for about 75 of them near the end of the text. (I counted that myself.) More importantly, there is also a 60 page section at the end of the book with fairly complete solutions to all of the exercises. Some seem a bit terse, but they do all seem to be addressed. This makes the text a really good choice for individual study. A person with the self-discipline to make an honest effort at each exercise would be glad of this feature.
Latitude and Longitude Coordinates on a Sphere, from Pressley
Some of the exercises I saw were pretty hard. That is, an undergraduate with the background listed in the preface would likely struggle mightily with some of them, though for the initiated they would be not difficult.
Exercise 4.1.4: Show that a unit cylinder can be covered by a single surface patch, but that the unit sphere cannot. (The second part requires some point set topology.)
That parenthetical remark is in the text.
Exercise 4.1.5: Show that every open subset of a surface is a surface.
A person with a fair amount of experience can dispatch those, but a student making the transition from calculus and linear algebra is likely to spin their wheels a long time. Those exercises just go straight to the "fiddly bits."
We have the now standard, definition-theorem-proof style of exposition, and plenty of computer generated diagrams. Everything is labeled in one consecutive sequence in section.subsection.item style.
The examples are the ones you would expect, or, rather, they are the ones I expect. I recognize that I am not typical by a long way, having been trained in the subject formally, and now undertaking my seventh book review of this material. It is nice that the examples reoccur when new concepts arise.
Pressley has included more material than can be reasonably discussed in a one semester course. In particular, it looks as though you can, and should, pick a path through the text to one of the final three chapters. This gives the book a little flexibility, and it leaves more material to whet the appetites of ambitious students.
A Complaint, only partly about the book
I have a big complaint to make, but I don't think it is altogether fair to Prof Pressley. If anything, he is merely exemplifying a trend in mathematical exposition that has important uses, but which I have come to question about the construction of learning materials.
It is standard practice in mathematics to present things axiomatically. This has been going for a long time: I teach out of Euclid's Elements, and it happens there. You get formal definitions and axioms first, theorems later, examples and discussion sometimes.
We don't have to have the conversation about why this is so. We can all agree that it is generally a good thing to have the axiomatic structure in mathematical work.
I can't really fault Pressley for doing things this way. This is
How Mathematics Is Supposed To Be Written.
(Did you hear the echo? I heard an echo.) If you want to prove theorems you actually suspect are true, this is how it goes.
But I don't necessarily think it is the right thing for curricular materials. To really appreciate the nuances of mathematical definitions, one needs lots of examples. Even more, to even really feel a need for all of these crazy words (and find a will for keeping them straight), you have to have lots of funny friends that you can classify and organize with those words.
The upshot is that I believe that chapters 4 and 5 are in the wrong order.
Here is how the standard axiomatic approach leads to trouble for the potential reader in this text. Chapter 4 is about conveying the idea of a surface, and getting a definition down. But the first subsection 4.1 What is a surface?, opens with a page of definition-making in which we see the following terms:
open subset ( definition),
open ball
open interval
open disc (Pressley is from the UK, so we get British English spellings)
functions continuous at a point ( definition)
functions continuous in the large
homeomorphism
homeomorphic spaces
Then, and only then do we meet the first official definition, Definition 4.1.1, which gives the formal atlas of patches definition of a surface, essentially the notion of a topological 2-manifold. To Pressley's credit, he doesn't actually use the word manifold.
All of that is over in about a page of text, and then basic examples start. It seems to me that the preceding would be very hard to comprehend for anyone who hasn't already mastered some point-set topology, or at least been exposed the ideas of metric spaces. The young, aspiring geometer just won't have a sense of what those words are for, and why we have chosen to use all of them.
Before the chapter is out, we are introduced to regular surfaces, diffeomorphisms, the derivative of a function from one surface to another, orientability, the notion of a maximal atlas, the tangent plane as the set of all tangent vectors to curves through the given point which lie in the surface, and other things in this neighborhood of mathematics.
In a weird way, I think that all of this axiomatically clean development puts the cart before the horse for newcomers.
So, though Pressley has made an admirable attempt to be clear and helpful, I find I am dissatisfied. Basically this comes across as an extraordinarily friendly introduction, written so as to be useful more as a reference than as a learning tool. Again, not exactly the author's fault…
Verdict
A solid, mathematician's introduction to the subject. Perhaps misses its mark of being truly accessible to the pre-analysis crowd. Any real flaws here are flaws in almost any advanced mathematics textbook. Instructors should make themselves aware of the book, as it could be a reasonable choice for an instructor who puts in the effort to help students through the transition.
This is a decent selection for someone with more advanced training who wishes to learn the subject by studying independently.
I get the sense that you're bumping up against one of the fundamental problems in mathematics education: the way our learning of mathematics is best structured does not always match the way the mathematics itself is structured. Unfortunately, figuring out the learning part can be quite difficult, and perhaps the only place where we can claim to broadly understand it well is for basic arithmetic.
I know some mathematicians lament the "demise" of a strict axiomatic approach in high school geometry. I think that understanding an axiomatic system is important, and that's why I would typically spend several days teaching directly from Book I of Euclid's Elements. But my goal for doing so wasn't for students to really understand the mathematics of the propositions being proven — I only expected them to gain an appreciation for how an axiomatic system is developed. If my goal was for students to know and understand how to use the Pythagorean Theorem, there were certainly better options than simply marching my way to Proposition 47. I think you're looking for those kinds of options for topics in differential geometry, which I'm sure will take some creativity and risk-taking to develop and test. Good luck! | 677.169 | 1 |
GENERAL INFORMATION SECTION - GENERAL INFORMATION PROBLEM...
Rev: July 2009 Page 1 GENERAL INFORMATION PROBLEM SOLVING INSTRUCTIONS AND FORMAT The ability to present one's work clearly and systematically is the mark of a professional. Neatness, clarity and conciseness of presentation are professional habits expected on all work. The attached examples provide a guide for the format required both in doing assigned homework and working problems on quizzes and tests. The format is as follows: 1. Use engineer's problem paper for all homework with the following exception. All graphs and drawings are to be done on proper graph or vellum paper unless the instructor indicates that the engineer's problem paper is satisfactory. 2. Letter or print neatly with a sharp 2-H lead or mechanical pencil. 3. Headings are to be printed on eachpage as shown in the examples. This includes the course, problem set number, name, date, and the page number. All pages are to be stapled together and handed in unfolded. 4. The layoutof the problem: GIVEN, REQUIRED, and SOLUTION should follow the examples given for each problem submitted. The solution portion should be worked in clear sequential steps with the answer shown at the right side of the page and double underlined. Allow enough space between steps to easily follow work. 5. Each problem should show the: a. appropriate equation(s), then numerical substitutions with units, b. source and page number of reference in obtaining information not given in the problem, c. values used for obtaining an answer from a graph or table, d. proper number of significant digits and units in the answer, e. symbol of the variable/constant being solved also double underlined with the answer. 6. Significant figures should follow these basic rules: Addition & Subtraction- digits can be retained only as far right as the least accurate number, Multiplication & Division- the answer can have no more significant figures than the number having the least significant figure, which is used in the calculation for the answer. 7. Graphs should include a complete title, labeled ordinates and abscissas including units. Plotted points representing measured data should be shown by a symbol whose size circumscribes the range of error of the measurements. Calculated values should be plotted as points. Curves should be drawn with a straight-edge or french curve depending on the shape of the plotted values with the exception of contour lines which should be drawn freehand.
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Page 2 Rev: July 2009 GENERAL INFORMATION FIELD BOOK INSTRUCTIONS A field book will be used to record data and draw sketches of the work details for most of the laboratory exercises in this course. Identification and clarity of details in recording information into the field book is of utmost importance along with the use of a consistent format. Information recorded in the field book is frequently used by other than the note keeper of the survey crew. Obscure data, sketches, and illegible information causes wasted time and resources in the office. In addition, later trips to the survey site may be required which are costly and could have been avoided.
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This note was uploaded on 01/13/2011 for the course BRAE 237 taught by Professor Mastin during the Winter '10 term at Cal Poly. | 677.169 | 1 |
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Algebra 1 SOL Practice Sets
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1.64 MB | 14 pages
PRODUCT DESCRIPTION
This resource includes 5 algebra 1 SOL Practice sets based on the 2009 Virginia Standards of Learning. The first three practice sets are 10 questions each and the last two are 20 questions each for a total of 70 SOL practice questions!! An answer key is included. The 10 question sets can each be printed on one page front to back. The two 20 question sets can be printed on two pages front to back. The sets include both multiple choice and TEI questions. The questions cover all of the algebra 1 standards and were designed with the released SOLs in mind.
In my classroom I assign these, one per week, in the weeks leading up to the SOL. Other teachers use them in class as warm-ups, SOL review, or mini quizzes. They can also be used after the SOL when you are remediating students for retakes | 677.169 | 1 |
Many classical and modern results and quadratic forms are brought together in this book. The treatment is self-contained and of a totally elementary nature requiring only a basic knowledge of rings, fields, polynomials, and matrices, such that the works of Pfister, Hilbert, Hurwitz and others are easily accessible to non-experts and undergraduates alike. The author deals with many different approaches to the study of squares; from the classical works of the late 19th century, to areas of current research. Anyone with an interest in algebra or number theory will find this a most fascinating volume. | 677.169 | 1 |
Items tagged with tutor
Students using Maple often have different needs than non-students. Students need more than just a final answer; they are looking to gain an understanding of the mathematical concepts behind the problems they are asked to solve and to learn how to solve problems. They need an environment that allows them to explore the concepts and break problems down into smaller steps.
The Student packages in Maple offer focused learning environments in which students can explore and reinforce fundamental concepts for courses in Precalculus, Calculus, Linear Algebra, Statistics, and more. For example, Maple includes step-by-step tutors that allow students to practice integration, differentiation, the finding of limits, and more. The Integration Tutor, shown below, lets a student evaluate an integral by selecting an applicable rule at each step. Maple will also offer hints or show the next step, if asked. The tutor doesn't only demonstrate how to obtain the result, but is designed for practicing and learning.
For this blog post, I'd like to focus in on an area of great interest to students: showing step-by-step solutions for a variety of problems in Maple.
Several commands in the Student packages can show solution steps as either output or inline in an interactive pop-up window. The first few examples return the solution steps as output.
Precalculus problems:
The Student:-Basics sub-package provides a collection of commands to help students and teachers explore fundamental mathematical concepts that are core to many disciplines. It features two commands, both of which return step-by-step solutions as output.
The ExpandSteps command accepts a product of polynomials and displays the steps required to expand the expression:
with(Student:-Basics):
ExpandSteps( (a^2-1)/(a/3+1/3) );
The LinearSolveSteps command accepts an equation in one variable and displays the steps required to solve for that variable.
with(Student:-Basics):
LinearSolveSteps( (x+1)/y = 4*y^2 + 3*x, x );
This command also accepts some nonlinear equations that can be reduced down to linear equations.
Calculus problems:
The Student:-Calculus1 sub-package is designed to cover the basic material of a standard first course in single-variable calculus. Several commands in this package provide interactive tutors where you can step through computations and step-by-step solutions can be returned as standard worksheet output.
Tools like the integration, differentiation, and limit method tutors are interactive interfaces that allow for exploration. For example, similar to the integration-methods tutor above, the differentiation-methods tutor lets a student obtain a derivative by selecting the appropriate rule that applies at each step or by requesting a complete solution all at once. When done, pressing "Close" prints out to the Maple worksheet an annotated solution containing all of the steps.
The Student:-Calculus1 sub-package is not alone in offering this kind of step-by-step solution finding. Other commands in other Student packages are also capable of returning solutions.
Linear Algebra Problems:
The Student:-LinearAlgebra sub-package is designed to cover the basic material of a standard first course in linear algebra. This sub-package features similar tutors to those found in the Calculus1 sub-package. Commands such as the Gaussian Elimination, Gauss-Jordan Elimination, Matrix Inverse, Eigenvalues or Eigenvectors tutors show step-by-step solutions for linear algebra problems in interactive pop-up tutor windows. Of these tutors, a personal favourite has to be the Gauss-Jordan Elimination tutor, which were I still a student, would have saved me a lot of time and effort searching for simple arithmetic errors while row-reducing matrices.
This tutor makes it possible to step through row-reducing a matrix by using the controls on the right side of the pop-up window. If you are unsure where to go next, the "Next Step" button can be used to move forward one-step. Pressing "All Steps" returns all of the steps required to row reduce this matrix.
When this tutor is closed, it does not return results to the Maple worksheet, however it is still possible to use the Maple interface to step through performing elementary row operations and to capture the output in the Maple worksheet. By loading the Student:-LinearAlgebra package, you can simply use the right-click context menu to apply elementary row operations to a Matrix in order to step through the operations, capturing all of your steps along the way!
An interactive application for showing steps for some problems:
While working on this blog post, it struck me that we did not have any online interactive applications that could show solution steps, so using the commands that I've discussed above, I authored an application that can expand, solve linear problems, integrate, differentiate, or find limits. You can interact with this application here, but note that this application is a work in progress, so feel free to email me (maplepm (at) Maplesoft.com) any strange bugs that you may encounter with it.
More detail on each of these commands can be found in Maple's help pages.
For people new to Maple, an easy way to learn how to code may be Tutor Syntax, that is the technique of generating code by selecting everything in a Tutor dialog Maple Command window and then copying it to a current session.
The code can then be edited to tailor it to user's needs. A problem with this technique is the copied syntax contains many single quotes (' '). The single quotes, which are usually not needed if the program in current session is short and specific, increase the difficulty when editing copied syntax. The attached file (mby2.mw) shows a method which removes the single quotes by using the SubstitueAll command in the StringTools package. One drawback: any strings inside the original syntax (e.g., "#78000E") must be removed before the syntax can be converted to a string.
Is there a better way to generate code from Tutor dialog so it can be edited (without single quotes) in user's current session?
With the RationalFunctionTutor generated then modified code (see egn2.mw attached). Is it possible to make linestyle=spacedash for the two vertical asymtotes? By selecting the graph and choosing from the plot menu the linestyle can be changed, but is there a way to do this with commands in the program code?
I once opened Tools-Tutors-Precalculus-Polynomials. Now every time I click the button of 'execute the entire worksheet', the same Tutor window will pop up before Maple executes my codes. I wanna know how to prevent the window to pop up?
Hi, I am trying to figure out how to find the maxima and minima of the function f(x)=(x^3-10*x^2-2*x+1)/(4*x^3+5*x+1) and its derivative.
a) Find the local maxima and minima of f.
b) Find the vertical asymptote
I graphed the function and located the local max to be (-1,1) and local min (1,-1) however, I can't figure out how to plot the derivative of the function. I tried to use the diff expression but it leaves the graph empty when i attempt to plot it.
Also, just curious how am i able to join text words and maple commands together? For example, I want to be able to say "The derivative of the original function is [maple command here]" ? I tried to switch from text to math however nothing worked.
Thanks in advance! | 677.169 | 1 |
Omtale
Guided Lecture Notes for Precalculus
This note-taking guide assists in taking thorough, organized, and understandable notes as they watch the Author in Action videos. The Notes ask students to complete definitions, procedures, and examples based on the content of the videos and book. By directing students into essential material, students can focus and retain the most important concepts. These notes help increase the success of your students in any course structure. | 677.169 | 1 |
Core Physics is the perfect tool for students, professionals in the fields of mechanical and electrical engineering and physics enthusiasts alike. The simplicity and functionality of the user interface is the basis of the design framework for Core Physics. The application uses the data provided to solve mathematical physics problems. Just enter your data into the fields provided, leaving only the field which you want to calculate blank.
Let's say that you want to find the volume of a cylindrical drum or that you want to calculate the distance between two points on a map.
Core Geometry is a mathematical application which aims to solve geometrical problems quickly and reliably. Simply input the know values into the fields correlating to the 2d and 3d diagrams provided for each problem. | 677.169 | 1 |
MAC 1114
Test 2 Preparation Package
Due: _
Instructions:
Use this test preparation package to prepare for the nal exam in this class (and youre encouraged to adapt
it to your other courses too!).
Make sure you complete every item and set up a consultation
To use Sine Regression on the TI-84
Data must first be typed into Lists. See the data for months and temperatures Table
12 p. 587.
Press:
STAT
1:EDIT
Clear lists by arrowing to the top, press clear, press enter
Then type in data. Use L1 for the x values (
Trigonometry Advice
Showing 1 to 1 of 1
Dr. Zaragoza explains math in a way that i've never been taught before. He forces you to partner with people at your table and teach each other. He wants your "mind body and spirit" to understand trigonometry. I laughed at first, but he does a really great job of doing just that. | 677.169 | 1 |
MATHEMATICS
Mathematics plays an important role in life and society at many levels. At Mazenod there is a strong tradition of providing courses that cater for the diverse needs of students.
In Years 7 to 10, there is a solid core of material based around the CSF II documents. This is supplemented by an enrichment program for Years 8 to 10 students that deepens and broadens the skills and understanding of the most able students. At the same time there is, in co-operation with the Learning Centre, a program remediation that is being expanded and refined to help the weaker students gain skill and confidence.
At Years 11 and 12, the entire range of VCE mathematics courses is offered, each one of which is designed for a different career pathway. For students intending to pursue a trade or small business there is Foundation Mathematics, a thematic course specifically designed to develop skills in real world situations. For students intending careers in Commerce or Psychology, the Further Mathematics pathway provides them with a variety of appropriate skills in statistics, graphs, finance, and trigonometry. For students heading toward the sciences or engineering, the Mathematical Methods-Specialist courses provide a thorough preparation.
In addition to the standard curriculum, students at Year 12 can enrol in a Mathematics Enhancement program run by Monash University. Upon completion, this course gives the students credit for a first year mathematics unit at any university in the country.
Many of the skills learned in mathematics involve the use of materials and equipment. This includes cards for probability, inclinometers for surveying and geometry, calculators and computer software. At Mazenod the Mathematics department has a strong emphasis on the use of as many different materials as possible across the curriculum. This includes the use of the new Computer Algebra System calculators from Year 9 onwards.
As well as the course-work, there are a number of extra-curricula activities available to students at the College. The two most notable are the Annual Westpac Mathematics Competition and the Mathematics Talent Quest. In both of these arenas the College has a successful history. | 677.169 | 1 |
8pyiw8mktg85m8925no
FHSST Authors
The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Mathematics Grades 10 - 12
Version 0 September 17, 2008
ii
iii Copyright 2007 "Free High School Science Texts" Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no FrontCover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
STOP!!!!
Did you notice the FREEDOMS we've granted you?
Our copyright license is different! It grants freedoms rather than just imposing restrictions like all those other textbooks you probably own or use.
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In Grade 10 we studied exponential numbers and learnt that there were six laws that made working with exponential numbers easier. There is one law that we did not study in Grade 10. This will be described here.
17.2
Laws of Exponents
In Grade 10, we worked only with indices that were integers. What happens when the index is not an integer, but is a rational number? This leads us to the final law of exponents, an =
m
√ n am
(17.1)
17.2.1
Exponential Law 7: a n =
m
√ n
am
We say that x is an nth root of b if xn = b. For example, (−1)4 = 1, so −1 is a 4th root of 1. Using law 6, we notice that m m (a n )n = a n ×n = am (17.2) therefore a n must be an nth root of am . We can therefore say √ m a n = n am √ where n am is the nth root of am (if it exists). For example, 23 =
2 m
(17.3)
√ 3 22
A number may not always have a real nth root. For example, if n = 2 and a = −1, then there is no real number such that x2 = −1 because x2 can never be a negative number. Extension: Complex Numbers There are numbers which can solve problems like x2 = −1, but they are beyond the scope of this book. They are called complex numbers.
It is also possible for more than one nth root of a number to exist. For example, (−2)2 = 4 and 22 = 4, so both -2 and 2 are 2nd (square) roots of 4. Usually if there is more than one root, we choose the positive real solution and move on. 251
how many will there be in half of a year? What will be the population be in 10 years and in 100 years ? Answer Step 1 : P opulation = Initial population×(1+growth percentage)time period in months 253
. Simplify: (a) (x0 ) + 5x0 − (0. in 1 day and in 1 week? Answer Step 1 : P opulation = Initial population×(1+growth percentage)time period in hours Therefore.CHAPTER 17. you used exponentials to calculate different types of interest.GRADE 11
17.5 + 8 3
7 12m 9 11 8m− 9 2
(c)
(b) s 2 ÷ s 3
1
1
(d) (64m6 ) 3
2
2. Re-write the expression as a power of x: x x √ x x x
17.3
Exponentials in the Real-World
In Chapter 8.8)168 = 7. determine how many there will be in 5 hours. in this case: P opulation = 10(1. EXPONENTS .8)24 = 13 382 588 Step 4 : in 1 week = 168 hours P opulation = 10(1. If there are 10 bacteria. for example on a savings account or on a loan and compound growth.3
Exercise: Applying laws Use all the laws to: 1. where n = number of hours Step 2 : In 5 hours P opulation = 10(1.
Worked Example 83: Exponentials in the Real world Question: A type of bacteria has a very high exponential growth rate at 80% every hour. If there are a total 821 of this type of fish and their growth rate is 2% each month.
Worked Example 84: More Exponentials in the Real world Question: A species of extremely rare.25)−0.687 × 1043 Note this answer is given in scientific notation as it is a very big number.8)n .8)5 = 188 Step 3 : In 1 day = 24 hours P opulation = 10(1. deep water fish has an extremely long lifespan and rarely have children.
8) √ √ If we have a fraction which has a √ denominator which looks like a + b. (That's why denominators get rationalised. SURDS .) √ √ 5x x − 16 x x √ ( x)(5x − 16) x
Worked Example 89: Rationalising the Denominator Question: Rationalise the following:
5x−16 √ y−10
258
.1
CHAPTER 18. √ √ c a− b c √ √ ×√ √ = √ (18.1. Note that √x = 1. then we can simply √ multiply both top and bottom by a − b achieving a rational denominator. √ x 5x − 16 √ ×√ x x
Step 2 : There is no longer a surd in the denominator.18. This √ "rationalises" the surd in the denominator. We will now see how this can be achieved. This is because √ √ √ √ ( a + b)( a − b) = a − b (18. which has a surd in the denominator as a fraction which has a rational denominator.9) √ a+ b a− b a+ b √ √ c a−c b = a−b or similarly c √ √ a− b = = √ √ c a+ b √ ×√ √ √ a+ b a− b √ √ c a+c b a−b (18. which have rational denominators instead of surd denominators.
Worked Example 88: Rationalising the Denominator
√ Question: Rationalise the denominator of: 5x−16 x Answer Step 1 : Get√ of the square root sign in the denominator rid √ To get rid of x in the denominator.6
Rationalising Denominators
It is useful to work with fractions. you can multiply it out by another x. √ √ Any expression of the form √ a + √b (where a and b are rational) can be changed into a rational √ √ number by multiplying by a − b (similarly a − b can be rationalised by multiplying by √ √ a + b).GRADE 11
18. The surd is expressed in the numerator which is the prefered way to write expressions.10) which is rational (since a and b are rational). thus the equation x becomes rationalised by multiplying by 1 and thus still says the same thing. It is possible to rewrite any fraction.
73 + 3.
Important: Simplification and Accuracy
It is best to simplify all expressions as much as possible before rounding-off answers.1: Two methods of writing 3 3 + 12 as Method 1 √ √ √ √ √ √ 3 3 + 12 = 3√3 + √ · 3 4 3 3 + 12 = 3√3 + 2 3 = 5 3 = 5 × 1. in a calculation that has many steps.660254038 . . Method 2 = = = 3 × 1.66 as an answer while Method 2 gives 8. For example. = 8.1. . there are two ways of doing this as described in Table 19. The answer of Method 1 is more accurate because the expression was simplified as much as possible before the answer was rounded-off. Write the answer to three decimal places.65 as an answer. = 8.Grade 11
We have seen that numbers are either rational or irrational and we have see how to round-off numbers.Chapter 19
Error Margins .
Worked Example 91: Simplification and Accuracy √ √ Question: Calculate 3 54 + 3 16. This maintains the accuracy of your answer.66 a decimal number.732050808 .65
In the example we see that Method 1 gives 8. In general. it is best to leave the rounding off right until the end. before using your calculator to work out the answer in decimal notation. √ √ Table 19. .19 + 3. However. it is best to simplify any expression as much as possible. if you were asked to write √ √ 3 3 + 12 as a decimal number correct to two decimal places. .46 5. Answer Step 1 : Simplify the expression 261
.46 8.
GRADE 11 Extension: Significant Figures In a number. For example. so it would be 6. rounding as you go. It is important to know when to estimate a number and when not to. and to instead use symbols to represent certain irrational numbers (such as π).CHAPTER 19.83. each non-zero digit is a significant figure. approximating them only at the very end of a calculation. ERROR MARGINS .
263
. but 2000. but if you wish to write it to 3 significant figures it would mean removing the 7 and rounding up. For example 6. then it is often good enough to approximate to a few decimal places. If it is necessary to approximate a number in the middle of a calculation. Estimating a number works by removing significant figures from your number (starting from the right) until you have the desired number of significant figures. the number 2000 has 1 significant figure (the 2). Zeroes are only counted if they are between two non-zero digits or are at the end of the decimal part.0 has 5 significant figures. It is usually good practise to only estimate numbers when it is absolutely necessary.827 has 4 significant figures.
GRADE 11
264
. ERROR MARGINS .CHAPTER 19.
For example..1 Introduction
In Grade 10. .Grade 11
20. 4. has a linear formula of the kind ax + b. 1. the first differences) of a quadratic sequence form a sequence where there is a constant difference between consecutive terms. . where the difference between consecutive terms was constant. 4. Let us see why . 2. In the above example. 4.}. .1). (20.1) is a quadratic sequence. 2. 3. . . which is formed by taking the differences between consecutive terms of (20. the sequence of {1.1)
a4 − a3 a5 − a4
=7−4 =3 = 11 − 7 = 4
We then work out the second differences. is a quadratic sequence. which is simply obtained by taking the difference between the consecutive differences {1. .2
What is a quadratic sequence?
Definition: Quadratic Sequence A quadratic sequence is a sequence of numbers in which the second differences between each consecutive term differ by the same amount. then: a2 − a1 a3 − a2 =2−1 =4−2 =1 =2 (20.} obtained above: 3−2 = 4−3 = . you learned about arithmetic sequences. called a common second difference. 3. Thus. . . 2−1 = 1 1 1
We then see that the second differences are equal to 1.
265
. . If we take the difference between consecutive terms. 11..Chapter 20
Quadratic Sequences . 2.. Note that the differences between consecutive terms (that is. 7. In this chapter we learn about quadratic sequences.
20..
38. . . So the next two terms in the sequence willl be: 38 + 19 = 57 57 + 23 = 80 So the sequence will be: 5. which leads to n + 2.. 17. 22.. The start of the formula will therefore be 2n2 . . . 15 Step 2 : Find the 2nd differences between the terms. then b then c. 23. 31. 71. .. 10. 6. 11. 38. 27. you have to get the value of term 1. .. 7. the 2nd difference is 2a. If n = 1. 30. which is can be written as n + 2. . . 4. Can you calculate the common second difference for each of the above examples?
Worked Example 93: Quadratic sequence Question: Write down the next two terms and find a formula for the nth term of the sequence 5. the formula for the nt h term is 2n2 + n + 2.. 38. ..
General Case If the sequence is quadratic. 23. 15. 266
. . 9.GRADE 11 Exercise: Quadratic Sequences The following are also examples of quadratic sequences: 3. 25. the second difference is 4. 21.2
CHAPTER 20.. which is 5 in this particular sequence. 36. . 2. We know that the first difference is 4. The difference between term 2( 12) and 8 is 4. the differences between each term will be: 15 + 4 = 19 19 + 4 = 23 Step 3 : Finding the next two terms. 12.e. 12. QUADRATIC SEQUENCES .. . i. 80 Step 4 : We now need to find the formula for this sequence. 7. 12. 31. . 49. . 82. 16. Then 2n2 = 8. when n = 2. the nt h term should be Tn = an2 + bn + c TERMS 1st difference 2nd difference a+b+c 3a + b 2a 4a + 2b + c 5a + b 2a 9a + 3b + c 7a + b
In each case.20. Answer Step 1 : Find the first differences between the terms.. So for the sequence 5. 26. 57. The difference between 2n2 = 2 and original number (5) is 3.. So continuing the sequence. 15. . i.e. 10. 50. 23. Step 5 : We now need to work out the next part of the sequence. . This fact can be used to find a. Check is it works for the second term.
Extension: Depreciation You may be wondering why we need to calculate depreciation.1 Introduction
In Grade 10. and thereby reduce their taxable income. the two methods of calculating depreciation are simple and compound methods. The terminology used for simple depreciation is straight-line depreciation and for compound depreciation is reducing-balance depreciation. the ideas of simple and compound interest was introduced. they will base the price on something called book value.Chapter 21
Finance . and the older the car. One way of calculating a depreciation amount would be to assume that the car has a limited useful life. In this chapter we will be extending those ideas. In the straight-line method the value of the asset is reduced by the same constant amount each year.
21. or depreciating. Just like interest rates. And from there on the value keeps falling. The book value of the car is the value of the car taking into account the loss in value due to wear. because it is now "second-hand".3
Simple Depreciation (it really is simple!)
Let us go back to the second hand cars. This means that the value of an asset does not decrease by a constant amount each year. If you master the techniques in this chapter.
21. We call this loss in value depreciation. Determining the value of assets (as in the example of the second hand cars) is one reason. Simple depreciation assumes that the value of 271
.Grade 11
21. A lower taxable income means that the company will pay less income tax to the Revenue Service. so it is a good idea to go back to Chapter 8 and revise what you learnt in Grade 10. and in this section we will look at two ways of how this is calculated.2
Depreciation
It is said that when you drive a new car out of the dealership. but the decrease is most in the first year. you will understand about depreciation and will learn how to determine which bank is offering the better interest rate. it loses 20% of its value. If you buy a second hand (or should we say pre-owned!) car from a dealership. In the compound depreciation method the value of the asset is reduced by the same percentage each year. but there is also a more financial reason for calculating depreciation . usually the cheaper it is.tax! Companies can take depreciation into account as an expense. age and use. then by a smaller amount in the second year and by even a smaller amount in the third year. and so on. Second hand cars are cheaper than new cars.
For example. FINANCE .21. What we are saying is that after 5 years you will have to buy a new car.GRADE 11
the car decreases by an equal amount each year.15 5 required 272
. Note that the difference between the simple interest calculations and the simple depreciation calculations is that while the interest adds value to the principal amount. let us say the limited useful life of a car is 5 years. If we replace the word interest with the word depreciation and the word principal with the words initial value we can use the same formula: Total depreciation after n years = n × (P × i) Then the book value of the asset after n years is: Initial Value . 5 years The value of the car is then: End End End End End of of of of of Year Year Year Year Year 1 2 3 4 5 R60 R60 R60 R60 R60 000 000 000 000 000 1×(R12 2×(R12 3×(R12 4×(R12 5×(R12 000) 000) 000) 000) 000) = = = = = R48 R36 R24 R12 R0 000 000 000 000
This looks similar to the formula for simple interest: Total Interest after n years = n × (P × i) where i is the annual percentage interest rate and P is the principal amount.Total depreciation after n years = = P − n × (P × i)
P (1 − n × i)
For example. which means that the old one will be valueless at that point in time.4) R60 000(0.a. on a staight-line depreciation.3
CHAPTER 21. and the cost of the car today is R60 000.6) R36 000
Worked Example 96: Simple Depreciation method Question: A car is worth R240 000 now. what is it worth in 5 years' time ? Answer Step 1 : Determine what has been provided and what is required P i n A = = = is R240 000 0. If it depreciates at a rate of 15% p. the depreciation amount reduces value! P (1 − n × i) R60 000(1 − 2 × 20%) R60 000(1 − 0. Therefore. the amount of depreciation is calculated: R60 000 = R12 000 per year. the book value of the car after two years can be simply calculated as follows: Book Value after 2 years = = = = = as expected.
CHAPTER 21. FINANCE . The annual depreciation amount is then calculated as (R60 000 . Step 2 : Value of the photocopier after 1 year 12 000 − 4 000 = R8 000 Step 3 : Value of the machine after 2 years 8 000 − 4 000 = R4 000 Step 4 : Write the final answer 4 000 − 4 000 = 0 After 3 years the photocopier is worth nothing
Extension: Salvage Value Looking at the same example of our car with an initial value of R60 000. R60 000 . and will only apply the depreciation to the value of the asset that we expect not to recoup. What amount will he fill in on his tax form after 1 year. but now instead of depreciating the full value of the asset. what if we suppose that we think we would be able to sell the car at the end of the 5 year period for R10 000? We call this amount the "Salvage Value" We are still assuming simple depreciation over a useful life of 5 years.25 60 000 = 240 000(1 − 0. the value of the photocopier will go down by 12 000 ÷ 3 = R4 000 per year. For the tax return the owner depreciates this asset over 3 years using a straight-line depreciation method.15 × 5)
21. after 2 years and then after 3 years ? Answer Step 1 : Understanding the question The owner of the business wants the photocopier to depreciates to R0 after 3 years.3
Step 4 : Write the final answer In 5 years' time the car is worth R60 000
Worked Example 97: Simple Depreciation Question: A small business buys a photocopier for R 12 000.e.Salvage Value Useful Life
273
. we will take into account the salvage value.GRADE 11 Step 2 : Determine how to approach the problem A Step 3 : Solve the problem A = = = 240 000(1 − 0.R10 000 = R50 000. i. Thus. the for simple (straight line) depreciation: Annual Depreciation = Initial Value .R10 000) / 5 = R10 000 In general.75) 240 000 × 0.
What is the value of the truck after 8 years ? 2.80
We can now write a general formula for the book value of an asset if the depreciation is compounded. It has now been valued at R2 300. Due to weathering. so after two years. Fiona buys a DsTV satellite dish for R3 000.21.00 608. A business buys a truck for R560 000.8) = R48 000 At the beginning of the second year. the car is worth: Book Value after second year = P (1 − n × i) = R48 000(1 − 1 × 20%) = R48 000(1 − 0. if our second hand car has a limited useful life of 5 years and it has an initial value of R60 000. After how long will the satellite dish be worth nothing ?
21.00 400.8) = R38 400 We can tabulate these values. Initial Value . FINANCE .1) 274
. His grandpa is quite pleased with the offer.4
CHAPTER 21. Shrek wants to buy his grandpa's donkey for R800.2) = R60 000(0.Total depreciation after n years = P (1 − i)n (21. Over a period of 10 years the value of the truck depreciates to R0 (using the straight-line method).00 576.4
Compound Depreciation
The second method of calculating depreciation is to assume that the value of the asset decreases at a certain annual rate.00 720. the car is now worth R48 000. Grandpa bought the donkey 5 years ago. End End End End End of of of of of first year second year third year fourth year fifth year R60 R48 R38 R30 R24 000(1 − 1 × 20%)=R60 000(1 − 1 × 20%)=R60 400(1 − 1 × 20%)=R60 720(1 − 1 × 20%)=R60 576(1 − 1 × 20%)=R60 000(1 − 1 × 20%)1 000(1 − 1 × 20%)2 000(1 − 1 × 20%)3 000(1 − 1 × 20%)4 000(1 − 1 × 20%)5 = = = = = R48 R38 R30 R24 R19 000. its value depreciates simply at 15% per annum. What rate of simple depreciation does this represent ? 4. Seven years ago. but that the initial value of the asset this year.GRADE 11
Exercise: Simple Depreciation 1. What did grandpa pay for the donkey then ? 3. seeing that it only depreciated at a rate of 3% per year using the straight-line method. For example. After 1 year.2) = R48 000(0. is the book value of the asset at the end of last year. Rocco's drum kit cost him R 12 500. the car is worth: Book Value after first year = P (1 − n × i) = R60 000(1 − 1 × 20%) = R60 000(1 − 0. then the interest rate of depreciation is 20% (100%/5 years).
12)5
Step 4 : Write the final answer There would be approximately 1 690 flamingos in 5 years' time. What is the depreciated value of the tractor after 5 years ? Answer Step 1 : Determine what has been provided and what is required 275
.8)2 = R38 400 as expected. If there is now 3 200 flamingos in the wetlands of the Bergriver mouth.88)5 3 200 × 0.
Worked Example 99: Compound Depreciation Question: Farmer Brown buys a tractor for R250 000 and depreciates it by 20% per year using the compound depreciation method.2)2
Note that the difference between the compound interest calculations and the compound depreciation calculations is that while the interest adds value to the principal amount. how many will there be in 5 years' time ? Answer to three significant numbers.527731916 1688.
= R60 000(1 − 20%)2 = R60 000(1 − 0.4
= R60 000(0.742134 3 200(1 − 0. Answer Step 1 : Determine what has been provided and what is required P i n A = = = is R3 200 0.a. FINANCE .GRADE 11 For example.CHAPTER 21. the depreciation amount reduces value!
Worked Example 98: Compound Depreciation Question: The Flamingo population of the Bergriver mouth is depreciating on a reducing balance at a rate of 12% p. the book value of the car after two years can be simply calculated as follows: Book Value after 2 years = P (1 − i)n
21.12 5 required
Step 2 : Determine how to approach the problem A = Step 3 : Solve the problem A = = = 3 200(0.
21.5% per annum as people migrate to the cities. and calculating what it will be worth in the future. 3. Calculate the value of x correct to two decimal places. 2008 the value of my Kia Sorento is R320 000. 2. Calculate the decrease in population over a period of 5 years if the initial population was 2 178 000. The population of Bonduel decreases at a rate of 9. A 20 kg watermelon consists of 98% water. On January 1. the calculations examined what the total money would be at some future date. We call these future values.2 5 required
Step 2 : Determine how to approach the problem A = Step 3 : Solve the problem A = 250 000(0.8)5 = 250 000 × 0.21. If it is left outside in the sun it loses 3% of its water each day. What is the value of the car on January 1. A computer depreciates at x% per annum using the reducing-balance method. Whether the money was borrowed or invested.1
Present Values or Future Values of an Investment or Loan
Now or Later
When we studied simple and compound interest we looked at having a sum of money now. 2012.5. Four years ago the value of the computer was R10 000 and is now worth R4 520.5
CHAPTER 21.32768 = 81 920 Step 4 : Write the final answer Depreciated value after 5 years is R 81 920 250 000(1 − 0. Each year after that. the cars value will decrease 20% of the previous years value. 276
. How much does in weigh after a month of 31 days ? 4.GRADE 11
P i n A
= = = is
R250 000 0. FINANCE .5
21.2)5
Exercise: Compound Depreciation 1.
In Equation 21. I will have: A = = = P · (1 + i)n
R1 000(1 + 10%)5 R1 610. we can solve for P instead of A. I need to invest: P = = = A · (1 + i)−n R1 610. We can test this as follows. There we started with an amount of R1 000 and looked at what it would grow 277
. the interest rate (i as a rate per annum) and the term (n in years) is: A = P · (1 + i)n (21.51(1 + 10%)−5 R1 000
We end up with R1 000 which . which relates the open balance (P ). This is called a present value. until we see what it is worth right now. FINANCE . Do you see that? Of course we could apply the same techniques to calculate a present value amount under simple interest rate assumptions .if you think about it for a moment . P = A/(1 + i)n (21.2) Using simple algebra. if R1 000 is needed at some future time.CHAPTER 21. For example. and work out what it is worth now. If I have R1 000 now and I invest it at 10% for 5 years.
A = P (1 + i × n) Solving for P gives: P = A/(1 + i × n)
(21.GRADE 11
21. the future value is what that amount will accrue to by some specified future date. if I know I have to have R1 610. we start off with a sum of money and we let it grow for n years.is what we started off with.51 in 5 years time. and we "unwind" the interest . and come up with: P = A · (1 + i)−n This can also be written as follows. however. how much must be invested to ensure the money grows to R1 000 at that future date? The equation we have been using so far in compound interest. In Equation 21.we just need to solve for the opening balance using the equations for simple interest.5) (21.3)
Now think about what is happening here.5
It is also possible.6)
Let us say you need to accumulate an amount of R1 210 in 3 years time. if R1 000 is deposited into a bank account now.in other words.51
at the end. and a bank account pays Simple Interest of 7%. How much would you need to invest in this bank account today? P = = = A 1+n·i R1 210 1 + 3 × 7% R1 000
Does this look familiar? Look back to the simple interest worked example in Grade 10. However.in other words we take off interest for n years. the closing balance (A). but the first approach is usually preferred. to look at a sum of money in the future.2.3 we have a sum of money which we know in n years time.4) (21. BUT. then the present value can be found by working backwards .
8.a. A=R3 000.21. start off with the basic equation that you should recognise very well: A = P · (1 + i)n If this were an algebra problem. you have always known what interest rate to use in the calculations. and you were told to "solve for i". A loan has to be returned in two equal semi-annual instalments. After a 20-year period Josh's lump sum investment matures to an amount of R313 550. make sure you use the compound interest rate formula!
Exercise: Present and Future Values 1.
21.45%
Ouch! That is not a very generous neighbour you have. n=8/12=0. This means that: i = = (R3 000/R2 500)1/0. and how long the investment or loan will last. You have then either taken a known starting point and calculated a future value.666667 − 1 31.a. If the rate of interest is 16% per annum.5 years time. How much did he invest if his money earned interest at a rate of 13. But here are other questions you might ask: 1. compounded quarterly for the next five years and 7. it is easy to derive any time you need it! So let us look at the two examples mentioned above. Check that you agree that P =R2 500.4% p.GRADE 11 to in 3 years' time using simple interest rates. I will need R450 for some university textbooks in 1.compounded half yearly for the first 10 years. find the sum borrowed. So unless you are explicitly asked to calculate a present value (or opening balance) using simple interest rates.6
CHAPTER 21. What interest rate do I need to earn to meet this goal? Each time that you see something different from what you have seen before.
In practice. I currently have R400.6
Finding i
By this stage in your studies of the mathematics of finance. FINANCE . present values are usually always calculated assuming compound interest. 1. however.a. What interest is she charging me? 2.2% p. compounded monthly for the remaining period ? 2. 278
. I want to borrow R2 500 from my neighbour. Now we have worked backwards to see what amount we need as an opening balance in order to achieve the closing balance of R1 210. you should be able to show that: A/P = (1 + i)n (1 + i) = (A/P )1/n i = (A/P )1/n − 1
You do not need to memorise this equation. compounded semi-annually and each instalment is R1 458. or taken a known future value and calculated a present value.65% p.666667. who said I could pay back R3 000 in 8 months time.
Trial and Error
By this stage you should be seeing a pattern.7
Finding n .7
This means that as long as you can find a bank which pays more than 8. Check that P =R400.CHAPTER 21. Always keep this in mind keep years with years to avoid making silly mistakes. if we know what the starting sum of money is and what it grows to. Solving for n.6). This section will calculate n by trial and error and by using a calculator.
Worked Example 100: Term of Investment . n=1. After 5 years an investment doubled in value.1 (on page 38) for some ideas as to how to go about finding n. This time we are going to solve for n. we expressed n as a number of years ( 12 years.5 − 1 8.
Exercise: Finding i 1.5) and i (in section 21.5). Determine the annual rate of depreciation if it is calculated on the reducing balance method. A machine costs R45 000 and has a scrap value of R9 000 after 10 years. and if we know what interest rate applies .17%
21. at the end of which our account is worth R4 044.17% interest.5% compound interest for an unknown period of time. We have our standard formula. 2. we can write: A = A = P P (1 + i)n (1 + i)n
Now we have to examine the numbers involved to try to determine what a possible value of n is. P (in section 21. you should have the money you need!
8 Note that in both examples. not 8 because that is the number of months) which means i is the annual interest rate. FINANCE .69. In other words. A=R450.5 i = = (R450/R400)1/1. How long did we invest the money? Answer Step 1 : Determine what is given and what is required 279
. which has a number of variables: A = P · (1 + i)n We have solved for A (in section 8.Trial and Error Question: If we invest R3 500 into a savings account which pays 7.GRADE 11 2.then we can work out how long the money needs to be invested for all those other numbers to tie up. At what annual rate was interest compounded ?
21. The proper algebraic solution will be learnt in Grade 12. Refer to Table 5.
Step 2 : Determine how to approach the problem We know that: A = A = P Step 3 : Solve the problem R4 044.e. Although it has not been explicitly stated.21. So how do we compare a monthly interest rate. Step 4 : Write final answer The R3 500 was invested for about 2 years.
21. After how many years will the book value of the two models be the same ? 2.5%)n (1.5 2.0 1. A company buys two typs of motor cars: The Acura costs R80 600 and the Brata R101 700 VAT included.8 • P =R3 500 • A=R4 044. where the interest is quoted as a per annum amount. 1. also compunded annually. The fuel in the tank of a truck decreases every minute by 5.69 = R3 500 1. 12 times per year.7&. say.5%
We are required to find n.075n 1. Interest however.075 1. compunded annually of 15. The Acura depreciates at a rate.156 1. may be paid more than just once a year.GRADE 11
• i=7.3% per year and the Brata at 19.075)n P (1 + i)n (1 + i)n
We now use our calculator and try a few values for n. Possible n 1.0 2. per year.69
CHAPTER 21.115 1.156 = (1 + 7.8
Nominal and Effective Interest Rates
So far we have discussed annual interest rates. i. Calculate after how many minutes there will be less than 30l in the tank if it originally held 200l. FINANCE . we have assumed that when the interest is quoted as a per annum amount it means that the interest is once a year. to an annual interest rate? This brings us to the concept of the effective annual interest rate.5% of the amount in the tank at that point in time.198
Exercise: Finding n . for example we could receive interest on a monthly basis. 280
.5 We see that n is close to 2.Trial and Error 1.
regardless of the differences in how frequently the interest is paid.83
which is the same as the answer obtained for 12 months. So we can calculate the amount that would be accumulated by the end of 1-year as follows: Closing Balance after 12 months = = = P × (1 + i)n Remember. The difference is due to interest on interest.1
The General Formula
So we know how to convert a monthly interest rate into an effective annual interest. Note that this is greater than simply multiplying the monthly rate by 12 (12 × 1% = 12%) due to the effects of compounding. This way. The effective annual interest rate is an annual interest rate which represents the equivalent per annum interest rate assuming compounding. Specifically. this means that the effective annual rate for a monthly rate i12 = 1% is: i = = = = (1 + i12)12 − 1 (1 + 1%)12 − 1 0.683%
If we recalculate the closing balance using this annual rate we get: Closing Balance after 1 year = = = P × (1 + i)n R1 000 × (1 + 12. We have introduced this notation here to distinguish between the annual interest rate. Another. We use i12 to denote the monthly interest rate.8
One way to compare different rates and methods of interest payments would be to compare the Closing Balances under the different options. Similarly. we can compare apples-with-apples. For example. It is the annual interest rate in our Compound Interest equation that equates to the same accumulated balance after one year. FINANCE .683%)1 R1 126.8.CHAPTER 21.GRADE 11
21. we have used n = 12 months to calculate the balance at the end of one year. for a given Opening Balance. a savings account with an opening balance of R1 000 offers a compound interest rate of 1% per month which is paid at the end of every month. more widely used. or a semi-annual interest rate or an interest rate of any frequency for that matter into an effective annual interest rate.83. we need to solve for i in the following equation: P × (1 + i)1 = P × (1 + i12)12
(1 + i) = (1 + i12)12 divide both sides by P i = (1 + i12)12 − 1 subtract 1 from both sides
For the example. but it is an important point!
21. We can calculate the accumulated balance at the end of the year using the formulae from the previous section. we can convert a quarterly interest. But be careful our interest rate has been given as a monthly rate. So we need to solve for the effective annual interest rate so that the accumulated balance is equal to our calculated amount of R1 126. and use the interest rate relevant to the time period.12683 12. We have seen this before.83
Note that because we are using a monthly time period. so we need to use the same units (months) for our time period of measurement. i.
R1 000 × (1 + 1%)12 R1 126. 281
. the trick to using the formulae is to define the time period. way is to calculate and compare the "effective annual interest rate" on each option.
55%. the interest will be paid four times per year (every three month). and the difference is due to the effects of interest-on-interest. that a nominal interest rate of 12% per annum paid monthly.7)
21. monthly interest rate = Nominal interest Rate per annum number of periods per year (21. is not to state the interest rate as say 1% per month.
Worked Example 101: Nominal Interest Rate Question: Consider a savings account which pays a nominal interest at 8% per annum. equates to an effective annual interest rate of 12. Answer Step 1 : Determine what is given and what is required We are given that a savings account has a nominal interest rate of 8% paid quarterly. (21. Calculate (a) the interest amount that is paid each quarter.2
De-coding the Terminology
Market convention however. So if you are given an interest rate expressed as an annual rate but paid more frequently than annual.GRADE 11
For a quarterly interest rate of say 3% per quarter. the monthly interest rate on 12% interest per annum paid monthly. we first need to calculate the actual interest paid per period in order to calculate the effective annual interest rate.8
CHAPTER 21. i
Step 2 : Determine how to approach the problem We know that: quarterly interest rate = Nominal interest Rate per annum number of quarters per year 282
. paid quarterly. is: monthly interest rate = = = Nominal interest Rate per annum number of periods per year 12% 12 months 1% per month
The same principle apply to other frequencies of payment. i4 • the effective annual interest rate.8. This annual amount is called the nominal amount. and so i = 12. We know from a previous example. We are required to find: • the quarterly interest rate.21. for interest paid at a frequency of T times per annum.68%. The market convention is to quote a nominal interest rate of "12% per annum paid monthly" instead of saying (an effective) 1% per month. the follow equation holds: P (1 + i) = P (1 + iT )T where iT is the interest rate paid T times per annum. In general. So (1 + i) = (1.8)
For example. This is the effective annual interest rate. FINANCE . We can calculate the effective annual interest rate by solving for i: P (1 + i) = P (1 + i4)4 where i4 is the quarterly interest rate. and (b) the effective annual interest rate. but rather to express this amount as an annual amount which in this example would be paid monthly.03)4 .
If you save R100 in such an account now. for a nominal interest rate of 8% paid quarterly. Step 2 : Recall relevant formulae We know that monthly interest rate = Nominal interest Rate per annum number of periods per year
for converting from nominal interest rate to effective interest rate. how much would the amount have accumulated to in 3 years' time? Answer Step 1 : Determine what is given and what is required Interest rate is 18% nominal paid monthly.GRADE 11 and P (1 + i) = P (1 + iT )T where T is 4 because there are 4 payments each year.CHAPTER 21.8
quarterly interest rate
= = =
Step 4 : Calculate the effective annual interest rate The effective annual interest rate (i) is calculated as: (1 + i) = (1 + i4)4 (1 + i) = (1 + 2%)4 i = (1 + 2%)4 − 1 = 8. paid monthly. We need the accumulated value. so i12 = = = Nominal annual interest rate 12 18% 12 1.
Worked Example 102: Nominal Interest Rate Question: On their saving accounts. There are 12 months in a year.24%. Step 3 : Calculate the monthly interest rate Nominal interest Rate per annum number of periods per year 8% 4 quarters 2% per quarter
21. FINANCE . we have A = P × (1 + i)n Step 3 : Calculate the effective interest rate There are 12 month in a year. so P = 100. so n = 3. We are working with a yearly time period. we have 1 + i = (1 + iT )T and for cacluating accumulated value. The amount we have saved is R100. A. Echo Bank offers an interest rate of 18% nominal.5% per month 283
.24% Step 5 : Write the final answer The quarterly interest rate is 2% and the effective annual interest rate is 8.
(Remember to round off to the the nearest cent.56%)3 = 100 × 1.)
Exercise: Nominal and Effect Interest Rates 1. here are the key formulae that we derived and used during this chapter. Cebela is quoted a nominal interest rate of 9. FINANCE . iT = Nominal Interest Rate T 284
. it is the application that is useful.75% p.e.7091
Step 5 : Write the final answer The accumulated value is R170. 2. normally the effective rate per annum period for which the investment is made the interest rate paid T times per annum. we have 1+i = i = = = = Step 4 : Reach the final answer = P × (1 + i)n
CHAPTER 21.
21.56%
A
= 170.5%)1 2 − 1 (1. While memorising them is nice (there are not many). Calculate the effective rate equivalent to a nominal interest rate of 8.a.GRADE 11
(1 + i12)1 2 (1 + i12)1 2 − 1 (1 + 1.15% per annum compounded every four months on her investment of R 85 000.
21. Calculate the effective rate per annum.91.21.91
= 100 × (1 + 19. they are paid a salary to use the right methods to solve financial problems. compounded monthly. i.1
P i n iT
Definitions
Principal (the amount of money at the starting point of the calculation) interest rate.9.015)1 2 − 1 19. Financial experts are not paid a salary in order to recite formulae.9
Formulae Sheet
As an easy reference.9 and then.
4. straight-line depreciation B the car depreciates at 12% p.9. according to the reducing-balance method. How much will Maggie receive in total after 5 years? 5. Maggie invests R12 500. Shrek buys a Mercedes worth R385 000 in 2007. A How much money will she owe Hilton Fashion World after two years ? B What is the effective rate of interest that Hilton Fashion World is charging her ?
285
.a. 2. A Determine the book value of the computer after 3 years if depreciation is calculated according to the straight-line method. She owes Hilton Fashion world R5 000 and the shop agrees to let Paris pay the bill at a nominal interest rate of 24% compounded monthly. Paris opens accounts at a number of clothing stores and spends freely. He is quoted a nominal interest rate of 7.10
End of Chapter Exercises
1. B Find the rate.GRADE 11
21. but after 18 months makes a withdrawal of R20 000.2% per annum compounded monthly. FINANCE .00 for 5 years at 12% per annum compounded monthly for the first 2 years and 14% per annum compounded semi-annually for the next 3 years. Greg enters into a 5-year hire-purchase agreement to buy a computer for R8 900. 3. She gets heself into terrible debt and she cannot pay off her accounts.10
21. C Suppose Tintin invests his money for a total period of 4 years. A computer is purchased for R16 000. Tintin invests R120 000. reducing-balance depreciation. What will the value of the Mercedes be at the end of 2013 if A the car depreciates at 6% p. Calculate the required monthly payment for this contract. that would yield the same book value as in 3a after 3 years.2
Equations
Simple Increase : A = Compound Increase : A = Simple Decrease : A = Compound Decrease : A = Ef f ective Annual Interest Rate(i) : (1 + i) = P (1 + i × n) P (1 + i)n P (1 − i × n) P (1 − i)n (1 + iT )T
21. The interest rate is quoted as 11% per annum based on simple interest. It depreciates at 15% per annum.CHAPTER 21.a. A Calculate the effective rate per annum correct to THREE decimal places. how much will he receive at the end of the 4 years? 6. B Use the effective rate to calculate the value of Tintin's investment if he invested the money for 3 years.
FINANCE .10
CHAPTER 21.GRADE 11
286
.21.
b = −5 and c = −12.2
Solution by Factorisation
The solving of quadratic equations by factorisation was discussed in Grade 10. with a = 2.
22. Answer Step 1 : Determine whether the equation has common factors This equation has no common factors.
Worked Example 103: Solution of Quadratic Equations Question: Solve the equation 2x2 − 5x − 12 = 0. Step 2 : Determine if the equation is in the form ax2 + bx + c with a > 0 The equation is in the required form. Here is an example to remind you of what is involved. This multiplies out to 2x2 + (s + 2v)x + sv We see that sv = −12 and s + 2v = −5. This chapter extends on that work. quadratic equations. the basics of solving linear equations. exponential equations and linear inequalities were studied.1 Introduction
In grade 10.Chapter 22
Solving Quadratic Equations Grade 11
22. but it is easy to solve numerically. 287
. Step 3 : Factorise the quadratic 2x2 − 5x − 12 has factors of the form: (2x + s)(x + v) with s and v constants to be determined. All the options for s and v are considered below. We look at different methods of solving quadratic equations. This is a set of simultaneous equations in s and v.
but it gives one a good understanding about some of the solutions of the quadratic equations.GRADE 11
22. There various possibilities are summarised in the figure below. We have already seen that whether the roots exist or not depends on whether this factor ∆ is negative or positive. SOLVING QUADRATIC EQUATIONS .imaginary roots ∆ ≥ 0 .real roots ∆>0 unequal roots ∆=0 equal roots
∆ a perfect square .CHAPTER 22.rational roots 297
∆ not a perfect square irrational roots
. The discriminant is defined as: ∆ = b2 − 4ac. In this case there are solutions to the equation f (x) = 0 given by the formula √ √ −b ± b2 − 4ac −b ± ∆ x= = (22.
The Nature of the Roots
Real Roots (∆ ≥ 0) Consider ∆ ≥ 0 for some quadratic function f (x) = ax2 + bx + c.
What is the Discriminant of a Quadratic Equation?
Consider a general quadratic function of the form f (x) = ax2 + bx + c.Advanced This section is not in the syllabus. (22.19) 2a 2a Since the square roots exists (the expression under the square root is non-negative.5
Worked Example 111: Fraction roots
3 Question: Find an equation with roots − 2 and 4 Answer Step 1 : Product of two brackets Notice that if x = − 3 then 2x + 3 = 0 2 Therefore the two brackets will be:
(2x + 3)(x − 4) = 0 Step 2 : Remove brackets The equation is: 2x2 − 5x − 12 = 0
Extension: Theory of Quadratic Equations .) These are the roots of the function f (x). ∆ ∆ < 0 .18) This is the expression under the square root in the formula for the roots of this function.
22. The roots of f (x) are rational if ∆ is a perfect square (a number which is the square of a rational number). Nov. [IEB. 2003. if ∆ is not a perfect square. Imaginary Roots (∆ < 0) If ∆ < 0. determine all pairs (b. Discuss the nature of the roots. [IEB. HG] If b and c can take on only the values 1. 2002. in this √ case. B When will the roots of the equation be equal? 7. 2005.advanced exercises
Exercise: From past papers 1.
A Find a value of k for which the roots are equal.5
CHAPTER 22. b and p. B Find one rational value of k. Nov. We therefore say that the roots of f (x) are imaginary (the graph of the function f (x) does not intersect the x-axis). Nov. Nov. [IEB. Otherwise. Nov. c) such that x2 + bx + c = 0 has real roots. HG] A Prove that the roots of the equation x2 − (a + b)x + ab − p2 = 0 are real for all real values of a. from the formula. ∆ is rational. 2 or 3. SOLVING QUADRATIC EQUATIONS .20)
Extension: Theory of Quadratics . 298
. HG] The equation x2 + 12x = 3kx2 + 2 has real roots. b 2a (22. 6. HG] Consider the equation: k= x2 − 4 2x − 5 where x =
5 2
2. HG] Given: x2 + bx − 2 + k(x2 + 3x + 2) = 0 (k = −1) A Show that the discriminant is given by: ∆ = k 2 + 6bk + b2 + 8 B If b = 0. [IEB. HG] Show that k 2 x2 + 2 = kx − x2 has non-real roots for all real values for k. 3. find the value(s) of k for which the roots are equal. [IEB. then the roots are irrational. discuss the nature of the roots of the equation. 2003. Nov.
5. 2004.
4. p.GRADE 11 Equal Roots (∆ = 0) If ∆ = 0. these are given by x=− Unequal Roots (∆ > 0) There will be 2 unequal roots if ∆ > 0. then the roots are equal and. Nov. for which the above equation has rational roots. q and r are positive real numbers and form a geometric sequence. HG] In the quadratic equation px2 + qx + r = 0. [IEB. 2005. since. B Find an integer k for which the roots of the equation will be rational and unequal. 2001. [IEB. C If b = 2. A Find the largest integral value of k. then the solution to f (x) = ax2 + bx + c = 0 contains the square root of a negative number and therefore there are no real solutions.
a different approach is needed. From the answers we have five regions to consider. SOLVING QUADRATIC INEQUALITIES .GRADE 11 equation −x2 − 3x + 5 = x + 3x − 5 = ∴x = = x1 x2 = =
2
23. We know that the roots of the function correspond to the x-intercepts of the graph.2
0 0 −3 ± (3)2 − 4(1)(−5) 2(1)
√ −3 ± 29 2√ −3 − 29 2√ −3 + 29 2
Step 2 : Determine which ranges correspond to the inequality We need to figure out which values of x satisfy the inequality. We can see that this is a parabola with a maximum turning point that intersects the x-axis at x1 and x2 . Let g(x) = −x2 − 3x + 5. by drawing a rough sketch of the graph of the function. 7 6 5 4 3 2 1 x1 −4 −3 −2 −1 −1 1 x2
It is clear that g(x) > 0 for x1 Step 4 : Write the final answer and represent the solution graphically −x2 − 3x + 5 > 0 for x1 x1 x2
When working with an inequality where the variable is in the denominator. 303
.CHAPTER 23. A B C D x1 x2 E
Step 3 : Determine whether the function is negative or positive in each of the regions We can use another method to determine the sign of the function over different regions.
In this chapter. As in Grade 10. the solutions to the system of equations in (24.Chapter 24
Solving Simultaneous Equations Grade 11
In grade 10.
Method: Graphical solution to a system of simultaneous equations with one linear and one quadratic equation 1. 2. the solution will be found both algebraically and graphically. The only difference between a system of linear simultaneous equations and a system of simultaneous equations with one linear and one quadratic equation.0) and (-4. had the highest power equal to 1).1)
24. Therefore. making y the subject of each equation. y = 0 and x = −4. gives: y = 2x − 4 y = 4 − x2 Plotting the graph of each equation.1
Graphical Solution
The method of graphically finding the solution to one linear and one quadratic equation is identical to systems of linear simultaneous equations. The parabola and the straight line intersect at two points: (2. Make y the subject of each equation.-12). gives a straight line for the first equation and a parabola for the second equation. you will learn how to solve sets of simultaneous equations where one is linear and one is a quadratic. An example of a system of simultaneous equations with one linear equation and one quadratic equation is: y − 2x = −4 x +y =4
2
(24. is that the second system will have at most two solutions. y = 12 307
.e. Draw the graphs of each equation as defined above.1) is x = 2. For the example. The solution of the set of simultaneous equations is given by the intersection points of the two graphs. you learnt how to solve sets of simultaneous equations where both equations were linear (i. 3.
how much money will Anna earn if she helps her mother 5 times to wash the dishes and helps her father 2 times to wash the car.
A mathematical model is an equation (or a set of equations for the more difficult problems) that describes are particular situation. sociology and political science). biology. Then we have: Total earned = x × R3 + y × R5
25. The first step to modelling is to write the equation.
Definition: Mathematical Model A mathematical model is a method of using the mathematical language to describe the behaviour of a physical system. but there has not been much application of what you have learnt. and electrical engineering) but also in the social sciences (such as economics. engineers. To model population growth 2.Grade 11
Up until now. This chapter builds introduces you to the idea of a mathematical model which uses mathematical concepts to solve real-world problems. if Anna receives R3 for each time she helps her mother wash the dishes and R5 for each time she helps her father cut the grass. In computer games 313
. To model effects of air pollution 3. you have only learnt how to solve equations and inequalities. Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics.Chapter 25
Mathematical Models . we say. To calculate how much Anna will earn we see that she will earn :
5 + 2
= R15 +R10 = R25
×R5 for cutting the grass
×R3 for washing the dishes
If however. For example. To model effects of global warming 4. physicists. what is the equation if Anna helps her mother x times and her father y times.1
Real-World Applications: Mathematical Models
Some examples of where mathematical models are used in the real-world are: 1. and economists use mathematical models most extensively. computer scientists. that describes the situation.
Distance is measured in meters and time is measured in seconds.00 for four cups of cappuccino and three cups of filter coffee. Kevin scored 80 more than in the second game. write an equation describing how many times they both sneezed? 2. In the third game. How long will it take both of them to paint a room together?
1 4. physics. calculate how much each type of coffee costs? 7. what must he score on the fourth game? 6.25. His total score for the first two games was 208. Step 2 : List all known and unknown information • v0 = 10 m · s−1 • t = 2s 314
. t is described by the following equation: s = 5t2 + v0 t In this equation. It rains half as much in July as it does in December. d. mathematically: 1. Billy can paint a room in 2 hours. 3. If he wants an average score of 146. Today. If it rains y mm in July. In the sciences (e. Jack and Jill both have colds. MATHEMATICAL MODELS . Arthur was 5 more than 3 as old as Lee was.
Worked Example 118: Mathematical Modelling of Falling Objects Question: When an object is dropped or thrown downward. Find the integers if their total is 24. chemistry. Write an equation that describes the following real-world situations. Jack sneezes twice for each sneeze of Jill's.GRADE 11
5. The product of two integers is 95.00 more than a cup of filter coffee. If a cup of cappuccino costs R3. the distance. In medicine to track the progress of a disease
Activity :: Investigation : Simple Models In order to get used to the idea of mathematical models. Erica has decided to treat her friends to coffee at the Corner Coffee House. v0 is the initial velocity. Kevin has played a few games of ten-pin bowling. In simulators that are used to train people in certain jobs. that it falls in time. We are also given the initial velocity and time and are required to calculate the distance travelled. If Jill sneezes x times. 25 years ago. Erica paid R54. write an expression relating the rainfall in July and December. doctors and soldiers 7. like pilots. Lee is 26 less than twice Arthur's age. Use the equation to find how far an object will fall in 2 s if it is thrown downward at an initial velocity of 10 m·s−1 ? Answer Step 1 : Identify what is given for each problem We are given an expression to calculate distance travelled by a falling object in terms of initial velocity and time. In the first game Kevin scored 110 less than the third game. Zane can paint a room in 4 hours.1
CHAPTER 25. in m·s−1 .g. biology) to understand how the natural world works 6. How old is Lee?
5. try the following simple models.
Distance is measured in meters and time is measured in seconds.
315
. Step 2 : List all known and unknown information • t =? s • v0 = 0 m · s−1 • s = 2000 m
Step 3 : Substitute values into expression s = 2000 = 2000 = t2 ∴ t = = = 5t2 + v0 t 5t2 + (0)(2) 5t2 2000 5 400 20 s
Step 4 : Write the final answer The object will take 20 s to reach the ground if it is dropped from a height of 2000 m. in m·s−1 . the distance.1
Step 4 : Write the final answer The object will fall 40 m in 2 s if it is thrown downward at an initial velocity of 10 m·s−1 . We are also given the initial velocity and time and are required to calculate the distance travelled. that it falls in time. d. v0 is the initial velocity. MATHEMATICAL MODELS .GRADE 11 • s =? m Step 3 : Substitute values into expression s = = = = = 5t2 + v0 t 5(2)2 + (10)(2) 5(4) + 20 20 + 20 40
25. The initial velocity is 0 m·s−1 ? Answer Step 1 : Identify what is given for each problem We are given an expression to calculate distance travelled by a falling object in terms of initial velocity and time. t is described by the following equation: s = 5t2 + v0 t In this equation. Use the equation find how long it takes for the object to reach the ground if it is dropped from a height of 2000 m.
Worked Example 119: Another Mathematical Modelling of Falling Objects Question: When an object is dropped or thrown downward.CHAPTER 25.
. How long does it take the car to travel 300 m?
Worked Example 120: More Mathematical Modelling Question: A researcher is investigating the number of trees in a forest over a period of n years. Do you think this model. How many trees.1
CHAPTER 25.GRADE 11
Activity :: Investigation : Mathematical Modelling The graph below shows the how the distance travelled by a car depends on time. 31 . How far does the car travel in 20 s? 2. the following data model emerged: Year 1 2 3 4 Number of trees in hundreds 1 3 9 27
1. are there in the SIXTH year if this pattern is continued? 2. Answer Step 1 : Find the pattern The pattern is 30 ..25. Step 2 : Trees in year 6 year6 = hundreds = 243hundreds = 24300 Step 3 : Algebraic expression for year n number of trees = 3n−1 hundreds Step 4 : Conclusion No The number of trees will increase without bound to very large numbers. MATHEMATICAL MODELS . will continue indefinitely? Give a reason for your answer. After investigating numerous data.
400 Distance (m) 300 200 100 0 0 10 20 30 Time (s) 40
1. 32 . in hundreds. Determine an algebraic expression that describes the number of trees in the nt h year in the forest. which determines the number of trees in the forest. thus the forestry authorities will if necessary cut down some of the trees from time to time. 3. . Use the graph to answer the following questions. 33 . Therefore. 316
. three to the power one less than the year.
2
Worked Example 121: Setting up an equation Question: Currently the subsription to a gym for a single member is R1 000 annually while family membership is R1 500. The tennis ball is thrown at an initial velocity of 5 m·s−1 . Distance is measured in meters and time is measured in seconds. The table below lists the times that Sheila takes to walk the given distances.CHAPTER 25. the distance. If the relationship between the distances and times are linear. v0 is the initial velocity.
Single Family Step 2 : Set up an equation
Now 1 000 1 500
After increase 1 000+x 1 500+x
1 000 + x = Step 3 : Solve the equation
5 (1 500 + x) 7
7 000 + 7x = 2x = x Step 4 : Write down the answer Therefore the increase is R250. in m·s−1 . When an object is dropped or thrown downward. find the equation of the straight line. If this is done then the single membership will cost 7 of the family membership. t is described by the following equation: s = 5t2 + v0 t In this equation. Time (minutes) Distance (km) Plot the points. The gym is considering raising all member5 shipfees by the same amount. using any two points. 2. MATHEMATICAL MODELS . Use the equation to find how long it takes a tennis ball to reach the ground if it is thrown downward from a hot-air balloon that is 500 m high. =
7 500 + 5x 500 250
25. Determine the proposed increase. that it falls in time. Answer Step 1 : Summarise the information in a table Let the proposed increase be x.GRADE 11
25. Then use the equation to answer the following questions: 317 5 1 10 2 15 3 20 4 25 5 30 6
.2
End of Chatpter Exercises
1. d.
You are in the lobby of a business building waiting for the lift. what would the graph of her distances and times look like?
3. MATHEMATICAL MODELS . E At what value of current will the power supplied be a maximum? 4. read off how much power is supplied to the circuit when the current is 10 amperes? Use the equation to confirm your answer. the number of people in the lift and how often it will stop: If N people get into a lift at the lobby and the number of floors in the building is F . B Draw a graph of P = 12I − 0.5I 2 where I is the current in amperes. if it stopped 12 times and there are 17 floors? 5.2
CHAPTER 25. The ends are right-angled triangles having sides 3x. A Since both power and current must be greater than 0. The length of the block is y. A If the building has 16 floors and there are 9 people who get into the lift. 4x and 5x. You are late for a meeting and wonder if it will be quicker to take the stairs. C What is the maximum current that can be drawn? D From your graph.25. find the limits of the current that can be drawn by the circuit. A wooden block is made as shown in the diagram.GRADE 11 A How long will it take Sheila to walk 21 km? B How far will Sheila walk in 7 minutes? If Sheila were to walk half as fast as she is currently walking. to define the extent of the graph. then the lift can be expected to stop F −F times. There is a fascinating relationship between the number of floors in the building. how many times is the lift expected to stop? B How many people would you expect in a lift. The power P (in watts) supplied to a circuit by a 12 volt battery is given by the formula P = 12I − 0.5I 2 and use your answer to the first question. F −1 F
N
3x
4x
y
Show that y=
300 − x2 x
318
. The total surface area of the block is 3 600 cm2 .
The room temperature. travelling at an average speed of x km per hour. Now deduce that the total cost. for a 2 000 km trip is given by: C(x) = 256000 + 40x x
8. C. The box has a volume of 480 cm3 . is given by: x 55 + litres per kilometre P (x) = 2x 200 Assume that the petrol costs R4. a cooling system is allowed to operate for 10 minutes.008t3 − 0.CHAPTER 25. according to the formula: 1 T (t) = 30 + 4t − t2 2 t ∈ [1. A washing powder box has the shape of a rectangular prism as shown in the diagram below. MATHEMATICAL MODELS .00 per hour (travelling time). T after t minutes is given in ◦ C by the formula: T = 28 − 0. 10] A At what rate (rounded off to TWO decimal places) is the temperature falling when t = 4 minutes? B Find the lowest room temperature reached during the 10 minutes for which the cooling system operates. varies with time t (in hours). 10.2
6.00 per litre and the driver earns R18. A stone is thrown vertically upwards and its height (in metres) above the ground at time t (in seconds) is given by: h(t) = 35 − 5t2 + 30t Find its initial height above the ground. in Rands. During an experiment the temperature T (in degrees Celsius).16t where t ∈ [0.
Washing powder
Show that the total surface area of the box (in cm2 ) is given by: A = 8x + 960x−1 + 240
Extension: Simulations A simulation is an attempt to model a real-life situation on a computer so that it 319
. by drawing a graph. 7. After doing some research. a transport company has determined that the rate at which petrol is consumed by one of its large carriers. In order to reduce the temperature in a room from 28◦ C.GRADE 11
25. a breadth of 4 cm and a length of x cm. B During which time interval was the temperature dropping? 9. 10]
A Determine an expression for the rate of change of temperature with time.
320
. In the past. problem solving skills and dispositions of children. MATHEMATICAL MODELS . Simulation can be used to show the eventual real effects of alternative conditions and courses of action. however.GRADE 11 can be studied to see how the system works.25. Simulation is used in many contexts. safety engineering. Other contexts include simulation of technology for performance optimization. training and education. problem solve and role play. predictions may be made about the behaviour of the system. video has been used for teachers and education students to observe. including the modeling of natural systems or human systems in order to gain insight into their functioning.2
CHAPTER 25. and pre-service and in-service teachers. Simulation in education Simulations in education are somewhat like training simulations. By changing variables. testing. ANVs are cartoon-like video narratives of hypothetical and realitybased stories involving classroom teaching and learning. ANVs have been used to assess knowledge. a more recent use of simulations in education include animated narrative vignettes (ANV). They focus on specific tasks.
You should have also found that the value of p affects whether the turning point of the graph is above the x-axis (p < 0) or below the x-axis (p > 0). You should have also found that the value of q affects whether the turning point is to the left of the y-axis (q > 0) or to the right of the y-axis (q < 0). If a > 0 then we have: (x + p)2 a(x + p)2 a(x + p)2 + q f (x) ≥ 0 ≥ 0 (The square of an expression is always positive) (Multiplication by a positive number maintains the nature of the inequality)
≥ q ≥ q
322
.1. choose your own values of p and q to plot 5 different graphs (on the same set of axes) of y = a(x+ p)2 + q to deduce the effect of a.2
CHAPTER 26. the graph makes a frown and if a > 0 then the graph makes a smile.1
Domain and Range
For f (x) = a(x + p)2 + q. QUADRATIC FUNCTIONS AND GRAPHS . The range of f (x) = a(x + p)2 + q depends on whether the value for a is positive or negative. 2. the domain is {x : x ∈ R} because there is no value of x ∈ R for which f (x) is undefined.26. The axes of symmetry for each graph is shown as a dashed line. 3.
From your graphs. you should have found that a affects whether the graph makes a smile or a frown.GRADE 11 E e(x) = (x + 2)2 Use your results to deduce the effect of p. If a < 0. These different properties are summarised in Table 26. p<0 p>0 a>0 a<0 a>0 a<0
q≥0
q≤0
26. Following the general method of the above activities.9. plot the following graphs: A f (x) = (x − 2)2 + 1 B g(x) = (x − 1)2 + 1 C h(x) = x2 + 1 D j(x) = (x + 1)2 + 1 E k(x) = (x + 2)2 + 1 Use your results to deduce the effect of q. The axes of symmetry are shown as dashed lines.1: Table summarising general shapes and positions of functions of the form y = a(x + p)2 + q. Table 26.2. On the same set of axes. We will consider these two cases separately. This is shown in Figure 10.
Find the x.2
CHAPTER 26.and y-intercepts of the function f (x) = (x − 4)2 − 1.8) is only valid if − a > 0 which means that either q < 0 or a < 0. 3. We know that if a > 0 then the range of f (x) = a(x + p)2 + q is {f (x) : f (x) ∈ [q. If however.∞)} and if a < 0 then the range of f (x) = a(x + p)2 + q is {f (x) : f (x) ∈ (−∞. 2
324
.
For example. the x-intercepts of g(x) = (x − 1)2 + 2 is given by setting y = 0 to get: g(x) 0 = (xint − 1)2 + 2 −2 = (xint − 1)2 which is not real. if a < 0.and y-intercepts of the graph of f . 3. if q < 0 and a > 0 then − a is also positive. then the lowest value that f (x) can take on is q. Given: f (x) = −x2 + 4x − 3.
Exercise: Turning Points 1.q]}. (26. 2. Therefore. 2. the graph of g(x) = (x − 1)2 + 2 does not have any x-intercepts. since if q > 0 and a > 0 then − a is negative and in this case the graph lies above the x-axis and therefore does not intersect the x-axis. then − a is positive and the graph is hat shaped and should have two x-intercepts. Given: f (x) = −x2 + 4x − 3. QUADRATIC FUNCTIONS AND GRAPHS . and the graph should intersect with the x-axis.GRADE 11
q However.q). if a > 0. Calculate the co-ordinates of the turning point of f . Find the intercepts with both axes of the graph of f (x) = x2 − 6x + 8. So. Determine the turning point of the graph of f (x) = x2 − 6x + 8 .26. This is q consistent with what we expect.
26. Solving for the value of x at which f (x) = q gives: q 0 0 0 x = a(x + p)2 + q = a(x + p)2 = (x + p)2 = x+p = −p
∴ x = −p at f (x) = q.3
Turning Points
The turning point of the function of the form f (x) = a(x + p)2 + q is given by examining the range of the function. Similarly. Find the turning point of the following function by completing the square: y = 1 (x + 2)2 − 1. The co-ordinates of the (minimal) turning point is therefore (−p.q). Calculate the x. then the highest value that f (x) can take on is q and the co-ordinates of the (maximal) turning point is (−p. q Similarly. q > 0 q and a < 0.2. = (x − 1)2 + 2
Exercise: Intercepts 1.
domain and range 3. Using the fact that the maximum value that f (x) achieves is -3.4
Axes of Symmetry
There is one axis of symmetry for the function of the form f (x) = a(x + p)2 + q that passes through the turning point and is parallel to the y-axis. The x-coordinate is determined as follows: 1 − (x + 1)2 − 3 2 1 − (x + 1)2 − 3 + 3 2 1 − (x + 1)2 2 1 Divide both sides by − 2 : (x + 1)2 = −3 = 0 = 0 = 0 = 0 = −1
Take square root of both sides: x + 1 ∴ x 325
. The range of the graph is determined as follows: (x + 1)2 1 − (x + 1)2 2 ≥ ≤ 0 0 −3 −3
1 − (x + 1)2 − 3 ≤ 2 ∴ f (x) ≤
Therefore the range of the graph is {f (x) : f (x) ∈ (−∞.CHAPTER 26.
26.2. QUADRATIC FUNCTIONS AND GRAPHS . turning point 2 and axis of symmetry. this is the axis of symmetry. Since the x-coordinate of the turning point is x = −p. Write down the equation of the axis of symmetry of the graph of y = 3(x − 2)2 + 1 3. − 3]}. sign of a 2. sketch the graph of g(x) = − 1 (x + 1)2 − 3.2. x-intercept For example. Write down an example of an equation of a parabola where the y-axis is the axis of symmetry. y-intercept 5. f (x) = a(x + p)2 + q.5
Sketching Graphs of the Form f (x) = a(x + p)2 + q
In order to sketch graphs of the form.2
26. turning point 4. we determine that a < 0. The domain of the graph is {x : x ∈ R} because f (x) is defined for all x ∈ R.GRADE 11
26. then the y-coordinate of the turning point is -3. Firstly. Mark the intercepts. This means that the graph will have a maximal turning point.
Exercise: Axes of Symmetry 1. we need to calculate determine four characteristics: 1. Find the equation of the axis of symmetry of the graph y = 2x2 − 5x − 18 2.
6
Writing an equation of a shifted parabola
Given a parabola with equation y = x2 − 2x − 3. Or else the y-axis shifts one unit to the left. B The x .3
End of Chapter Exercises
1. Therefore the new equation will become: y = (x − 1)2 − 2(x − 1) − 3
= x2 − 2x + 1 − 2x + 2 − 3 = x2 − 4x
If the given parabola is shifted 3 units down. find the values of the constants a and k.CHAPTER 26. QUADRATIC FUNCTIONS AND GRAPHS . Give the equation of the new graph originating if: A The graph of f is moved three units to the left.axis is moved down three.3
26. 5. A parabola with turning point (-1. (3. If (2. the new equation will become: (Notice the x-axis then moves 3 units upwards) y+3 = y = x2 − 2x − 3
x2 − 2x − 6
26. The graph in the figure is represented by the equation f (x) = ax2 + bx. -4) is shifted vertically by 4 units upwards.2. then the range of f (x) = a(x + p)2 + q is {f (x) : f (x) ∈ (−∞. The graph of the parabola is shifted one unit to the right. 2.7) is the turning point of f (x) = −2x2 − 4ax + k. The coordinates of the turning point are (3.q]}.9).GRADE 11
26. What are the coordinates of the turning point of the shifted parabola ?
327
. Given: f : x = x2 − 2x3. Show that if a < 0. 3. Show that a = −1 and b = 6.9)
4.
3
CHAPTER 26.GRADE 11
328
.26. QUADRATIC FUNCTIONS AND GRAPHS .
1 Introduction
In Grade 10.
Activity :: Investigation : Functions of the Form y = 1.
a x+p
1
2
3
4
5
+ q.Chapter 27
Hyperbolic Functions and Graphs Grade 11
27. you will learn a little more about the graphs of functions. In this chapter. 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 Figure 27.1: General shape and position of the graph of a function of the form f (x) = The asymptotes are shown as dashed lines. On the same set of axes. plot the following graphs: 329
a x+p
+q
. you studied graphs of many different forms. a x+p
27.2
Functions of the Form y =
+q
This form of the hyperbolic function is slightly more complex than the form studied in Grade 10.
HYPERBOLIC FUNCTIONS AND GRAPHS . the function is undefined for x = −p. 2.1
For y = −p}. plot the following graphs: A f (x) = B g(x) = C h(x) = D j(x) = E k(x) =
1 x−2 + 1 1 x−1 + 1 1 x+0 + 1 1 x+1 + 1 1 x+2 + 1
Use your results to deduce the effect of p. The axes of symmetry for each graph is shown as a dashed line.GRADE 11 A a(x) = B b(x) = C c(x) = D d(x) = E e(x) =
−2 x+1 −1 x+1 0 x+1 +1 x+1 +2 x+1
+1 +1 +1 +1 +1
Use your results to deduce the effect of a.2. 3.27.
Domain and Range
a x+p
+ q.
q>0
q<0
27. Table 27.
You should have found that the value of a affects whether the graph is located in the first and third quadrants of Cartesian plane. p<0 p>0 a>0 a<0 a>0 a<0
a x+p +q. You should have also found that the value of p affects whether the x-intercept is negative (p > 0) or positive (p < 0). The domain is therefore {x : x ∈ R.1. On the same set of axes. choose your own values a of a and p to plot 5 different graphs of y = x+p + q to deduce the effect of q. Following the general method of the above activities.1: Table summarising general shapes and positions of functions of the form y = The axes of symmetry are shown as dashed lines. You should have also found that the value of q affects whether the graph lies above the x-axis (q > 0) or below the x-axis (q < 0).2
CHAPTER 27.x = 330
. These different properties are summarised in Table 27.
x = −1} and the range to be {g(x) : g(x) ∈ (−∞. Draw the graph of the function defined by y = and intercepts with the axes.
Exercise: Asymptotes Given:h(x) =
1 x+4
Write down the equation of the vertical asymptote of the graph y =
− 2. y-intercept 4. Sketch the graph of h showing clearly the asymptotes and ALL intercepts with the axes. domain and range 2. Plot the graph of the hyperbola defined by y = x for −4 ≤ x ≤ 4. 1 2.∞)}.2.
5. Therefore the range is {g(x) : g(x) ∈ (−∞. Based on the graph of y = x .Determine the equations of the asymptotes of h.
4.
We have determined the domain to be {x : x ∈ R.2) ∪ (2. Therefore the asymptotes are x = −p and y = q.
8 x−8 +4. Explain your method.CHAPTER 27. Given:h(x) = x+4 − 2. f (x) = determine four characteristics: 1.2) ∪ (2. x-intercept For example. determine the equation of the graph with asymptotes y = 2 and x = 1 and passing through the point (2.
1 x−1 . we need to calculate
In order to sketch graphs of functions of the form. 3).
Exercise: Graphs
1 1.GRADE 11
27.
333
. What is the new equation then ? 1 2. We also see that g(x) is undefined at y = 2. Therefore the asymptotes are at x = −1 and y = 2. Suppose the hyperbola is shifted 3 units to the right and 1 unit down. Draw the graph of y =
5 x−2.
2 For example.
27.5
+ 2.3
End of Chapter Exercises
2 1.∞)}. The y-intercept is yint = 4 and the x-intercept is xint = −2. the domain of g(x) = x+1 + 2 is {x : x ∈ R. 8 1 3. HYPERBOLIC FUNCTIONS AND GRAPHS .
Indicate the asymptotes
27. asymptotes 3. Indicate the new horizontal asymptote. Draw the graph of y = x and y = − x+1 + 3 on the same system of axes. sketch the graph of g(x) =
2 x+1
a x+p
+ 2. Mark the intercepts and asymptotes.4
Sketching Graphs of the Form f (x) =
a x+p
+q
+ q. x = −1} because g(x) is undefined at x = −1. Draw the graph of y = x + 2.3
We saw that the function was undefined at x = −p and for y = q.
From this we deduce that the asymptotes are at x = −1 and y = 2.
plot the following graphs: A B C D E a(x) = −2 · b(x+1) + 1 b(x) = −1 · b(x+1) + 1 c(x) = −0 · b(x+1) + 1 d(x) = −1 · b(x+1) + 1 e(x) = −2 · b(x+1) + 1
Use your results to deduce the effect of a.2
Functions of the Form y = ab(x+p) + q
This form of the exponential function is slightly more complex than the form studied in Grade 10.
28. On the same set of axes.1 Introduction
In Grade 10.1: General shape and position of the graph of a function of the form f (x) = ab(x+p) +q.Grade 11
28. In this chapter. you studied graphs of many different forms.
Activity :: Investigation : Functions of the Form y = ab(x+p) + q 1.Chapter 28
Exponential Functions and Graphs . plot the following graphs: 335
. you will learn a little more about the graphs of exponential functions.
4 3 2 1 −4 −3 −2 −1 1 2 3 4
Figure 28. 2. On the same set of axes.
These different properties are summarised in Table 28. then the range is {f (x) : f (x) ∈ [q. if a > 0.1
Domain and Range
For y = ab(x+p) + q.1: Table summarising general shapes and positions of functions of the form y = ab(x+p) + q.2. Following the general method of the above activities. the domain is {x : x ∈ R}.GRADE 11 f (x) = 1 · b(x+1) − 2 g(x) = 1 · b(x+1) − 1 h(x) = 1 · b(x+1) 0 j(x) = 1 · b(x+1) + 1 k(x) = 1 · b(x+1) + 2
Use your results to deduce the effect of q.2 A B C D E
CHAPTER 28. The axes of symmetry for each graph is shown as a dashed line. The range of y = ab(x+p) + q is dependent on the sign of a. 3. the function is defined for all real values of x. p<0 a>0 a<0 a>0 p>0 a<0
q>0
q<0
28.1. Therefore. choose your own values of a and q to plot 5 different graphs of y = ab(x+p) + q to deduce the effect of p.28. EXPONENTIAL FUNCTIONS AND GRAPHS . Table 28. You should have also found that the value of p affects the position of the x-intercept. You should have also found that the value of q affects the position of the y-intercept. 336
. If a > 0 then: b(x+p) a · b(x+p) a · b(x+p) + q ≥ 0 0 q q
f (x)
≥
≥ ≥
Therefore.∞)}.
You should have found that the value of a affects whether the graph curves upwards (a > 0) or curves downwards (a < 0).
2.and y-intercepts of the graph of y = 1 (1.2. the graph of g(x) = 3 · 2x+1 + 2 does not have any x-intercepts. Give the x.GRADE 11
Which only has a real solution if either a < 0 or Q < 0. y-intercept 338
. domain and range 2. Therefore the range is {g(x) : g(x) ∈ (−∞. Otherwise.5)x+3 − 0. Therefore the asymptotes are x = −p and y = q. Therefore.28.8)x−1 − 3 ?
28.75. x = −1} because g(x) is undefined at x = −1. 2. Give the equation of the asymptote of the graph of y = 3x − 2. We saw that the function was undefined at x = −p and for y = q. They are determined by examining the domain and range.
Exercise: Asymptotes 1. We also see that g(x) is undefined at y = 2.3
Asymptotes
There are two asymptotes for functions of the form y = ab(x+p) + q. we need to calculate determine four characteristics: 1. Give the y-intercept of the graph of y = bx + 2. f (x) = ab(x+p) + q. 2
28. EXPONENTIAL FUNCTIONS AND GRAPHS .
From this we deduce that the asymptotes are at x = −1 and y = 2. 2.
Exercise: Intercepts 1. For example.2
CHAPTER 28. For example.∞)}. What is the equation of the horizontal asymptote of the graph of y = 3(0. the domain of g(x) = 3 · 2x+1 + 2 is {x : x ∈ R. the graph of the function of form y = ab(x+p) + q does not have any x-intercepts. the x-intercept of g(x) = 3 · 2x+1 + 2 is given by setting x = 0 to get: y 0 = 3 · 2x+1 + 2 = 3 · 2xint +1 + 2
−2 = 3 · 2xint +1 −2 2xint +1 = 2
which has no real solution.4
Sketching Graphs of the Form f (x) = ab(x+p) + q
In order to sketch graphs of functions of the form.2) ∪ (2.
75. the value of f (13). A Determine the value of a.064
2. D If the graph of f is shifted 2 units to the right to give the function h. C Determine the value of x. B Determine the value of f (−15) correct to FIVE decimal places.4 0.5 2 1. 3. correct to TWO decimal places.4 1. C Determine.16 B 6.bx (a = 0) has the point P(2.25 3.5 1 0. A If b = 0.2x (a is a constant) passes through the origin.GRADE 11 x -2 -1 0 1 2 A 7. The graph of f (x) = 1 + a.3
CHAPTER 28.16 0.4 0.
340
. EXPONENTIAL FUNCTIONS AND GRAPHS .16 C 2.25 2. 0.5) lies on the graph of f . if P (x. D Describe the transformation of the curve of f to h if h(x) = f (−x). calculate the value of a. B Hence write down the equation of f . write down the equation of h.28.5 1 0.144) on f . The graph of f (x) = a.
In Grade 11.
29.
341
.-5).1: The average gradient between two points on a curve is the gradient of the straight line that passes through the points.
What happens to the gradient if we fix the position of one point and move the second point closer to the fixed point?
Activity :: Investigation : Gradient at a Single Point on a Curve The curve shown is defined by y = −2x2 − 5. The position of point A varies.1 Introduction
In Grade 10.-1)
x
Figure 29.2
Average Gradient
We saw that the average gradient between two points on a curve is the gradient of the straight line passing through the two points. and are introduced to the idea of a gradient of a curve at a point.Grade 11
29.7) y
C(-1. Point B is fixed at co-ordinates (0. we investigated the idea of average gradient and saw that the gradient of some functions varied over different intervals. Complete the table below by calculating the y-coordinates of point A for the given x-coordinates and then calculate the average gradient between points A and B. we further look at the idea of average gradient. A(-3.Chapter 29
Gradient at a Point .
A y (a) y (b)
A C x C x
(c) y y
(d)
C A
x
C A
x
Figure 29. We therefore introduce the idea of a gradient at a single point on a curve. There comes a point when A and C overlap (as shown in (c)). g(2)) 342
.5 2 yA average gradient
CHAPTER 29. At this point the line is a tangent to the curve.GRADE 11 y B x
A
What happens to the average gradient as A moves towards B? What happens to the average gradient as A away from B? What is the average gradient when A overlaps with B?
In Figure 29.5 -1 -0. The gradient at a point on a curve is simply the gradient of the tangent to the curve at the given point.2 xA -2 -1.2. the gradient of the straight line that passes through points A and C changes as A moves closer to C.5 0 0. g(a+h)) on a curve g(x) = x2 . GRADIENT AT A POINT . the straight line only passes through one point on the curve.
Worked Example 122: Average Gradient Question: Find the average gradient between two points P(a. Then find the average gradient between P(2.29.2: The gradient of the straight line between A and C changes as the point A moves along the curve towards C. At the point when A and C overlap. g(a)) and Q(a + h. Such a line is known as a tangent to the curve.5 1 1.
Step 4 : Calculate the average gradient between P(2..2
= = = = =
(29. g(4)) We can use the result in (29. g(a)) and Q(a+ h. g(2)) and Q(4. g(4)).1)
The average gradient between P(a. If we have a curve defined by f (x) then for two points P and Q with P(a. g(a+ h)) on the curve g(x) = x2 is 2a + h. Answer Step 1 : Label x points x1 = a x2 = a + h Step 2 : Determine y coordinates Using the function g(x) = x2 .
We now see that we can write the equation to calculate average gradient in a slightly different manner.GRADE 11 and Q(4. We do this by looking at the definitions of P and Q. Finally. When the point Q overlaps with the point P h = 0 and the average gradient is given by 2a. h gets smaller. When point P moves closer to point Q. The x coordinate of P is a and the x coordinate of Q is a + h therefore if we assume that a = 2 then if a + h = 4. 343
.CHAPTER 29. we can determine: y1 = g(a) = a2 y2 = g(a + h) = (a + h)2 = a2 + 2ah + h2 Step 3 : Calculate average gradient y2 − y1 x2 − x1 (a2 + 2ah + h2 ) − (a2 ) (a + h) − (a) 2 a + 2ah + h2 − a2 a+h−a 2ah + h2 h h(2a + h) h 2a + h
29. f (a + h)). Then the average gradient is: 2a + h = 2(2) + (2) = 6 Step 5 : When P moves closer to Q. GRADIENT AT A POINT .1). which gives h = 2.. f (a)) and Q(a + h. This means that the average gradient also gets smaller. explain what happens to the average gradient if P moves closer to Q. then the average gradient between P and Q on f (x) is: y2 − y1 x2 − x1 = = f (a + h) − f (a) (a + h) − (a) f (a + h) − f (a) h
This result is important for calculating the gradient at a point on a curve and will be explored in greater detail in Grade 12. but we have to determine what is a and h.
344
. B Determine the gradient of the curve of f where x = 2.
End of Chapter Exercises
A Determine the average gradient of the curve f (x) = x(x + 3) between x = 5 and x = 3.
2. B Hence.3
CHAPTER 29.3
1. determine the equation of the tangent line at A. A(1.3) is a point on f (x) = 3x2 . B Hence.GRADE 11
29. state what you can deduce about the function f between x = 5 and x = 3. Given: f (x) = 2x2 . A Determine the average gradient of the curve between x = −2 and x = 1.29. A Determine the gradient of the curve at point A. GRADIENT AT A POINT . 3.
Grade 11
30. In this chapter we look at optimisation problems with two variables and where the possible solutions are restricted. If the farmer has two crops then the objective function f (x. the objective function is the yield and it is dependent on the amount of crops planted. These values are unknown at the beginning of the problem.
30. For the stock broker.2. maximise or minimise) is called the objective function. in the case of the farmer.
30.2
Objective Function
The objective function is a mathematical combination of the decision variables and represents the function that we want to optimise (i. for example the rate at which water is consumed or the number of birds living in a certain park. For example. These are examples of optimisation problems. We will only be looking at objective functions which are functions of two variables. We have seen optimisation problems of one variable in Chapter 40.1
Decision Variables
The aim of an optimisation problem is to find the values of the decision variables.1 Introduction
In everyday life people are interested in knowing the most efficient way of carrying out a task or achieving a goal. a farmer might want to know how many crops to plant during a season in order to maximise yield (produce) or a stock broker might want to know how much to invest in stocks in order to maximise profit.y) is the amount of profit earned by investing x rand in the first stock and y rand in the second. Decision variables usually represent things that can be changed.e. 345
. You were then required to find the highest (maximum) or lowest (minimum) possible value of some function.
30.Chapter 30
Linear Programming . the objective function f (x. For example.2. where there were no restrictions to the answer.y) is the yield. where x represents the amount of the first crop planted and y represents the amount of the second crop planted. assuming that there are two stocks to invest in. where by optimising we mean finding the maxima or minima of a function.2
Terminology
There are some basic terms which you need to become familiar with for the linear programming chapters.
30.
30. are often placed on the variables being optimised.2
CHAPTER 30. Examples of linear constraints are: x+y ≤0 −2x = 7 √ y≤ 2
30. For example. or restrictions. Sometimes there will be more 346
. x ≤ 20 means that only x values which are less than or equal to 20 are allowed. Other constraints might be that the farmer cannot plant more of the second crop than the first crop and that no more than 20 units of the first crop can be planted. the constraints x≥0
y ≥ 0. Any point in the feasible region is called a feasible point. therefore the constraints would be: x≥0
y ≥ 0.GRADE 11
30. Similarly.2. LINEAR PROGRAMMING .
Once we have determined the feasible region the solution of our problem will be the feasible point where the objective function is a maximum / minimum. he cannot plant a negative number of crops. For the example of the farmer.4
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. the constraint x≥y means that only values of x that are greater than or equal to the y values are allowed.3
Constraints
Constraints. These constraints can be written as: x≥y
x ≤ 20 Constraints that have the form ax + by ≤ c or ax + by = c are called linear constraints.y) in the xyplane then we call the set of all points in the xy-plane that satisfy our constraints the feasible region. If we think of the variables x and y as a point (x.2.2. mean that only values of x and y that are positive are allowed.5
The Solution
Important: Points that satisfy the constraints are called feasible solutions.
Important: The constraints are used to create bounds of the solution.
30.5. Identify the decision variables in the problem.4
Method of Linear Programming
Method: Linear Programming 1.GRADE 11
30. Write constraint equations 3. How should she divide her land so that she can earn the most profit? Let m represent the area of mielies grown and let p be the area of potatoes grown. Solve the problem
30. LINEAR PROGRAMMING .5
30. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes. 2.
30. Words Mathematical description x equals a x=a x is greater than a x>a x is greater than or equal to a x≥a x is less than a x<a x is less than or equal to a x≤a x must be at least a x≥a x must be at most b x≤a
Worked Example 123: Writing constraints as equations Question: Mrs Nkosi grows mielies and potatoes on a farm of 100 m2 . We shall see how we can solve this problem. Mrs Nkosi grows mielies and potatoes on a farm of 100 m2 .1: Phrases and mathematical equivalents. Write objective function as an equation 4. Nkosi and her farm.1. She expects to make a profit of R650 per m2 for her mielies and R1 500 per m2 on her sorgum.1
Skills you will need
Writing Constraint Equations
You will need to be comfortable with converting a word description to a mathematical description for linear programming.CHAPTER 30. Table 30.3
Example of a Problem
A simple problem that can be solved with linear programming involves Mrs. She has accepted orders that will need her to grow at least 40 m2 of mielies and at least 30 m2 of potatoes. She has accepted orders that will need her to grow at least 40 m2 of mielies and at least 347
.3
than one feasible point where the objective function is a maximum/minimum — in this case we have more than one solution. Some of the words that are used is summarised in Table 30.
30.5
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11 30 m2 of potatoes. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes. Answer Step 1 : Identify the decision variables There are two decision variables: the area used to plant mielies (m) and the area used to plant potatoes (p). Step 2 : Identify the phrases that constrain the decision variables • grow at least 40 m2 of mielies • area of farm is 100 m2
• grow at least 30 m2 of potatoes • demand is twice as much for mielies as for potatoes
Exercise: constraints as equation Write the following constraints as equations: 1. Michael is registering for courses at university. Michael needs to register for at least 4 courses. 2. Joyce is also registering for courses at university. She cannot register for more than 7 courses. 3. In a geography test, Simon is allowed to choose 4 questions from each section. 4. A baker can bake at most 50 chocolate cakes in 1 day. 5. Megan and Katja can carry at most 400 koeksisters.
30.5.2
Writing the Objective Function
If the objective function is not given to you as an equation, you will need to be able to convert a word description to an equation to get the objective function. You will need to look for words like: • most profit • least cost • largest area 348
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11
30.5
Worked Example 124: Writing the objective function Question: The cost of hiring a small trailer is R500 per day and the cost of hiring a big trailer is R800 per day. Write down the objective function that can be used to find the cheapest cost for hiring trailers for 1 day. Answer Step 1 : Identify the decision variables There are two decision variables: the number of big trailers (nb ) and the number of small trailers (ns ). Step 2 : Write the purpose of the objective function The purpose of the objective function is to minimise cost. Step 3 : Write the objective function The cost of hiring ns small trailers for 1 day is: 500 × ns The cost of hiring nb big trailers for 1 day is: 800 × nb Therefore the objective function, which is the total cost of hiring ns small trailers and nb big trailers for 1 day is: 500 × ns + 800 × nb
Worked Example 125: Writing the objective function Question: Mrs Nkosi expects to make a profit of R650 per m2 for her mielies and R1 500 per m2 on her potatoes. How should she divide her land so that she can earn the most profit? Answer Step 1 : Identify the decision variables There are two decision variables: the area used to plant mielies (m) and the area used to plant potatoes (p). Step 2 : Write the purpose of the objective function The purpose of the objective function is to maximise profit. Step 3 : Write the objective function The profit of planting m m2 of mielies is: 650 × m The profit of planting p m2 of potatoes is: 1500 × p Therefore the objective function, which is the total profit of planting mielies and potatoes is: 650 × m + 1500 × p
349
30.5
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11
Exercise: Writing the objective function 1. The EduFurn furniture factory manufactures school chairs and school desks. They make a profit of R50 on each chair sold and of R60 on each desk sold. Write an equation that will show how much profit they will make by selling the chairs and desks? 2. A manufacturer makes small screen GPS's and wide screen GPS's. If the profit on small screen GPS's is R500 and the profit on wide screen GPS's is R250, write an equation that will show the possible maximum profit.
30.5.3
Solving the Problem
The numerical method involves using the points along the boundary of the feasible region, and determining which point has the optimises the objective function.
The question is How do you find the feasible region? We will use the graphical method of solving a system of linear equations to determine the feasible. We draw all constraints as graphs and mark the area that satisfies all constraints. This is shown in Figure 30.1 for Mrs. Nkosi's farm. Now we can use the methods we learnt previously to find the points at the vertices of the feasible region. In Figure 30.1, vertex A is at the intersection of p = 30 and m = 2p. Therefore, the coordinates of A are (30,60). Similarly vertex B is at the intersection of p = 30 and m = 100−p. Therefore the coordinates of B are (30,70). Vertex C is at the intersection of m = 100 − p and 1 2 m = 2p, which gives (33 3 ,66 3 ) for the coordinates of C. If we now substitute these points into the objective function, we get the following: m 60 70 2 66 3 p 30 30 1 33 3 Profit 81 000 87 000 89 997
2 2 Therefore Mrs. Nkosi makes the most profit if she plants 66 3 m2 of mielies and 66 3 m2 of potatoes. Her profit is R89 997. 350
Worked Example 126: 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. 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 351
GRADE 11 The cost of manufacturing a kettle is R60 and a toaster is R50. The first section is on Algebra and the second section is on Geometry. Step 6 : Calculate cost at each vertex At vertex A. You are given a test consisting of two sections. the coordinates of vertex A is (3. the cost is: C = = = = 60xk + 50yt 60(10) + 50(30) 600 + 1500 2100 60xk + 50yt 60(30) + 50(10) 1800 + 500 2300
Step 7 : Write the final answer The cheapest combination of prizes is 10 kettles and 30 toasters. 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
Step 5 : Determine vertices of feasible region From the graph. costing the company R2 100. You are not allowed to answer more than 10 questions from any section.6
End of Chapter Exercises
1. The time allowed 352
. but you have to answer at least 4 Algebra questions. LINEAR PROGRAMMING .1) and the coordinates of vertex B are (1.6
CHAPTER 30.
30.3).30. the cost is: C = = = = At vertex B.
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11
30.6
is not more than 30 minutes. An Algebra problem will take 2 minutes and a Geometry problem will take 3 minutes each to solve. If you answer xA Algebra questions and yG Geometry questions, A Formulate the constraints which satisfy the above constraints. B The Algebra questions carry 5 marks each and the Geometry questions carry 10 marks each. If T is the total marks, write down an expression for T . 2. A local clinic wants to produce a guide to healthy living. The clinic intends to produce the guide in two formats: a short video and a printed book. The clinic needs to decide how many of each format to produce for sale. Estimates show that no more than 10 000 copies of both items together will be sold. At least 4 000 copies of the video and at least 2 000 copies of the book could be sold, although sales of the book are not expected to exceed 4 000 copies. Let xv be the number of videos sold, and yb the number of printed books sold. A Write down the constraint inequalities that can be deduced from the given information. B Represent these inequalities graphically and indicate the feasible region clearly. C The clinic is seeking to maximise the income, I, earned from the sales of the two products. Each video will sell for R50 and each book for R30. Write down the objective function for the income. D Determine graphically, by using a search line, the number of videos and books that ought to be sold to maximise the income. E What maximum income will be generated by the two guides? 3. A patient in a hospital needs at least 18 grams of protein, 0,006 grams of vitamin C and 0,005 grams of iron per meal, which consists of two types of food, A and B. Type A contains 9 grams of protein, 0,002 grams of vitamin C and no iron per serving. Type B contains 3 grams of protein, 0,002 grams of vitamin C and 0,005 grams of iron per serving. The energy value of A is 800 kilojoules and the of B 400 kilojoules per mass unit. A patient is not allowed to have more than 4 servings of A and 5 servings of B. There are xA servings of A and yB servings of B on the patients plate. A Write down in terms of xA and yB i. The mathematical constraints which must be satisfied. ii. The kilojoule intake per meal. B Represent the constraints graphically on graph paper. Use the scale 1 unit = 20mm on both axes. Shade the feasible region. C Deduce from the graphs, the values of xA and yB which will give the minimum kilojoule intake per meal for the patient. 4. A certain motorcycle manufacturer produces two basic models, the 'Super X' and the 'Super Y'. These motorcycles are sold to dealers at a profit of R20 000 per 'Super X' and R10 000 per 'Super Y'. A 'Super X' requires 150 hours for assembly, 50 hours for painting and finishing and 10 hours for checking and testing. The 'Super Y' requires 60 hours for assembly, 40 hours for painting and finishing and 20 hours for checking and testing. The total number of hours available per month is: 30 000 in the assembly department, 13 000 in the painting and finishing department and 5 000 in the checking and testing department. The above information can be summarised by the following table: Department Assembley Painting and Finishing Checking and Testing Hours for 'Super X' 150 50 10 Hours for Super 'Y' 60 40 20 Maximum hours available per month 30 000 13 000 5 000
Let x be the number of 'Super X' and y be the number of 'Super Y' models manufactured per month. 353
30.6
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11 A Write down the set of constraint inequalities. B Use the graph paper provided to represent the constraint inequalities. C Shade the feasible region on the graph paper. D Write down the profit generated in terms of x and y. E How many motorcycles of each model must be produced in order to maximise the monthly profit? F What is the maximum monthly profit?
5. A group of students plan to sell x hamburgers and y chicken burgers at a rugby match. They have meat for at most 300 hamburgers and at most 400 chicken burgers. Each burger of both types is sold in a packet. There are 500 packets available. The demand is likely to be such that the number of chicken burgers sold is at least half the number of hamburgers sold. A Write the constraint inequalities. B Two constraint inequalities are shown on the graph paper provided. Represent the remaining constraint inequalities on the graph paper. C Shade the feasible region on the graph paper. D A profit of R3 is made on each hamburger sold and R2 on each chicken burger sold. Write the equation which represents the total profit, P, in terms of x and y. E The objective is to maximise profit. How many, of each type of burger, should be sold to maximise profit? 6. Fashion-cards 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. Fashion-cards 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 Calculate the maximum weekly profit. 7. To meet the requirements of a specialised diet a meal is prepared by mixing two types of cereal, Vuka and Molo. The mixture must contain x packets of Vuka cereal and y packets of Molo cereal. The meal requires at least 15 g of protein and at least 72 g of carbohydrates. Each packet of Vuka cereal contains 4 g of protein and 16 g of carbohydrates. Each packet of Molo cereal contains 3 g of protein and 24 g of carbohydrates. There are at most 5 packets of cereal available. The feasible region is shaded on the attached graph paper. A Write down the constraint inequalities. B If Vuka cereal costs R6 per packet and Molo cereal also costs R6 per packet, use the graph to determine how many packets of each cereal must be used for the mixture to satisfy the above constraints in each of the following cases: i. The total cost is a minimum. ii. The total cost is a maximum (give all possibilities). 354
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11
30.6
6 6 Number of packets of Molo 5 5 4 4 3 3 2 2 1 1 0 0
00
11 22 33 44 55 Number of packets of Vuka
66
8. A bicycle manufacturer makes two different models of bicycles, namely mountain bikes and speed bikes. The bicycle manufacturer works under the following constraints: No more than 5 mountain bicycles can be assembled daily. No more than 3 speed bicycles can be assembled daily. It takes one man to assemble a mountain bicycle, two men to assemble a speed bicycle and there are 8 men working at the bicycle manufacturer. Let x represent the number of mountain bicycles and let y represent the number of speed bicycles. A Determine algebraically the constraints that apply to this problem. B Represent the constraints graphically on the graph paper. C By means of shading, clearly indicate the feasible region on the graph. D The profit on a mountain bicycle is R200 and the profit on a speed bicycle is R600. Write down an expression to represent the profit on the bicycles. E Determine the number of each model bicycle that would maximise the profit to the manufacturer.
355
30.6
CHAPTER 30. LINEAR PROGRAMMING - GRADE 11
356
Islamic geometry (c.
Method: Surface Area of a Pyramid The surface area of a pyramid is calculated by adding the area of each face together.2
Right Pyramids.1. 700 .Grade 11
31.20th centuries (c.1500)
31. Examples of pyramids are shown in Figure 31.1: Examples of a square pyramid.1500) A Thabit ibn Qurra B Omar Khayyam C Sharafeddin Tusi 2. Right Cones and Spheres
A pyramid is a geometric solid that has a polygon base and the base is joined to an apex. Geometry in the 17th .Chapter 31
Geometry .
357
.
Figure 31.1 Introduction
Activity :: Extension : History of Geometry Work in pairs or groups and investigate the history of the development of geometry in the last 1500 years. The works of the following people or cultures can be investigated: 1. Describe the various stages of development and how different cultures used geometry to improve their lives. 700 . a triangular pyramid and a cone.
show that the surface √ area is πr2 + πr r2 + h2 .2 Worked Example 127: Surface Area
CHAPTER 31. Answer Step 1 : Draw a picture
h
a
h
r
r
Step 2 : Identify the faces that make up the cone The cone has two faces: the base and the walls.31. 358
. a
2πr = circumference This curved surface can be cut into many thin triangles with height close to a (a is called a slant height).GRADE 11
Question: If a cone has a height of h and a base of radius r. Therefore: a= r 2 + h2
Step 4 : Calculate the area of the circular base Ab = πr2 Step 5 : Calculate the area of the curved walls Aw = πra = πr Step 6 : Calculate surface area A A = = Ab + Aw πr2 + πr r 2 + h2 r 2 + h2
Method: Volume of a Pyramid The volume of a pyramid is found by: V = 1 A·h 3
where A is the area of the base and h is the height. GEOMETRY . The base is a circle of radius r and the walls can be opened out to a sector of a circle. The area of these triangles will add up to 1 ×base×height 2 which is 1 × 2πr × a = πra 2 Step 3 : Calculate a a can be calculated by using the Theorem of Pythagoras.
3 1 2 a h 3
31. Surface area = Volume = 4πr2 4 3 πr 3
Exercise: Surface Area and Volume 359
. 3cm high with a side length of 2cm? Answer Step 1 : Determine the correct formula The volume of a pyramid is 1 V = A · h. so the volume of a cone is given by V = A square pyramid has volume V = where a is the side length.GRADE 11 A cone is a pyramid.2
Worked Example 128: Volume of a Pyramid Question: What is the volume of a square pyramid. 3 which for a square base means V = 1 a · a · h. 1 2 πr h. 3
3cm
2cm
2cm
Step 2 : Substitute the given values 1 ·2·2·3 3 1 · 12 3 4 cm3
= = =
We accept the following formulae for volume and surface area of a sphere (ball). GEOMETRY .CHAPTER 31.
The pyramid has a height of 12 cm.
AB PQ
=
BC QR
=
CD RS
=
DE ST
=
EA TP
S D
R C
then the polygons PQRST are similar. Water covers approximately 71% of the Earth's surface.
31. All corresponding angles must be congruent. A right triangular pyramid is placed on top of a right triangular prism. GEOMETRY . The prism has an equilateral triangle of side length 20 cm as a base. B Find the area of each face of the pyramid.
A
If
P E
T
Q
B
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1. 2. B = Q. C Find the total surface area of the object.3
Similarity of Polygons
In order for two polygons to be similar the following must be true: 1. Calculate the volumes and surface areas of the following solids: *Hint for (e): find the perpendicular height using Pythagoras. ˆ=T ˆ E and 2.GRADE 11 1. A Find the total volume of the object.3
CHAPTER 31. and has a height of 42 cm. A = P . D = S.
a) b) c) d)
3 6 4 14 5 13
e)
24 7 a sphere a hemisphere a cone a hemisphere on top of a cone 24 a pyramid with a square base
2. what is the total area of land (area not covered by water)? 3. All corresponding sides must be in the same proportion to each other.31.
ABCDE
and
Worked Example 129: Similarity of Polygons Question: 360
. Taking the radius of the Earth to be 6378 km. C = R.
area △ABC = A 1 · h · BC = area △DBC 2 D
h
B
C
• Triangles on the same side of the same base. D
B
C
Theorem 1. If area △ ABC = area △ BDC. lie between parallel lines.31. with equal areas. then AD A BC.GRADE 11
h1 area △ABC ∴ area △DEF
= h2 1 BC × h1 BC = = 2 1 EF EF × h2 2 A D
h1 h2
B
C
E
F
• A special case of this happens when the bases of the triangles are equal: Triangles with equal bases between the same parallel lines have the same area. Proportion Theorem:A line drawn parallel to one side of a triangle divides the other two sides proportionally.4
CHAPTER 31. E A A A
h1 h2
D
D B D C E 362
E
B
C B
C
. GEOMETRY .
y2 ).
J 2 L y M 7 I K y-2
31.
D B C E F
7. AD = 9 m. when two points are given. DF = 18m.
E
36 42
D G
21 F
H
6. y1 ) and (x2 .5
K
CHAPTER 31.31. BF = 25 m. Calculate the lengths of BC.5
31. CF. One option is to find the equation of a straight line. AB = 13 m. and we know that the general form of the equation for a straight line is: 368
.5. calculate y. GEOMETRY .1
Co-ordinate Geometry
Equation of a Line between Two Points
There are many different methods of specifying the requirements for determining the equation of a straight line. and find the ratio
A
DE AC .GRADE 11
J
H
L
I
5. CD. Find FH in the following figure. Assume that the two points are (x1 . CE and EF. If LM
JK.
GRADE 11
31.2
Equation of a Line through One Point and Parallel or Perpendicular to Another Line
Another method of determining the equation of a straight-line is to be given one point.5.4(a). 371
. we see that the line makes an angle θ with the x-axis. then m If the lines are perpendicular.3
Inclination of a Line
In Figure 31.CHAPTER 31.12) (31. GEOMETRY . then m × m0 = m0 = −1 (31. 2) and (5. and to be told that the line is parallel or perpendicular to another line. y1 ). we can then use the given point together with: y − y1 = m(x − x1 ) to determine the equation of the line. 8) is y = 3 17 4x + 4 . (x1 .13)
Once we have determined a value for m. For example. 1) into: y − y1 y−1 = m(x − x1 ) = 2(x − (−1) = 2(x + 1) = 2x + 2
y−1 y−1 y y
= 2x + 2 + 1 = 2x + 3
31. 1). This angle is known as the inclination of the line and it is sometimes interesting to know what the value of θ is. If the equation of the unknown line is y = mx + c and the equation of the second line is y = m0 x + c0 . First we determine m.5
y − y1
=
y − (2) = y = = = =
m(x − x1 ) 3 (x − (−3)) 4 3 (x + 3) + 2 4 3 3 x+ ·3+2 4 4 9 8 3 x+ + 4 4 4 17 3 x+ 4 4
Step 4 : Write the final answer The equation of the straight line that passes through (−3. m=2 The equation is found by substituting m and (−1. Since the line we are looking for is parallel to y = 2x − 1.5. then we know the following: If the lines are parallel. find the equation of the line that is parallel to y = 2x − 1 and that passes through (−1.
31.
4) through points (7. we note that if the gradient changes.)
31.GRADE 11
31.6
Firstly. m= But. 3) and (1. so we suspect that the inclination of a line is related to the gradient. Find the inclination of the following lines
3. GEOMETRY .CHAPTER 31.4(a) we see that tan θ ∴m = = ∆y ∆x tan θ ∆y ∆x
For example. we know m = 1 ∴ tan θ ∴θ = = 1 45◦
Exercise: Co-ordinate Geometry 1. to find the inclination of the line y = x. in Figure 31. What happens to the coordinates of a point that is rotated by 90◦ or 180◦ around the origin?
Activity :: Investigation : Rotation of a Point by 90◦ 373
. Show that the line y = k for any constant k is parallel to the x-axis. 4) 1 parallel to y = 2 x + 3 passing through (−1. (Hint: Show that the inclination of this line is 0◦ . −3) and (0. (Hint: Show that the inclination of this line is 90◦ . then the value of θ changes (Figure 31.6. 2) perpendicular to 2y + x = 6 passing through the origin y = 2x − 3 y = 1x − 7 3 4y = 3x + 8 y = − 2 x + 3 (Hint: if m is negative θ must be in the second quadrant) 3 3y + x − 3 = 0
2. Find the equations of the following lines A B C D E A B C D E through points (−1.) 4. we say that it is rotated. 3) 1 perpendicular to y = − 2 x + 3 passing through (−1. Show that the line x = k for any constant k is parallel to the y-axis.6
31.1
Transformations
Rotation of a Point
When something is moved around a fixed point. We know that the gradient is a ratio of a change in the y-direction to a change in the x-direction.4(b)).
GEOMETRY .GRADE 11
Complete the table31 if it was rotated to the position of point C? What about point B rotated to the position of D?
D E F
C B A H
G
Activity :: Investigation : Rotation of a Point by 180◦
Complete the table. if it was rotated to the position of point E? What about point F rotated to the position of B? C B A F G H
D E
From these activities you should have come to the following conclusions: 374
. by filling in the coordinates of the points shown in the figure. by filling in the coordinates of the points shown in the figure.
5) and B′ (5. -y)
Exercise: Rotation 1. y)
• 180◦ rotation: The image of a point P(x. Give the co-ordinates of X′ . GEOMETRY . −y). 375
.4) B OB is rotated to OB′ with B(-2. (ii) Draw a diagram showing the direction of rotation. y)
31. y) rotated anticlockwise through 90◦ around the origin is P'(−y. −y). y) rotated through 180◦ around the origin is P'(−x.GRADE 11 y
P(x. x)
y
P(x. For each of the following rotations about the origin: (i) Write down the rule.6
• 90◦ clockwise rotation: The image of a point P(x. y) → (y. y) → (−x.
x
P"'(-x.2) and A′ (-2. x). y) rotated clockwise through 90◦ around the origin is P'(y.
P'(y. −x). B Rotate ∆XYZ through 180◦ about the origin to give ∆X′′ Y′′ Z′′ . We write the rotation as (x.4) 2. We write the rotation as (x. A OA is rotated to OA′ with A(4. y) → (−y.CHAPTER 31. −x). The co-ordinates are given on the picture.2) C OC is rotated to OC′ with C(-1. x). Give the co-ordinates of X′′ . Copy ∆XYZ onto squared paper. We write the rotation as (x. Y′′ and Z′′ . Y′ and Z′ . -x)
x
y
P(x.
x
P"(-y.-4) and C′ (1. A Rotate ∆XYZ anti-clockwise through an angle of 90◦ about the origin to give ∆X′ Y′ Z′ . y)
• 90◦ anticlockwise rotation: The image of a point P(x.
PointGRADE 11
X(4.4)
Z(-4. if the square ABCD was enlarged by a factor 2? F 1 B −1 C −1 G A 1 D H E
Activity :: Investigation : Enlargement of a Polygon 2 376
.31.6. by filling in the coordinates of the points shown in the figure.2
Enlargement of a Polygon 1
When something is made larger.-1)
Y(-1. What happens to the coordinates of a polygon that is enlarged by a factor k?
Activity :: Investigation : Enlargement of a Polygon Complete the table.-4)
31. we say that it is enlarged. GEOMETRY .
GRADE 11
Exercise: Transformations
1. Draw this on the same grid. 1) Copy polygon STUV onto squared paper and then answer the following questions.31. C If the area of △ABC is m times the area of △A'B'C'. △ABC is an enlargement of △A'B'C' by a constant factor of k through the origin.
3
2
S T
1
0 -3 -2 -1 -1 0 1 2 3 4 5
V
-2
U
-3
A What are the co-ordinates of polygon STUV? B Enlarge the polygon through the origin by a constant factor of c = 2. what is m?
5
A
4
3
A'
2
B B'
1
0 -5 -4 -3 -2 -1 -1 0 1 2 3 4 5
-2
C'
-3
-4
C
-5
378
. GEOMETRY . C What are the co-ordinates of the vertices of S'T'U'V' ? 2.6
CHAPTER 31. Label it S'T'U'V'. A What are the co-ordinates of the vertices of △ABC and △A'B'C' ? B Giving reasons. calculate the value of k.
obtaining polygon M'N'P'Q'.6
4
M
3
2
P
1
N Q
0 -2 -1 -1 0 1 2 3 4 5
3.
379
.GRADE 11
5
31. E Find the inclination of OM".CHAPTER 31. GEOMETRY .
-2
A What are the co-ordinates of the vertices of polygon MNPQ? B Enlarge the polygon through the origin by using a constant factor of c = 3. C What are the co-ordinates of the new vertices? D Now draw M"N"P"Q" which is an anticlockwise rotation of MNPQ by 90◦ around the origin. Draw this on the same set of axes.
6
CHAPTER 31.GRADE 11
380
. GEOMETRY .31.
y = sin(kθ).Chapter 32
Trigonometry . People A Lagadha (circa 1350-1200 BC) B Hipparchus (circa 150 BC) C Ptolemy (circa 100) D Aryabhata (circa 499) E Omar Khayyam (1048-1131) F Bhaskara (circa 1150) G Nasir al-Din (13th century) H al-Kashi and Ulugh Beg (14th century) I Bartholemaeus Pitiscus (1595)
32.
Exercise: Functions of the Form y = sin(kθ) On the same set of axes.2. k is a constant and has different effects on the graph of the function.2
32. Describe the various stages of development and how different cultures used trigonometry to improve their lives. plot the following graphs: 1. The works of the following people or cultures can be investigated: 1.1 for the function f (θ) = sin(2θ).1 History of Trigonometry
Work in pairs or groups and investigate the history of the development of trigonometry.1
Graphs of Trigonometric Functions
Functions of the form y = sin(kθ)
In the equation.Grade 11
32.5θ 381
. The general shape of the graph of functions of this form is shown in Figure 32. a(θ) = sin 0. Cultures A Ancient Egyptians B Mesopotamians C Ancient Indians of the Indus Valley 2.
The curve y = sin(x) is shown in gray. There are many x-intercepts. The range of f (θ) = sin(kθ) is {f (θ) : f (θ) ∈ [−1. Notice that in ◦ the case of the sine graph. k These different properties are summarised in Table 32. Table 32.5θ Use your results to deduce the effect of k.1. c(θ) = sin 1.1: Table summarising general shapes and positions of graphs of functions of the form y = sin(kx).
You should have found that the value of k affects the periodicity of the graph.32.5θ 4. Intercepts For functions of the form. the details of calculating the intercepts with the y axis are given.1]}. y = sin(kθ).1: Graph of f (θ) = sin(2θ) with the graph of g(θ) = sin(θ) superimposed in gray. TRIGONOMETRY . b(θ) = sin 1θ 3. The y-intercept is calculated by setting θ = 0: y yint = sin(kθ)
= sin(0) = 0 382
. the domain is {θ : θ ∈ R} because there is no value of θ ∈ R for which f (θ) is undefined.
2.GRADE 11
1
−270 −180 −90
90
180
270
−1
Figure 32. e(θ) = sin 2. k>0 k<0
Domain and Range For f (θ) = sin(kθ).2
CHAPTER 32. the period (length of one wave) is given by 360 . d(θ) = sin 2θ 5.
k>0 k<0
Domain and Range For f (θ) = cos(kθ). The range of f (θ) = cos(kθ) is {f (θ) : f (θ) ∈ [−1. The curve y = cos(x) is shown in gray.2
32.5θ 2.2
Functions of the form y = cos(kθ)
In the equation.
Exercise: Functions of the Form y = cos(kθ) On the same set of axes. k is a constant and has different effects on the graph of the function. y = cos(kθ). 1
−270 −180 −90
90
180
270
−1
Figure 32.GRADE 11
32.2 for the function f (θ) = cos(2θ). 383
. d(θ) = cos 2θ 5. The general shape of the graph of functions of this form is shown in Figure 32.5θ 4.2. plot the following graphs: 1. e(θ) = cos 2.CHAPTER 32. c(θ) = cos 1. b(θ) = cos 1θ 3. Table 32. the domain is {θ : θ ∈ R} because there is no value of θ ∈ R for which f (θ) is undefined. TRIGONOMETRY . k These different properties are summarised in Table 32.1]}.2. a(θ) = cos 0.
You should have found that the value of k affects the periodicity of the graph.2: Graph of f (θ) = cos(2θ) with the graph of g(θ) = cos(θ) superimposed in gray.5θ Use your results to deduce the effect of k. The period of ◦ the cosine graph is given by 360 .2: Table summarising general shapes and positions of graphs of functions of the form y = cos(kx).
a(θ) = tan 0. The general shape of the graph of functions of this form is shown in Figure 32.32. e(θ) = tan 2. As k decreases. TRIGONOMETRY . the graph is more tightly packed. once again. c(θ) = tan 1. the details of calculating the intercepts with the y axis are given. the graph is more spread out.2.∞)}.3 for the function f (θ) = tan(2θ). The ◦ period of the tan graph is given by 180 . d(θ) = tan 2θ 5.2 Intercepts
CHAPTER 32. y = cos(kθ). the domain of one branch is {θ : θ ∈ (− 90 . b(θ) = tan 1θ 3. plot the following graphs: 1.GRADE 11
For functions of the form.3
Functions of the form y = tan(kθ)
In the equation.3: The graph of tan(2θ) superimposed on the graph of g(θ) = tan(θ) (in gray). 384
◦ ◦
.
You should have found that. y = tan(kθ).
Exercise: Functions of the Form y = tan(kθ) On the same set of axes.
5
−360 −270 −180 −90
90
180
270
360
−5
Figure 32. As k increases. k k The range of f (θ) = tan(kθ) is {f (θ) : f (θ) ∈ (−∞. 90 )} because the function is k k ◦ ◦ undefined for θ = 90 and θ = 90 . k These different properties are summarised in Table 32. the value of k affects the periodicity of the graph.5θ 2.5θ Use your results to deduce the effect of k.3. The y-intercept is calculated as follows: y yint = cos(kθ) = cos(0) = 1
32. k is a constant and has different effects on the graph of the function. Domain and Range For f (θ) = tan(kθ). The asymptotes are shown as dashed lines.5θ 4.
GRADE 11
32.
Exercise: Functions of the Form y = sin(θ + p) On the same set of axes. tan kθ approaches infinity.3: Table summarising general shapes and positions of graphs of functions of the form y = tan(kθ). d(θ) = sin(θ + 90◦ ) 5.4: Graph of f (θ) = sin(θ + 30◦ ) with the graph of g(θ) = sin(θ) in gray. The general shape of the graph of functions of this form is shown in Figure 32.2. plot the following graphs: 2. TRIGONOMETRY . c(θ) = sin θ
4.2
Table 32. There are many x-intercepts. each one is halfway between the asymptotes.CHAPTER 32. there is no defined value of the function at the asymptote values. p is a constant and has different effects on the graph of the function. the details of calculating the intercepts with the x and y axis are given. y = sin(θ + p). 1
−270 −180 −90
90
180
270
−1
Figure 32. tan(kθ) tan(0) 0
32. b(θ) = sin(θ − 60◦ ) 1. 385
. e(θ) = sin(θ + 180◦ ) Use your results to deduce the effect of p. y = tan(kθ). In other words. The y-intercept is calculated as follows: y yint = = = Asymptotes The graph of tan kθ has asymptotes because as kθ approaches 90◦ . k>0 k<0
Intercepts For functions of the form.4 for the function f (θ) = sin(θ + 30◦ ).4
Functions of the form y = sin(θ + p)
In the equation. a(θ) = sin(θ − 90◦ )
3.
The general shape of the graph of functions of this form is shown in Figure 32.2. y = sin(θ + p). y = cos(θ + p).1]}. the graph shifts left and if p is negative tha graph shifts right. TRIGONOMETRY . plot the following graphs: 386
. The range of f (θ) = sin(θ + p) is {f (θ) : f (θ) ∈ [−1. The p value shifts the graph horizontally. the details of calculating the intercept with the y axis are given. p is a constant and has different effects on the graph of the function.2
CHAPTER 32. These different properties are summarised in Table 32.5: Graph of f (θ) = cos(θ + 30◦ ) with the graph of g(θ) = cos(θ) shown in gray. If p is positive. the domain is {θ : θ ∈ R} because there is no value of θ ∈ R for which f (θ) is undefined. Table 32.32. p>0 p<0
Domain and Range For f (θ) = sin(θ + p).5
Functions of the form y = cos(θ + p)
In the equation.4. The y-intercept is calculated as follows: set θ = 0◦ y yint = sin(θ + p) = sin(0 + p) = sin(p)
32.GRADE 11
You should have found that the value of p affects the y-intercept and phase shift of the graph.5 for the function f (θ) = cos(θ + 30◦ ). 1
−270 −180 −90
90
180
270
−1
Figure 32.
Exercise: Functions of the Form y = cos(θ + p) On the same set of axes. Intercepts For functions of the form.4: Table summarising general shapes and positions of graphs of functions of the form y = sin(θ + p).
the details of calculating the intercept with the y axis are given. y = cos(θ + p). positive values of p shift the cosine graph left while negative p values shift the graph right. b(θ) = cos(θ − 60◦ )
4.2
3.2.GRADE 11 1. a(θ) = cos(θ − 90◦ )
32.
You should have found that the value of p affects the y-intercept and phase shift of the graph.CHAPTER 32. y = tan(θ + p).6 for the function f (θ) = tan(θ + 30◦ ). These different properties are summarised in Table 32. p>0 p<0
Domain and Range For f (θ) = cos(θ + p). The general shape of the graph of functions of this form is shown in Figure 32. d(θ) = cos(θ + 90◦ ) 5. As in the case of the sine graph. The y-intercept is calculated as follows: set θ = 0◦ y yint = = = cos(θ + p) cos(0 + p) cos(p)
32. The range of f (θ) = cos(θ + p) is {f (θ) : f (θ) ∈ [−1.6
Functions of the form y = tan(θ + p)
In the equation. Table 32. p is a constant and has different effects on the graph of the function. c(θ) = cos θ
2.5: Table summarising general shapes and positions of graphs of functions of the form y = cos(θ + p). a(θ) = tan(θ − 90◦ ) 387
.5. e(θ) = cos(θ + 180◦ ) Use your results to deduce the effect of p. Intercepts For functions of the form. The curve y = cos θ is shown in gray.1]}. the domain is {θ : θ ∈ R} because there is no value of θ ∈ R for which f (θ) is undefined. plot the following graphs: 1.
Exercise: Functions of the Form y = tan(θ + p) On the same set of axes. TRIGONOMETRY .
y = tan(θ + p).6. the domain for one branch is {θ : θ ∈ (−90◦ − p.∞)}. TRIGONOMETRY . 388 = tan(θ + p) = tan(p)
.
You should have found that the value of p once again affects the y-intercept and phase shift of the graph.6: The graph of tan(θ + 30◦ ) with the graph of g(θ) = tan(θ) shown in gray. k>0 k<0
Domain and Range For f (θ) = tan(θ + p). Intercepts For functions of the form. e(θ) = tan(θ + 180◦ ) Use your results to deduce the effect of p. There is a horizontal shift to the left if p is positive and to the right if p is negative.GRADE 11
5
−360 −270 −180 −90
90
180
270
360
−5
Figure 32. c(θ) = tan θ
2. These different properties are summarised in Table 32. b(θ) = tan(θ − 60◦ )
4.90◦ − p} because the function is undefined for θ = −90◦ − p and θ = 90◦ − p. there is no defined value of the function at the asymptote values. Thus. the details of calculating the intercepts with the y axis are given. The y-intercept is calculated as follows: set θ = 0◦ y yint Asymptotes The graph of tan(θ + p) has asymptotes because as θ + p approaches 90◦ .6: Table summarising general shapes and positions of graphs of functions of the form y = tan(θ + p).
3.2
CHAPTER 32. d(θ) = tan(θ + 60◦ ) 5. tan(θ + p) approaches infinity. The range of f (θ) = tan(θ + p) is {f (θ) : f (θ) ∈ (−∞.32. Table 32.
Answer Step 1 : Write the equation so that all the terms with the unknown quantity (i. if tan θ + 0. The dotted line represents y = 0. θ) are on one side of the equation.5.4. This is true. arccos and arctan can also be used to solve trigonometric equations.5 is solved as sin θ ∴ θ = 0.7.GRADE 11
32. it does not tell the whole story.10.5 = 30◦
Worked Example 139: Question: Find θ.10: The sine graph. There are four points of intersection on this interval.5 1 arctan 1 45◦
Trigonometric equations often look very simple. Determine the solution using inverse trigonometric functions. with 0◦ < θ < 90◦ . cos−1 and tan−1 . Using inverse trigonometric functions. 401
. TRIGONOMETRY . We can take the inverse sine of both sides to find that θ = sin−1 (0.5 = tan θ = ∴ θ = = 1. If we were to extend the range of the sine graph to infinity we would in fact see that there are an infinite number of solutions to this equation! This difficulty (which is caused by the periodicity of the sine function) makes solving trigonometric equations much harder than they may seem to be. Then solve for the angle using the inverse function. there are four possible angles with a sine of 0.7) = 44.5 = arcsin 0.7 between −360◦ and 360◦. however.e. These are shown as second functions on most calculators: sin−1 .4
32.2
Algebraic Solution
The inverse trigonometric functions arcsin. y 1 x
−360
−180
180 −1
360
Figure 32.7). thus four solutions to sin θ = 0.7.7.42◦ .5 = 1. the equation sin θ = 0.CHAPTER 32. As you can see from figure 32. If we put this into a calculator we find that sin−1 (0. Consider solving the equation sin θ = 0. tan θ + 0.
Bearing this in mind we can already solve trigonometric equations within these ranges. when −360◦ ≤ x ≤ 360◦ ? Answer Step 1 : Draw the graph We take a look at the graph of sin x = 0.5.5 if it is given that x < 90◦ .5)
⇒ ⇒
∴x =
We can.
Worked Example 140: Question: Find the values of x for which sin x = 0.4
CHAPTER 32. This means that there are four solutions to the equation. Your calculator will always give you the smallest answer (i.5.GRADE 11
Any problem on trigonometric equations will require two pieces of information to solve.4) (32. so we draw in a line at y = 0. 2 Answer Because we are told that x is an acute angle.32.5.
Worked Example 141: Question: For what values of x does sin x = 0. of course.e. −210◦.5 30◦ 60◦
(32. TRIGONOMETRY . Step 3 : Read off the x values of those intercepts from the graph as x = −330◦. sin
x 2 x 2 x 2
= = =
0.2) (32.5 on the interval [−360◦. 402
. we can simply apply an inverse trigonometric function to both sides. The hard part is making sure you find all of the possible answers within the range. We want to know when the y value of the graph is 0. the one that lies between −90◦ and 90◦ for tangent and sine and one between 0◦ and 180◦ for cosine). solve trigonometric equations in any range by drawing the graph. y 1 x
−360
−180
180 −1
360
Step 2 : Notice that this line touches the graph four times. The first is the equation itself and the second is the range in which your answers must lie.3) (32.5 arcsin 0. 360◦ ]. 30◦ and 150◦.
We call them quadrants because they correspond to the four quadrants of the unit circle. only sine is positive in the 2nd .CHAPTER 32. We express 403
. 1st +VE +VE +VE 2nd +VE -VE -VE 3rd -VE -VE +VE 4th -VE +VE -VE
sin cos tan
Table 32.7 shows which trigonometric functions are positive and which are negative in each quadrant.3
Solution using CAST diagrams
The Sign of the Trigonometric Function The first step to finding the trigonometry of any angle is to determine the sign of the ratio for a given angle. Figure 32.4
1
1st
2nd
3rd
4th
90◦ 2nd +VE 1st +VE 0◦ /360◦ 3rd -VE 4th -VE 270◦
0
90 ◦
180 ◦
270 ◦
360 ◦
180◦
−1
+VE
+VE
-VE
-VE
Figure 32. We notice from figure 32.
32. A more convenient way of writing this is to note that all functions are positive in the 1st quadrant. In figure 32.
y 1 x
−360 −270 −180 −90
90
180
270
360
−1
This method can be time consuming and inexact.GRADE 11
32. All of this can be summed up in two ways. We shall now look at how to solve these problems algebraically. We shall do this for the sine function first and do the same for the cosine and tangent.11 we have split the sine graph into four quadrants. each 90◦ wide. TRIGONOMETRY .4. Table 32.12 shows similar graphs for cosine and tangent.11 that the sine graph is positive in the 1st and 2nd quadrants and negative in the 3rd and 4th . only tangent in the 3rd and only cosine in the 4th .11: The graph and unit circle showing the sign of the sine function.7: The signs of the three basic trigonometric functions in each quadrant.
32.46◦
However.
Magnitude of the trigonometric functions Now that we know in which quadrants our solutions lie. This diagram is known as a CAST diagram as the letters. for the tangent. To add multiples of the period we use 360◦ · n (where n is an integer) for sine and cosine and 180◦ · n. cosine and tangent.12: Graphs showing the sign of the cosine and tangent functions.GRADE 11
1
0
−1
8 6 4 2 0 90◦ 180◦ 270◦ 360◦ −2 −4 −6 −8 +VE +VE -VE -VE
1st
2nd
3rd
4th
1st
2nd
3rd
4th
90 ◦
180 ◦
270 ◦
360 ◦
+VE
-VE
+VE
-VE
Figure 32.of course. TRIGONOMETRY . The simpler version on the right is useful for ranges between 0◦ and 360◦ . This version is useful for equations which lie in large or negative ranges.3 17. 'C' and 'T' . The letter in each quadrant tells us which trigonometric functions are positive in that quadrant. Another useful diagram shown in figure 32. taken anticlockwise from the bottom right. n ∈ Z. cosine and tangent are all positive in this quadrant).4
CHAPTER 32. remembering that sine. we need to know which angles in these quadrants satisfy our equation.3 we can apply the inverse sine function to both sides of the equation to find– θ = = arcsin 0.13: The two forms of the CAST diagram and the formulae in each quadrant. For example. stand for sine. read C-A-S-T. The 'A' in the 1st quadrant stands for all (meaning sine. if we wish to solve sin θ = 0.13 gives the formulae to use in each quadrant when solving a trigonometric equation.
this using the CAST diagram (figure 32.
90◦
S
180
◦
A
0 /360
◦ ◦
S T
A C
180◦ − θ 180◦ + θ
θ 360◦ − θ
T
270◦
C
Figure 32. The version on the left shows the CAST diagram including the unit circle. Calculators give us the smallest possible answer (sometimes negative) which satisfies the equation. cosine and tangent are periodic (repeating) functions. and answers in other quadrants. The diagram is shown in two forms. We then look at the sign of the trigonometric function in order to decide in which quadrants we need to work (using the CAST diagram) and add multiples of the period to each.13). We get the rest of the answers by finding relationships between this small angle. To do this we use our small angle θ as a reference angle. 404
. θ. we know that this is just one of infinitely many possible answers. 'S'.
522. For example. Let us try to solve tan(2x − 10◦ ) = 2. sin θ is given as a positive amount (0. This is our reference angle. . If n = 0. Notice that we added the 10◦ and divided by 2 only at the end. θ = −342. .etc.GRADE 11
32. Step 4 : Find the specific solutions We can then find all the values of θ by substituting n = . .
32.2◦ ] 68. We therefore use θ and 180◦ − θ. If there is anything more complicated than this we need to be a little more careful.46◦ + 360◦ · n. n ∈ Z = 162. I: θ II : θ = 17. Our calculator tells us that arctan(2. .2◦ + 180◦ · n
39. We want solutions for positive tangent so using our CAST diagram we know to look in the 1st and 3rd quadrants. and add the 360◦ · n for the periodicity of sine.5) = 68.5 in the range −360◦ ≤ x ≤ 360◦ . Reference to the CAST diagram shows that sine is positive in the first and second quadrants. n ∈ Z
= 180◦ − 17. the θ in cos θ or the (2x − 7) in tan(2x − 7)).46◦ .1◦ + 90◦ · n. e.5 [68.4.46◦
180◦ − θ 180◦ + θ θ 360◦ − θ
S T
A C
Step 3 : Determine the general solution Our solution lies in quadrants I and II. θ = 17.54◦ If n = −1.3).46◦. n ∈ Z
This is called the general solution.g.CHAPTER 32. n ∈ Z
This is the general solution. θ = 377. .54◦ If n = 1.3 arcsin 0.46◦ We can find as many as we like or find specific solutions in a given interval by choosing more values for n. 162. TRIGONOMETRY . This is also divided by 2 in the last step to keep the equation balanced.2◦ + 180◦ · n 78.54◦ + 360◦ · n. Step 2 : Determine the reference angle The small angle θ is the angle returned by the calculator: sin θ ⇒θ ⇒θ = = = 0.3 Answer Step 1 : Determine in which quadrants the solution lies We look at the sign of the trigonometric function. Notice that we added 180◦ · n because the tangent has a period of 180◦ .4
General Solution Using Periodicity
Up until now we have only solved trigonometric equations where the argument (the bit after the function. − 1. has been θ. 1. .0. −197.46◦ + 360◦ · n. So to find the general solution we proceed as follows: tan(2x − 10◦ ) = I : 2x − 10◦ = 2x = x = 2. We chose quadrants I and III because tan 405
.4
Worked Example 142: Question: Solve for θ: sin θ = 0.54◦. 2.2◦.3 17.
7
More Complex Trigonometric Equations
Here are two examples on the level of the hardest trigonometric equations you are likely to encounter. Step 2 : Solve the quadratic equation Factorising yields (2y + 1)(y − 1) = 0 ∴ y = −0. let y = cos x.5 or cos x = 1 Both equations are valid (i. reject the second equation. Consider solving tan2 (2x + 1) + 3 tan (2x + 1) + 2 = 0 Here we notice that tan(2x + 1) occurs twice in the equation.CHAPTER 32. In that case you need to discard that solution. The solutions to either of these equations will satisfy the original quadratic. So.4. General solution: 407
. If you can solve these. Only solutions to the first equation will be valid. Then we have 2y 2 − y − 1 = 0 Note that with practice you may be able to leave out this step. TRIGONOMETRY .4
This gives two linear trigonometric equations. For example consider the same equation with cosines instead of tangents cos2 (2x + 1) + 3 cos (2x + 1) + 2 = 0 Using the same method we find that cos (2x + 1) = −1 or cos (2x + 1) = −2
The second solution cannot be valid as cosine must lie between −1 and 1.5 or y=1 Step 1 : Substitute back and solve the two resulting equations We thus get cos x = −0. 360◦ ] Answer Step 1 : Use a temporary variable We note that cos x occurs twice in the equation.e.
32. lie in the range of cosine). They require using everything that you have learnt in this chapter. The next level of complexity comes when we need to solve a trinomial which contains trigonometric functions. We can immediately write down the factorised form and the solutions: (y + 1)(y + 2) = 0 ⇒ y = −1 OR y = −2
Next we just substitute back for the temporary variable: tan (2x + 1) = −1 or tan (2x + 1) = −2
And then we are left with two linear trigonometric equations. Be careful: sometimes one of the two solutions will be outside the range of the trigonometric function. We must.GRADE 11
32. you should be able to solve anything!
Worked Example 144: Question: Solve 2 cos2 x − cos x − 1 = 0 for x ∈ [−180◦. therefore. hence we let y = tan(2x + 1) and rewrite: y 2 + 3y + 2 = 0 That should look rather more familiar. It is much easier in this case to use temporary variables.
TRIGONOMETRY . The two lighthouses are 0.GRADE 11 C
b
h
a
A
c
B
The area of △ABC can be written as: area △ABC = ˆ ˆ However. The lighthouses tell how close a boat is by taking bearings to the boat (remember – a bearing is an angle measured clockwise from north). we get: 2 ˆ ˆ ˆ sin B sin C sin A = = a b c This is known as the sine rule and applies to any triangle.67 km apart and one is exactly due east of the other. Use the sine rule to calculate how far the boat is from each lighthouse. h can be calculated in terms of A or B as: ˆ sin A = ∴ and ˆ sin B ∴ Therefore the area of △ABC is: 1 1 ˆ 1 ˆ c · h = c · b · sin A = c · a · sin B 2 2 2 Similarly.5
CHAPTER 32. h = = h a ˆ a · sin B h h b ˆ = b · sin A 1 c · h. by drawing the perpendicular between point B and line AC we can show that: 1 ˆ 1 ˆ c · b · sin A = a · b · sin C 2 2 Therefore the area of △ABC is: 1 1 ˆ 1 ˆ ˆ c · b · sin A = c · a · sin B = a · b · sin C 2 2 2 If we divide through by 1 a · b · c. one on either side of a beach. These bearings are shown. 410
.32. 2
Worked Example 146: Lighthouses Question: There is a coastline with two lighthouses.
67km) sin(15◦ ) sin(128◦ ) 0.GRADE 11
32.67km) sin(37◦ ) = sin(128◦ ) = 0.5
A
127◦
B
255◦
C
Answer We can see that the two lighthouses and the boat form a triangle. We can use the sine rule to find our missing lengths. Show that ˆ ˆ ˆ sin A sin B sin C = = a b c 411
. BC ˆ sin A BC AB ˆ sin C
= =
ˆ AB · sin A ˆ sin C (0. the distance of the boat from the two lighthouses. Since we know the distance between the lighthouses and we have two angles we can use trigonometry to find the remaining two sides of the triangle. TRIGONOMETRY .67 km 15
◦
B
C We need to know the lengths of the two sides AC and BC.CHAPTER 32.51 km
AC ˆ sin B AC
= = = =
AB ˆ sin C ˆ AB · sin B ˆ sin C (0.22 km
Exercise: Sine Rule 1. A 37◦ 128◦ 0.
In △ABC.
a2 = (c − d)2 + h2
(32. b is the side opposite B and c is the side opposite C. ˆ ˆ 5.
32. Consider △ABC which we will use to show that: ˆ a2 = b2 + c2 − 2bc cos A. C = 32◦ and AC = 23 m.5 is equivalent to:
CHAPTER 32. M = 100◦ and s = 23 cm.5.GRADE 11
b c = = . A = 116◦. Find the length of the side AB. ˆ ˆ ˆ sin A sin B sin C Note: either of these two forms can be used. C = 70. C
b
h
a
A
d
D c
c-d
B
In △DCB: from the theorem of Pythagoras. Find the length of the side m.3 ˆ = 56◦ .
The cosine rule relates the length of a side of a triangle to the angle opposite it and the lengths of the other two sides.7)
. M = 50◦ and m = 1 ˆ B △KLM in which K ◦ ˆ ˆ C △ABC in which A = 32.7 . S = 30◦ and RT = 120 km. R = 24◦ and r = 3. In △ACD: from the theorem of Pythagoras. 2. R = 19◦ . Find all the unknown sides and angles of the following triangles: ˆ ˆ A △PQR in which Q = 64◦ . Find the length of the side ST. In △KMS. ˆ ˆ 4. TRIGONOMETRY . K = 20◦ . ˆ = 43◦ .2
The Cosine Rule
Definition: The Cosine Rule The cosine rule applies to any triangle and states that: a2 b2 c2 = = = ˆ b2 + c2 − 2bc cos A 2 2 ˆ c + a − 2ca cos B 2 2 ˆ a + b − 2ab cos C
ˆ ˆ ˆ where a is the side opposite A.5◦ and a = 52.32. In △RST. Z = 40◦ and x = 50 ˆ D △XYZ in which X
a
ˆ ˆ 3.6)
b2 = d2 + h2 412
(32.
In order to eliminate d we look at △ACD, where we have: ˆ d cos A = . b So, Substituting this into (32.8), we get: ˆ a2 = b2 + c2 − 2bc cos A The other cases can be proved in an identical manner. (32.9) ˆ d = b cos A.
Definition: The Area Rule The area rule applies to any triangle and states that the area of a triangle is given by half the product of any two sides with the sine of the angle between them. That means that in the △DEF , the area is given by: A= 1 1 1 ˆ ˆ ˆ DE · EF sin E = EF · F D sin F = F D · DE sin D 2 2 2 F
E D In order show that this is true for all triangles, consider △ABC. C
b
h
a
A
c
B
The area of any triangle is half the product of the base and the perpendicular height. For △ABC, this is: 1 A = c · h. 2 ˆ as: However, h can be written in terms of A ˆ h = b sin A So, the area of △ABC is: 1 ˆ c · b sin A. 2
A=
Using an identical method, the area rule can be shown for the other two angles. 414
Exercise: The Area Rule Draw sketches of the figures you use in this exercise. 1. Find the area of △PQR in which: ˆ A P = 40◦ ; q = 9 and r = 25 ˆ B Q = 30◦ ; r = 10 and p = 7 ˆ C R = 110◦ ; p = 8 and q = 9 2. Find the area of: ˆ A △XYZ with XY= 6 cm; XZ= 7 cm and Z = 28◦ ˆ B △PQR with PR= 52 cm; PQ= 29 cm and P = 58,9◦ ˆ C △EFG with FG= 2,5 cm; EG= 7,9 cm and G = 125◦ 3. Determine the area of a parallelogram in which two adjacent sides are 10 cm and 13 cm and the angle between them is 55◦ . 4. If the area of △ABC is 5000 m2 with a = 150 m and b = 70 m, what are the ˆ two possible sizes of C?
1. Q is a ship at a point 10 km due South of another ship P. R is a ˆ ˆ lighthouse on the coast such that P = Q = 50◦ . Determine: A the distance QR B the shortest distance from the lighthouse to the line joining the two ships (PQ).
3. On a flight from Johannesburg to Cape Town, the pilot discovers that he has been flying 3◦ off course. At this point the plane is 500 km from Johannesburg. The direct distance between Cape Town and Johannesburg airports is 1 552 km. Determine, to the nearest km: A The distance the plane has to travel to get to Cape Town and hence the extra distance that the plane has had to travel due to the pilot's error. B The correction, to one hundredth of a degree, to the plane's heading (or direction).
A x a B
However the distance cannot be determined directly as a ridge lies between the two points. β. A surveyor is trying to determine the distance between points X and Z. TRIGONOMETRY . Find the area of WXYZ (to two decimal places):
X 3. Find the area of the shaded triangle in terms of x.GRADE 11 5.6
x
Y θ Z
3
Z
4
Y
7. From a point Y which is equidistant from X ˆ and Z.CHAPTER 32. he measures the angle XYZ ˆ A If XY= x and XYZ = θ. show that XZ= X x 2(1 − cos θ) B Calculate XZ (to the nearest kilometre) if x = 240 km and θ = 132◦ 6.5 W 120◦
32. θ and φ:
A x φ B
θ E D
β
α C
417
. α.
6
CHAPTER 32.32.GRADE 11
418
. TRIGONOMETRY .
33. .1. Measures of central tendency (mean.xn }. semi-inter-quartile range. with mean x (read as "x bar").for example. . . The behaviour of the entire data set is therefore not examined.
33. percentiles. . 419
. The ¯ variance of the population. you have played dice games or card games even before you came to school. ranges) provide information on the data values at the centre of the data set and provide information on the spread of the data. Your basic understanding of probability and chance gained so far will be deepened to enable you to come to a better understanding of how chance and uncertainty can be measured and understood. because data are sometimes unscrupulously misused and abused in order to try to prove or support a viewpoint.Chapter 33
Statistics .2. The graph represents the results of 100 tosses of a fair coin. You will also learn how to interpret data.2
Standard Deviation and Variance
The measures of central tendency (mean. 2 Population Variance Let the population consist of n elements {x1 . median and mode) and dispersion (range. which resulted in 45 heads and 55 tails.g.Grade 11
33. denoted by σ 2 . you will end up mastering further methods of collecting. and not always to accept the data at face value. The two important measures that are used are called the variance and the standard deviation of the data set. e.1
Variance
The variance of a data set is the average squared distance between the mean of the data set and each data value. Of course. quartiles. the activities involving probability will be familiar to most of you .x2 .1 Introduction
This chapter gives you an opportunity to build on what you have learned in previous Grades about data handling and probility. which is 1 (25 + 25) = 25. displaying and analysing data. The average of these two squared distances gives the variance. The work done will be mostly of a practical nature. Through problem solving and activities. organising. The squared distance between the heads value and the mean is (45 − 50)2 = 25 and the squared distance between the tails value and the mean is (55 − 50)2 = 25. the end points for range and data points that divide the data set into 4 equal groups for the quartiles. is the average of the square of the distance of each data value from the mean value. median and mode) and measures of dispersion (quartiles. percentiles. An example of what this means is shown in Figure 33. inter-quartile. A method of determining the spread of data is by calculating a measure of the possible distances between the data and the mean. The mean of the results is 50. variance and standard deviation) will be investigated. The information on the spread of the data is however based on data values at specific points in the data set.
n−1 (33. Sample Variance Let the sample consist of the n elements {x1 . The mean value of the tosses is shown as a vertical dotted line. with mean x.x2 . . STATISTICS . Numbers are going to fall above and below the mean and. Difference between Population Variance and Sample Variance As seen a distinction is made between the variance. denoted by s2 . The variance of the sample. For example. s2 of a sample extracted from the population. σ 2 . a parameter which helps to describe the population. . it is also not directly comparable with the mean and the data themselves. since the variance is looking for distance.2 60 55 50 45 40 35 30 25 20 15 10 5 0
CHAPTER 33. When dealing with a sample from the population the (sample) variance varies from sample to sample.xn }. it is not directly comparable with the mean and the data themselves.2)
Since the sample variance is squared. it would be counterproductive if those distances factored each other out. The difference between the mean value and each data value is shown. we can conclude that it is never negative because the squares are positive or zero. the 420
.GRADE 11
Tails-Mean Heads-Mean
Frequency (%)
Heads Tails Face of Coin
Figure 33.
σ2 =
(
(x − x))2 ¯ . with 45 heads and 55 tails.1)
Since the population variance is squared. is the average of the squared deviations from the ¯ sample mean: s2 = (x − x)2 ¯ . . The unit of variance is the square of the unit of observation. A common question at this point is "Why is the numerator squared?" One answer is: to get rid of the negative signs. . of a whole population and the variance. taken from the population. Its value is only of interest as an estimate for the population variance.33. n
(33. When dealing with the complete population the (population) variance is a constant.1: The graph shows the results of 100 tosses of a fair coin. Properties of Variance If the variance is defined.
3)
Sample Standard Deviation Let the sample consist of n elements {x1 . it cannot be directly compared to the data values or the mean value of a data set. . ¯ The standard deviation of the sample.x2 . For example to calculate the standard deviation of 57.2
Standard Deviation
Since the variance is a squared quantity. In statistics. subtract each number from the mean. denoted by σ. with mean x. It is therefore more useful to have a quantity which is the square root of the variance.
33. The standard deviation ¯ of the population. . For example. Standard deviation measures how spread out the values in a data set are. . . STATISTICS . .GRADE 11
33. as a summary of dispersion. 48. The standard deviation is always a positive number and is always measured in the same units as the original data. 53.4)
It is often useful to set your data out in a table so that you can apply the formulae easily. known as the standard deviation.xn }. (x − x)2 ¯ n
σ=
(33. the standard deviation will also be measured in metres. then the standard variation is high (further from zero). taken from the population. If the data values are highly variable. 50. is the square root of the average of the square of the distance of each data value from the mean value. 58. This fact is inconvenient and has motivated many statisticians to instead use the square root of the variance.2. If the data values are all similar.xn }. .CHAPTER 33. . More precisely. 421
.
Population Standard Deviation Let the population consist of n elements {x1 . you could set it out in the following way: sum of items number of items x n 448 6 56
mean = = = =
Note: To get the deviations.2
variance of a set of heights measured in centimeters will be given in square centimeters. 51. with mean x. denoted by s. 66. the standard deviation is the most common measure of statistical dispersion. then the standard deviation will be low (closer to zero). if the data are distance measurements in metres. 65.x2 . . is the square root of the average of the squared deviations from the sample mean: (x − x)2 ¯ n−1
s=
(33. This quantity is known as the standard deviation. it is a measure of the average distance between the values of the data in the set.
33. When dealing with a sample from the population the (sample) standard deviation varies from sample to sample. there is a distinction between the standard deviation.σ. Find the average of the squared differences. Why is this? Find out. for any set of data. Take the square root of the variance to obtain the standard deviation. that ¯ is (X − X) = 0.32
Difference between Population Variance and Sample Variance As with variance. Variance = ¯ (X − X)2 n 320 = 8 = 40
Standard deviation = = = = =
√ variance ¯ (X − X)2 n 320 8 √ 40 6. 5. When dealing with the complete population the (population) standard deviation is a constant. σ 2 . Calculate the mean value x. STATISTICS . ¯ ¯ 3. the standard deviation can be calculated as follows: 1.
Worked Example 149: Variance and Standard Deviation 422
. of a whole population and the standard deviation. s. of sample extracted from the population. Calculate the variance (add the squared results together and divide this total by the number of items). 4. In other words. This always happens. ¯ 2. σ. Calculate the squares of these differences. a parameter which helps to describe the population. This quantity is the variance. For each data value xi calculate the difference xi − x between xi and the mean value x.2 ¯ X 57 53 58 65 48 50 66 51 X = 448 ¯ Deviation (X − X) 1 -3 2 9 -8 -6 10 -5 x=0
CHAPTER 33.GRADE 11 ¯ Deviation squared (X − X)2 1 9 4 81 64 36 100 25 ¯ (X − X)2 = 320
Note: The sum of the deviations of scores about their mean is zero.
In this case.25 + 0.2. 5 and 1.3
Interpretation and Application
A large standard deviation indicates that the data values are far from the mean and a small standard deviation indicates that they are clustered closely around the mean. 14. In physical science for example.25 0.708.5
Step 5 : Calculate the standard deviation The (population) standard deviation is calculated by: σ = = 2. STATISTICS .CHAPTER 33. The value of the standard deviation can be considered 'large' or 'small' only in relation to the sample that is being measured. When deciding whether measurements agree with a theoretical prediction. For example. Their standard deviations are 7. and n=6. respectively.2. the population consists of 6 possible outcomes.5.25 2.3. 6. the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard 423
.25 + 2. Given a different sample. a standard deviation of 7 might be considered small.5 -1.25 + 6. The data set is therefore x = {1. 0.5 x=0 ¯ (X − X)2 6.GRADE 11 Question: What is the variance and standard deviation of the population of possibilities associated with rolling a fair die? Answer Step 1 : Determine how many outcomes make up the population When rolling a fair die. (0.25) 6 2.25 + 0. Step 2 : Calculate the population mean The population mean is calculated by: x = ¯ = 1 (1 + 2 + 3 + 4 + 5 + 6) 6 3.25 0. 8) has a mean of 7. the reported standard deviation of a group of repeated measurements should give the precision of those measurements.4.5 2.
Notice how this standard deviation is somewhere in between the possible deviations. 14). 8.25 2. and (6. Standard deviation may be thought of as a measure of uncertainty.
33. each of the three samples (0.25 6. 8. 14). a standard deviation of 7 may be considered large.5 0. 6.5 1.25 ¯ (X − X)2 = 17.6}. The third set has a much smaller standard deviation than the other two because its values are all close to 7.917
Step 4 : Alternately the population variance is calculated by: ¯ X 1 2 3 4 5 6 X = 21 ¯ (X − X) -2.917 1.25 + 2.5
33.2
Step 3 : Calculate the population variance The population variance is calculated by: σ2 = = = (x − x)2 ¯ n 1 (6.5 -0.
A When is the data seen as a population? B When is the data seen as a sample?
33. The raw data. the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. 420. C Assuming the data is a sample find the standard deviation of each city's prices.3. 130. 700. STATISTICS .52 3.3
Graphical Representation of Measures of Central Tendency and Dispersion
The measures of central tendency (mean. mode) and the measures of dispersion (range.69 3.2. 110. This makes sense since they fall outside the range of values that could reasonably be expected to occur if the prediction were correct and the standard deviation appropriately quantified. are given below: Cape Town Durban 3.4
Relationship between Standard Deviation and the Mean
The mean and the standard deviation of a set of data are usually reported together.96 3. 80. B Assuming that the data is a population find the standard deviation of each city's prices. in rands per litre.GRADE 11
deviations).
33. 220. A When is the data seen as a population? B When is the data seen as a sample? 4.
33. 250. This section presents methods of representing the summarised data using graphs. See prediction interval. 500. median. What is the standard deviation? 3. quartiles. Consider a set of data that gives the weights of 50 cats at a cat show. This is because the standard deviation from the mean is smaller than from any other point. 140. third quartile and maximum. D Giving reasons which city has the more consistently priced petrol? 2.97 3. 320. 270.81 4. percentiles.91 4.08 3. then we consider the measurements as contradicting the prediction. 200. 150. In a certain sense. Consider a set of data that gives the results of 20 pupils in a class.1
Five Number Summary
One method of summarising a data set is to present a five number summary.68
A Find the mean price in each city and then state which city has the lowest mean. 300. first quartile. The five numbers are: minimum. inter-quartile range) are numerical methods of summarising data.33.3
CHAPTER 33.88 3. semi-inter-quartile range.
Exercise: Means and standard deviations 1. median. 424
.76 3. Bridget surveyed the price of petrol at petrol stations in Cape Town and Durban. The following data represents the pocket money of a sample of teenagers.72 3.00 3.
4. first quartile median third quartile
minimum 1 2 3 4 Data Values
maximum 5
425
.775. Answer Step 1 : Determine the five number summary Minimum = 1. Inside the box there is some representation of central tendency. 4.25. centered in the box in the vertical direction. graphically. 2. The height of the box is arbitrary.5. as there is no y-axis.3
33. first quartile minimum data value -4 -2 0 Data Values 2 median third quartile maximum data value 4
Figure 33. 4.775 2 The five number summary is therefore: 1. 1. 4.25 Maximum = 4.5.GRADE 11
33.5 + 2.2
Box and Whisker Diagrams
A box and whisker diagram is a method of depicting the five number summary. 4.65 2 Data value between 9 and 10 = 1 (4.5) = 2.8. The box can lie horizontally (as shown) or vertically.25. 2. The whiskers which extend to the sides reach the minimum and maximum values. For a horizonatal diagram.1) = 3. 3.1. Step 2 : Draw a box and whisker diagram and mark the positions of the minimum. maximum and quartiles.5 2 Data value between 6 and 7 = 1 (3. Additionally. 4. 3. 4.95. 3.25.1}.75 + 4. with the median shown with a vertical line dividing the box into two.8) = 4. 5.5.2 + 4.2.CHAPTER 33.65.75. STATISTICS .2: Main features of a box and whisker diagram
Worked Example 150: Box and Whisker Diagram Question: Draw a box and whisker diagram for the data set x = {1.95 Position of first quartile = between 3 and 4 Position of second quartile = between 6 and 7 Position of third quartile = between 9 and 10 Data value between 3 and 4 = 1 (2. The main features of the box and whisker diagram are shown in Figure 33.95.1.2.3.5. a star or asterix is placed at the mean value. 2. the left edge of the box is placed at the first quartile and the right edge of the box is placed at the third quartile.
31. The cumulative frequency is calculated from a frequency table. What is the variance and standard deviation? Comment on the data's spread. 7 A What is the five-number summary of the data? B Since there is an odd number of data points what do you observe when calculating the five-numbers?
33. 60. 12. 8. 18. 11. 3. 5. 2. 20. 17. 22. 54. are a plot of cumulative frequency and are used to determine how many data values lie above or below a particular value in a data set.33. 22. 48. 24. Lisa works as a telesales person. 9 12.GRADE 11
1. 3. 45. also known as ogives. He sells the following number of computers each month: 27. etc have been plotted.2 and is drawn in Figure 33. 9. 5. 3. 12. 2 25. 49. Using an appropriate graphical method (give reasons) represent the data. Calculate the five number summary and make a box and whisker plot. 7. 34. 13. 3. 6. Find the median. 43. 3. 12. The cumulative frequency is plotted at the upper limit of the interval.3 Exercise: Box and whisker plots
CHAPTER 33. 23. Jason is working in a computer store.3). 23. 2. 14. 65. 17 14.3
Cumulative Histograms
Cumulative histograms. 11.
Exercise: Intervals 426
.2) (97. 19. 6. 19. by adding each frequency to the total of the frequencies of all data values before it in the data set. so the points (30. 11. Where are 95% of the results expected to lie?
4. Rose has worked in a florists shop for nine months. She sold the following number of wedding bouquets: 16. 1. since all frequencies will already have been added to the previous total. The number of rugby matches attended by 36 season ticket holders is as follows: 15. 13. 40 A B C D E F G Sum the data. This is different from the frequency polygon where we plot frequencies at the midpoints of the intervals. the cumulative frequencies for Data Set 2 are shown in Table 33. 16. 1.3. The data below show how much she sells each month. mode and mean. 2. 15. 8. For example. 39.1) (62. 12 Give a five number summary and a box and whisker plot of her sales. Notice the frequencies plotted at the upper limit of the intervals. 5.3. 26. 2. She keeps a record of the number of sales she makes each month. 45. 27. 43. 16 Give a five number summary and a box and whisker plot of his sales. 15. STATISTICS . 35. 5. The last value for the cumulative frequency will always be equal to the total number of data values.
4
33.4.
skewed right
skewed left
33. B For each of the sets calculate the mean and the five number summary. median. and mode are approximately equal to each other.median) > 0 then the data is positively skewed (skewed to the right). A Make a stem and leaf plot for each set.GRADE 11
33.2
Relationship of the Mean. the distribution can be assumed to be approximately symmetrical. With both the mean and median known the following can be concluded: • (mean . median. It can be skewed right or skewed left. This means that the median is close to the start of the data set. and mode to each other can provide some information about the relative shape of the data distribution. • (mean .median) < 0 then the data is negatively skewed (skewed to the left).4
CHAPTER 33.1
Distribution of Data
Symmetric and Skewed Data
The shape of a data set is important to know. Definition: Shape of a data set This describes how the data is distributed relative to the mean and median.median) ≈ 0 then the data is symmetrical • (mean . The test was out of 50. STATISTICS .33. It does not have to be exactly equal to be symmetric
• Skewed data is spread out on one side more than on the other.4. and Mode
The relationship of the mean. This means that the median is close to the end of the data set. Three sets of 12 pupils each had test score recorded.
Exercise: Distribution of Data 1.
• Symmetrical data sets are balanced on either side of the median. Median. 428
. Use the given data to answer the following questions. C For each of the classes find the difference between the mean and the median and then use that to make box and whisker plots on the same set of axes. If the mean.
3 2.1 1.4 0.1 4.9 3.6 3 1.4 2.8 2.2 0.4 0.8 1.4. Ohm's law describes the relationship between current and voltage in a conductor.CHAPTER 33.6 0.4 3.2: Cumulative Frequencies for Data Set 2. E Is set A skewed or symmetrical? F Is set C symmetrical? Why or why not? 2. Table 33.GRADE 11 Set 1 25 47 15 17 16 26 24 27 22 24 12 31 Set 2 32 34 35 32 25 16 38 47 43 29 18 25 Set 3 43 47 16 43 38 44 42 50 50 44 43 42
33.2 1. but one is skewed right and the other is skewed left.6 1 4 2. For example.4 2.9 0.2 2 1 1 3. we could have mass on the horizontal axis (first variable) and height on the second axis (second variable).6 0. Current Voltage Current Voltage 0 0.9 0.5
Table 33. or we could have current on the horizontal axis and voltage on the vertical axis.4 0. we would have to say that a straight line best describes this data.5 2.5
Scatter Plots
A scatter-plot is a graph that shows the relationship between two variables.3: Values of current and voltage measured in a wire.2 0.1 1.6 2. When we measure the voltage (dependent variable) that results from a certain current (independent variable) in a wire. we get the data points as shown in Table 33.1 4. Sketch the box and whisker plots and then invent data (6 points in each set) that meets the requirements.8 0. STATISTICS .6 2.4 2 1.4 1.4 1.3 4. like a piece of wire. 429
.2 2.7 3. we get the scatter plot shown in Figure 33.6 1.4 1. Two data sets have the same range and interquartile range. Ohm's Law is an important relationship in physics.3. If we are to come up with a function that best describes the data.5
When we plot this data as points. We say this is bivariate data and we plot the data from two different sets using ordered pairs.
33.9 1.6 2.
D State which of the three are skewed (either right or left).8 1.
B What pattern or trend do you observe? 3. Number of sweets (per week) 15 12 5 3 18 23 11 4 A B C D Average sleeping time (per day) 4 4. C Plot the data pairs D What do you observe about the plot? E Is there any pattern emerging? 2. Score (percent) 67 55 70 90 45 75 50 60 84 30 66 96 Time spent studying (minutes) 100 85 150 180 70 160 80 90 110 60 96 200
33. the cause or independent variable and the effect or dependent variable. STATISTICS .5 8 8.GRADE 11 Exercise: Scatter Plots 1.CHAPTER 33. Eight childrens sweet consumption and sleep habits were recorded. The data is given in the following table. Practice time (min) 154 390 130 70 240 280 175 103 Ranking 5 1 6 8 3 2 4 7
A Construct a scatter plot and explain how you chose the dependent (cause) and independent (effect) variables. The results are given below.5
A Draw a diagram labelling horizontal and vertical axes.5 3 2 5 8
What is the dependent (cause) variable? What is the independent (effect) variable? Construct a scatter plot of the data. What trend do you observe?
431
. The rankings of eight tennis players is given along with the time they spend practising. A class's results for a test were recorded along with the amount of time spent studying for it. B State with reasons.
GRADE 11
33. Focus on particular research questions
4. Omissions and biased selection of data
3. Selection of groups
Activity :: Investigation : Misuse of statistics
1.
16 14 12 10
earnings
8 6 4 2 0 2002
2003
2004
432
. Graphs need to be carefully analysed and questions must always be asked about 'the story behind the figures.' Common manipulations are:
1. Examine the following graphs and comment on the effects of changing scale.6
CHAPTER 33. Changing the scale to change the appearence of a graph
2.6
Misuse of Statistics
Statistics can be manipulated in many ways that can be misleading.33. STATISTICS .
STATISTICS .00
3. R14 300.7
CHAPTER 33. The following two diagrams (showing two schools contribution to charity) have been exaggerated. F Using the above information work out which bonus is more beneficial for the teachers. how many salaries are within one standard deviation of the mean. The monthly income of eight teachers are given as follows: R10 050.
R200.33. R12 140. B Use your graph to find the median speed and the interquartile range. R15 000. R12 900. R11 990. R9 800.00
R100 R100 R100 R200. A What is the mean income and the standard deviation? B How many of the salaries are within one standard deviation of the mean? C If each teacher gets a bonus of R500 added to their pay what is the new mean and standard deviation? D If each teacher gets a bonus of 10% on their salary what is the new mean and standard deviation? E Determine for both of the above. Explain how they are misleading and redraw them so that they are not misleading. R13 800.
436
. C What percent of cars travel more than 120km/h on this road? D Do cars generally exceed the speed limit?
2.GRADE 11 A Draw a graph to illustrate this information.
for each event The probability of rolling a 1 is 1 and the probability of rolling a 6 is 1 . is given by: P (A ∩ B) = P (A) × P (B) (34. 6 6 Therefore. P (A ∩ B). 437
. This chapter describes the differences and how each type of event is worked with.1 Introduction
In probability theory an event is either independent or dependent.1)
Worked Example 151: Independent Events Question: What is the probability of rolling a 1 and then rolling a 6 on a fair die? Answer Step 1 : Identify the two events and determine whether the events are independent or not Event A is rolling a 1 and event B is rolling a 6. the events are independent.Chapter 34
Independent and Dependent Events .
34. it doesn't affect the other one happening or not. Since the outcome of the first event does not affect the outcome of the second event. 6 6 Step 3 : Use equation 34. P (A) = 1 and P (B) = 1 .
Definition: Independent Events Two events A and B are independent if when one of them happens.
The probability of two independent events occurring. the event of getting a "1" when a die is rolled and the event of getting a "1" the second time it is thrown are independent. Step 2 : Determine the probability of the specific outcomes occurring.1 to determine the probability of the two events occurring together. For example.Grade 11
34.2
Definitions
Two events are independent if knowing something about the value of one event does not give any information about the value of the second event.
Step 2 : Determine the probability of the specific outcomes occurring.GRADE 11
P (A ∩ B)
= P (A) × P (B) 1 1 = × 6 6 1 = 36
1 36 . Since the outcome of the first event affects the outcome of the second event (because there are less coins to choose from after the first coin has been selected). 1 Therefore.
34.
The probability of rolling a 1 and then rolling a 6 on a fair die is
Consequently.34. there are only three 3 coins to choose from). two events are dependent if the outcome of the first event affects the outcome of the second event.1
Identification of Independent and Dependent Events
Use of a Contingency Table A two-way contingency table (studied in an earlier grade) can be used to determine whether events are independent or dependent. 1 R1 coin. P (A ∩ B) = P (A) × P (B) 1 2 = × 4 3 2 = 12 1 = 6
1 The probability of first selecting a R1 coin followed by selecting a R2 coin is 6 . 3 Step 3 : Use equation 34. P (A) = 4 and P (B) = 2 . INDEPENDENT AND DEPENDENT EVENTS .2. The same equation as for independent events are used. the events are dependent. 2 R2 coins and 1 R5 coin. 438
.2
CHAPTER 34.1 to determine the probability of the two events occurring together. but the probabilities are calculated differently. What is the probability of first selecting a R1 coin followed by selecting a R2 coin? Answer Step 1 : Identify the two events and determine whether the events are independent or not Event A is selecting a R1 coin and event B is next selecting a R2.
Worked Example 152: Dependent Events Question: A cloth bag has 4 coins. for each event The probability of first selecting a R1 coin is 1 and the probability of next selecting 4 a R2 coin is 2 (because after the R1 coin has been selected.
Was the medication's succes independent of gender? Explain.
For example we can draw and analyse a two-way contingency table to solve the following problem. Give a table for the independent of gender results.
439
. So P(A/C)=P(A). 120 males and 90 females responded.2
Definition: two-way contingency table A two-way contingency table is used to represent possible outcomes when two events are combined in a statistical analysis.P(positive result)= 210 = 0.GRADE 11
34. Answer Step 1 : Draw a contingency table Male 50 70 120 Female 40 50 90 Totals 90 120 210
Positive result No Positive result Totals
Step 2 : Work out probabilities 120 P(male). This leads to the following table: Positive result No Positive result Totals Male 22 98 120 Female 68 22 90 Totals 90 120 210
Use of a Venn Diagram We can also use Venn diagrams to check whether events are dependent or independent. Step 4 : Gender-independent results To get gender independence we need the positve results in the same ratio as the gender. or 4:3. The gender ratio is: 120:90.P(positive result)= 210 = 0. These two are quite different. Out of these 50 males and 40 females responded positively to the medication. INDEPENDENT AND DEPENDENT EVENTS .CHAPTER 34.57 90 P(female).24 Step 3 : Draw conclusion P(male and positive result) is the observed probability and P(male). 1.
Worked Example 153: Contingency Tables Question: A medical trial into the effectiveness of a new medication was carried out. So there is no evidence that the medications success is independent of gender. 2.43 50 P(male and positive result)= 210 = 0. where P(A/C) represents the probability of event A after event C has occured. so the number in the male and positive column would have to be 4 of the total number of patients responding positively 7 which gives 22.P(positive result) is the expected probability.
Definition: Independent events Events are said to be independent if the result or outcome of the event does not affect the result or outcome of another event.
it may be of some importance to most citizens to understand how odds and probability assessments are made. Accordingly. the probabilities are not assessed independently nor necessarily very rationally.10 chance of not meeting their quota on time.which have ripple effects in the economy as a whole. Governments typically apply probability methods in environmental regulation where it is called "pathway analysis". In each of the following contingency tables give the expected numbers for the events to be perfectly independent and decide if the events are independent or dependent. as typically the assessments of risk are one-time and thus require more fundamental probability models. Black hair Red hair Totals Brown eyes 50 70 120 Not Brown eyes 30 80 110 Point B 40 20 60 Totals 80 150 230 Totals 55 35 100 Totals 160 340 500 Totals 700 260 960
A
B
Busses left late Busses left on time Totals
Point A 15 25 40 Durban 130 140 270
C
Liked living there Did not like living there Totals
Bloemfontein 30 200 230
D
Improvement in health No improvement in health Totals
Multivitamin A 400 140 540
Multivitamin B 300 120 420
2. It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society.3
End of Chapter Exercises
1. especially in a democracy. A good example is the application of game theory. less likely sends prices up or down. itself based strictly on probability. Use a Venn diagram and a contingency table to show the information and decide if the events are independent.
34. which makes probability measures a political matter.CHAPTER 34. Also there is a 0. INDEPENDENT AND DEPENDENT EVENTS . An assessment by a commodity trade that a war is more likely vs. A company has a probability of 0. and how they contribute to reputations and to decisions.4 of meeting their quota on time and a probability of 0. 441
. Many consumer products. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing. on policy. and signals other traders of that opinion.25 of meeting their quota late.GRADE 11 Extension: Applications of Probability Theory Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices . A law of small numbers tends to apply to all such choices and perception of the effect of such choices. It is not correct to say that statistics are involved in the modelling itself. such as automobiles and consumer electronics.3
34. Another significant application of probability theory in everyday life is reliability. to the Cold War and the mutual assured destruction doctrine. Accordingly. and are often measuring well-being using methods that are stochastic in nature.g. utilize reliability theory in the design of the product in order to reduce the probability of failure. e. The probability of failure is also closely associated with the product's warranty. "the probability of another 9/11". and on peace and conflict. and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole.
find P(F). Event Y's probability is 0. All those doing Computer Science do English. A study was undertaken to see how many people in Port Elizabeth owned either a Volkswagen or a Toyota. 270 do Computer Science. Jane invested in the stock market. The probability that she will not lose all her money is 1. Draw a contingency table to show all events and decide if car ownership is independent. but not Typing or Computer Science B English but not Typing C English and Typing but not Computer Science D English or Typing
442
. 5. Event D is picking a vowel. One dark night a thief steals a car. A car sales person has pink. 3% owned both.31. If D and F are mutually exclusive events.24. The last ten letters of the alphabet were placed in a hat and people were asked to pick one of them. At Dawnview High there are 400 Grade 12's.39 and P(H and J)=0. Event X's probability is 0.94. P(H)=0.10. What is the probability that X or Y will occur (this inculdes X and Y occuring simultaneously)? 8.3 and P(D or F)=0. 4.43. 20 take Computer Science and Typing and 35 take English and Typing. lime-green and purple models of car A and purple.3
CHAPTER 34. Using a Venn diagram calculate the probability that a pupil drawn at random will take: A English.34. What is the probability that she will lose all her money? Explain. Calculate: A P(H') B P(H or J) C P(H' or J') D P(H' or J) E P(H' and J') 9.62. A What is the experiment and sample space? B Draw a Venn diagram to show this.GRADE 11
3. 25% owned a Toyota and 60% owned a Volkswagen. The probability of both occuring together is 0.32. C What is the probability of stealing either model A or model B? D What is the probability of stealing both model A and model B? 7. 6. Calculate the following probabilities: A P(F') B P(F or D) C P(neither E nor F) D P(D and E) E P(E and F) F P(E and D') 10. 300 do English and 50 do Typing. orange and multicolour models of car B. with P(D')=0. Event E is picking a consonant and Evetn F is picking the last four letters. INDEPENDENT AND DEPENDENT EVENTS . P(J)=0.
APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work. either commercially or non-commercially. Inc. Secondarily. The "Document". refers to any such manual or work. We recommend this License principally for works whose purpose is instruction or reference.
PREAMBLE
The purpose of this License is to make a manual. Suite 330. Such a notice grants a world-wide. or with modifications and/or translated into another language. or of legal. A "Modified Version" of the Document means any work containing the Document or a portion of it. because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. this License preserves for the author and publisher a way to get credit for their work. and is addressed as "you". Boston. either copied verbatim. a Secondary Section may not explain any mathematics.2002 Free Software Foundation. in any medium. This License is a kind of "copyleft". unlimited in duration. which means that derivative works of the document must themselves be free in the same sense. ethical or political position regarding them. regardless of subject matter or whether it is published as a printed book. that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. But this License is not limited to software manuals. which is a copyleft license designed for free software. with or without modifying it. while not being considered responsible for modifications made by others. You accept the license if you copy. commercial. 619
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ADDENDUM: How to use this License for your documents
To use this License in a document you have written. Permission is granted to copy. If your document contains nontrivial examples of program code. If the Document specifies that a particular numbered version of this License "or any later version" applies to it. with the Front-Cover Texts being LIST. | 677.169 | 1 |
For more cool math videos visit my site at or
MATHEMATICS SECTION SKILLS
General Description
The Mathematics Section of the THEA Test consists of approximately 50 multiple-choice questions
covering four general areas: fundamental mathematics, algebra, geometry, and problem solving.
The test questions focus on a student(s ability to perform mathematical operations and solve problems.
Appropriate formulas will be provided to help students perform some of the calculations
required by the test questions.
You may use a four-function (,
, , ), nonprogrammable calculator [with square root
(
(
and percent (%) keys]. See (The Test Session( on the current THEA program Web site, )
for more information.
Skill Descriptions
The Mathematics Section of the THEA Test is based on the skills listed below. Each skill is accompanied
by a description of the content that may be included on the test.
FUNDAMENTAL MATHEMATICS
Skill: Solve word problems involving integers, fractions, decimals, and units of measurement.
Includes solving word problems involving integers, fractions, decimals (including percents),
ratios and proportions, and units of measurement and conversions (including scientific
notation).
Skill: Solve problems involving data interpretation and analysis.
Includes interpreting information from line graphs, bar graphs, pictographs, and pie charts;
interpreting data from tables; recognizing appropriate graphic representations of various
data; analyzing and interpreting data using measures of central tendency (mean, median,
and mode); and analyzing and interpreting data using the concept of variability.
ALGEBRA
Skill: Graph numbers or number relationships.
Includes identifying the graph of a given equation or a given inequality, finding the slope and/
or intercepts of a given line, finding the equation of a line, and recognizing and interpreting
information from the graph of a function (including direct and inverse variation).
Skill: Solve one- and two-variable equations.
Includes finding the value of the unknown in a given one-variable equation, expressing one
variable in terms of a second variable in two-variable equations, and solving systems of two
equations in two variables (including graphical solutions).
Skill: Solve word problems involving one and two variables.
Includes identifying the algebraic equivalent of a stated relationship and solving word problems
involving one and two unknowns.
Skill: Understand operations with algebraic expressions and functional notation.
Includes factoring quadratics and polynomials; performing operations on and simplifying
polynomial expressions, rational expressions, and radical expressions; and applying principles
of functions and functional notation.
Skill: Solve problems involving quadratic equations.
Includes graphing quadratic functions and quadratic inequalities; solving quadratic equations
using factoring, completing the square, or the quadratic formula; and solving problems
involving quadratic models.
GEOMETRY
Skill: Solve problems involving geometric figures.
Includes solving problems involving two-dimensional geometric figures (e.g., perimeter and
area problems) and three-dimensional geometric figures (e.g., volume and surface area
problems) and solving problems using the Pythagorean theorem.
Skill: Solve problems involving geometric concepts.
Includes solving problems using principles of similarity, congruence, parallelism, and perpendicularity.
PROBLEM SOLVING
Skill: Apply reasoning skills.
Includes drawing conclusions using inductive and deductive reasoning.
Skill: Solve applied problems involving a combination of mathematical skills.
Includes applying combinations of mathematical skills to solve problems and to solve a
series of related problems.-Source nesinc.com | 677.169 | 1 |
Linear quadratic and exponential models worksheet answers - Elementary Arithmetic - High School Math - College Algebra - Trigonometry - Geometry - Calculus. But let's start at the beginning and work our way up through the. Please use this form if you would like to have this math solver on your website, free of charge. Name: The National Center and State Collaborative (NCSC) is a project led by five centers and 24 states to build an alternate assessment based on alternate achievement. From simplify exponential expressions calculator to division, we have got every aspect covered. Come to Algebra-equation.com and read and learn about operations. Ask Math Questions you want answered . . . Share your favorite Solution to a math problem . . . Share a Story about your experiences with Math which could inspire or. Printable Worksheets And Lessons . Function It All Up Step-by-step Lesson - You are given a table and asked to write a linear, quadratic, or exponential function for it. Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Online tutoring available for. Mrs. Snow's Math McNeil High School. Home; Class Info; Parent Letter; Algebra I; Algebra II. Algebra II Lesson Notes. These notes follow the Prentice Hall Algebra. A Guide to Unit Resources - Read First Note to Teachers: Free Algebra 1 worksheets created with Infinite Algebra 1. Printable in convenient PDF format..study guide for fetal monitoring certification Free Algebra 1 worksheets created with Infinite Algebra 1. Printable in convenient PDF format. Please use this form if you would like to have this math solver on your website, free of charge. Name: Ask Math Questions you want answered . . . Share your favorite Solution to a math problem . . . Share a Story about your experiences with Math which could inspire or. A Guide to Unit Resources - Read First Note to Teachers: Mrs. Snow's Math McNeil High School. Home; Class Info; Parent Letter; Algebra I; Algebra II. Algebra II Lesson Notes. These notes follow the Prentice Hall Algebra. From simplify exponential expressions calculator to division, we have got every aspect covered. Come to Algebra-equation.com and read and learn about operations. Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Online tutoring available for. - Elementary Arithmetic - High School Math - College Algebra - Trigonometry - Geometry - Calculus. But let's start at the beginning and work our way up through the. Printable Worksheets And Lessons . Function It All Up Step-by-step Lesson - You are given a table and asked to write a linear, quadratic, or exponential function for it. The National Center and State Collaborative (NCSC) is a project led by five centers and 24 states to build an alternate assessment based on alternate achievement.. clam ice shelter | 677.169 | 1 |
Chain and Product Rules Matching Puzzle
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
0.09 MB | 4, including solution pages
PRODUCT DESCRIPTION
This is a matching puzzle, also known as a Tarsia. Please see my other puzzles, most of which deal with trigonometry and calculus, with some precalculus thrown in, too.
There are two types of sides: functions denoted y = ..., and derivatives denoted y'=.... Students match functions with their respective derivatives. The final shape is a rhombus/diamond.
Instructions (time is approximately 40 - 50 minutes):
1. Print each puzzle (three pages), using single-sided printing.
2. Students cut out all triangles, from each of the two sheets.
3. Students match functions with their derivatives, back to back, like one would put together a jigsaw puzzle. The puzzle can be put together in only one way, as each expression, reduced or otherwise, occurs only once | 677.169 | 1 |
Main menu
Mathematics Program Goals and Objectives
Goal #1: Problem-Solving.
To develop student's ability to apply both conventional and creative techniques to the solution of mathematical problems. (ULG 1,5,6) Student Learning Objectives:Students graduating with a major in mathematics will
Be able to use problem-solving techniques to formulate a mathematical model for and solve a complex problem.
Be able to make appropriate use of technology in the solution of a mathematical problem.
Goal #2: Abstraction and Proof.
To develop student's ability to comprehend, formulate, and produce mathematical proof. (ULG 1,5) Student Learning Objectives:Students graduating with a major in mathematics will
Be able to read and comprehend a mathematical argument, identifying any flaws in its reasoning.
Be able to use mathematical reasoning to prove or disprove conjectures.
Be able to write formal mathematical proofs.
Be able to use abstraction and generalization to make and test conjectures and to revise them as necessary.
Goal #3: Communication.
To develop student's ability to communicate correct mathematical content in both written and oral form. (ULG 1,5) Student Learning Objectives:Students graduating with a major in mathematics will
Be able to communicate sound mathematical reasoning and solutions of mathematical problems in writing.
Be able to communicate sound mathematical reasoning and solutions of mathematical problems through oral presentations.
Be able to collaborate with peers to solve mathematical problems.
Use a variety of representations of mathematical ideas to support and deepen students' mathematical understanding. (Education)
Goal #4: Breadth.
To provide students with the understanding that mathematics comprises a broad array of interconnected concepts. (ULG 1,5,6) Student Learning Objectives:Students graduating with a major in mathematics will
Possess the mathematical content knowledge and skills from several foundational areas of mathematics, including calculus, algebra, discrete mathematics, and linear algebra.
Be aware of a wide array of other areas of mathematics and how they interrelate.
Recognize, use, and make connections between and among mathematical ideas and in contexts outside mathematics to build mathematical understanding. (Education)
Goal #5: Depth.
To engage students in an in-depth study of a single area of mathematics. (ULG 1,5) Student Learning Objectives:Students graduating with a major in mathematics will
Complete a capstone project and/or an extended study of an advanced area of undergraduate mathematics. | 677.169 | 1 |
For courses in linear algebra. With traditional linear algebra texts, the course is relatively easy for students during the early stages as material is presented in a familiar, concrete setting. However, when abstract concepts are introduced, students often hit a wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood and require time to assimilate. These concepts are fundamental to the study of linear algebra, so students' understanding of them is vital to mastering the subject. This text makes these concepts more accessible by introducing them early in a familiar, concrete Rn setting, developing them gradually, and returning to them throughout the text so that when they are discussed in the abstract, students are readily able to understand. | 677.169 | 1 |
This title is in stock with our Australian supplier and will be ordered in for you asap. We will send you a confirmation email with a Tracking Code to follow the progress of your parcel when it ships.
Papua New Guinea Mathematics Grade 9 Author: Ros O'Sullivan
ISBN: 9780195565461 Format: Paperback Published: 9 September 2009 Country of Publication: AU Dimensions (cm): 25.500 x 18.900 x 2.200 Description: Mathematics for Grade 9 student book meets all the requirements of the new Grade 9 Mathematics Syllabus. Units of work are organised according to the Mathematics syllabus, namely: 9.1 - Mathematics in Our Community; 9.2 - Patterns of Change; 9.3 Working with Data - Core and Option A: Random Events and Stimulation and Option B: Statistical Surveys; 9.4 - Design in 2D and 3D Geometry - Core and Option A: Construction and Option B: Deductive Reasoning. Each Unit features: topics and learning approaches that directly support the achievement of syllabus outcomes; help and information boxes to assist in explaining mathematical processes and concepts; a wide range of practice exercises and problem-solving activities; revision and assessment tasks at the end of each unit; appendices which include tables to assist students in undertaking exercises | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
My teacher is very helpful and very understanding. As long as you understand algebra, the course is easy. It may become a little challenging along the way, but nothing too hard. There is a lot of working with letters but there are numbers as well.
Course highlights:
I learned a lot about Trigonometric functions. We are also learning about matrices and how to do problems on them. I like working with the numbers more than the letters, but the letters aren't hard.
Hours per week:
6-8 hours
Advice for students:
Pay attention in class, if not you will get lost. If you do get lost, ask questions! Questions are what will help you get through the course. | 677.169 | 1 |
The Mathematics Faculty
The Mathematics Faculty
To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas. – Ivars Peterson
Curriculum
Year 7
In year 7, we cover topics such as place value, addition, subtraction, multiplication, division, fractions, percentages, substitution, sequences and angles. Our scheme of work focuses on helping the students understand the topics at a much deeper level. We use a variety of resources including My Maths which is an online maths resource.
Year 8
In Year 8, we cover topics such as prime numbers, indices, solving equations, ratio, proportion, percentage change, perimeter, area, volume, statistical graphs and averages. Our scheme of work focuses on helping the students understand the topics at a much deeper level. We use a variety of resources including My Maths which is an online maths resource.
Year 9
In Year 9, we cover topics such as probability, formulae, graphs, sequences, quadratics and extended number skills. We focus on developing students' ability to solve problems in a range of contexts and become independent learners. Students start studying the GCSE content from June, when they are introduced to some of the statistics topics which are part of the Stats GCSE specification. Our aim in Year 9 is to provide all students with a secure foundation of maths skills and confidence on which to build at KS4.
GCSE Statistics. The students complete a GCSE in Statistics at the end of the year 10 where they have the opportunity to gain an extra maths GCSE a year earlier than their GCSEs in Year 11. The overall grade is split with 75% of the grade coming from the exam and 25% of the grade coming from a piece of controlled assessment. This is an excellent opportunity for students to gain an insight in to how it feels to sit a proper GCSE in Maths.
Year 11
Students will cover all topics from the new syllabus for Maths GCSE which is graded 9 – 1. They will learn how to unpick complex problems and will learn to apply their maths skills to multi-step problems There are a huge range of resources available to students, including My Maths, Method Maths, Pixl Maths app. These are online based resources which have practice questions, lessons and exam style questions for the students to practice. The Pixl Maths app also have powerpoints and therapy videos available which can help students with their revision.
The current Year 12s are the last cohort of students to follow this model for A level mathematics. The new Maths A level will be taught for the first time in September 2017 and this has a slightly different structure to the existing model for A level Mathematics.
Year 13
Students study 3 modules in this year which combine with their Year 12 results to make up the A level qualification. The modules studied are Core 3, Core 4 and Mechanics 1. Students can also continue the further maths A level. Student will build on the foundations of key topics such as integration and differentiation and develop their knowledge further.
Extra – curricular
Maths Homework club runs every lunch time in the computer rooms in the Maths block. There is always a maths teacher available to help the students with any problems the students may have.
All students are welcome at any time to get support or extra work in maths from their teachers.
A trip is organised every year for some students in Year 10 to visit a college in Cambridge. The morning is spent looking around the college and the afternoon is spent at the Centre for Mathematical Sciences taking part in maths activities.
Gifted and Talented
MATHS CHALLENGES
Students have the opportunity to participate in the annual UK Maths Challenges at the Junior, Intermediate and Senior level. Some students regularly get through to the follow-on rounds. We also enter students in the team challenges which take place in Cambridge.
Two students in Year 9 regularly participate in the University of Hertfordshire Masterclasses.
Some students in Year 11 have the opportunity of studying for an 'Additional Maths' qualification, as well as their GCSE Maths.
FAQ
Yes, though they are not used every lesson, students should have their own Scientific calculator. Throughout their time at Fearnhill, we will teach them how to use it effectively in preparation for their GCSE exams. | 677.169 | 1 |
CliffsQuickReview Linear AlgebraCliffsQuickReview course guides cover the essentials of your toughest classes. Get a firm grip on core concepts and key material, and approach your exams with newfound confidence. CliffsQuickReview Linear Algebra demystifies the topic with straightforward explanations of the fundamentals. This comprehensive guide begins with a close look at vector algebra (including position vectors, the cross product, and the triangle inequality) and matrix algebra (including square matrices, matrix addition, and identity matrices). Once you have those subjects nailed down, you'll be ready to take on topics such as Linear systems, including Gaussian elimination and elementary row operations Real Euclidean vector spaces, including the nullspace of a matrix, projection into a subspace, and the Rank Plus Nullity Theorem The determinant, including definitions, methods, and Cramer's Rule Linear transformations, including basis vectors, standard matrix, kernal and range, and composition Eigenvalues and Eigenvectors, including definitions and illustrations, Eigenspaces, and diagonalization CliffsQuickReview Linear Algebra acts as a supplement to your textbook and to classroom lectures. Use this reference in any way that fits your personal style for study and review - the information is clearly arranged and offered in manageable units. Here are just a few of the features you'll find in this guide: A review of core concepts Clear diagrams and loads of formulas Easy to understand definitions and explanations Plenty of examples and detailed solutions With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades | 677.169 | 1 |
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Unformatted text preview: EIGENVALUES AND EIGENVECTORS 1. Definition They are defined in terms of each other. Let A be an n × n matrix. A vector v Ó = 0 is an eigenvector of A with eigenvalue λ if the equation Av = λv is satisfied. Note that eigenvectors are not uniquely defined: If v is an eigenvector then any scalar multiple of v is also an eigenvector. In fact, if u and v are two eigenvectors for the same eigenvalue λ , then any linear combination a · u + b · v is also an eigenvector with eigenvalue λ . 2. Applications of eigenvectors You might wonder why anyone would be interested in eigenvectors and eigenval- ues. Eigenvectors are vectors toward which the matrix A behaves like a scalar — namely the scalar λ (the corresponding eigenvalue). If we could find a basis consisting of eigenvectors, A would become a diagonal matrix in this basis: its action on basis-vectors would only be to multiply each of them by a scalar — which is what a diagonal matrix does. Diagonal matrices are interesting because the are easy to work with — they behave like scalars when you add or multiply them....
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch. | 677.169 | 1 |
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Unformatted text preview: MathPad 2 : A System for the Creation and Exploration of Mathematical Sketches Joseph J. LaViola Jr. Robert C. Zeleznik Brown University * Abstract We present mathematical sketching , a novel, pen-based, modeless gestural interaction paradigm for mathematics problem solving. Mathematical sketching derives from the familiar pencil-and-paper process of drawing supporting diagrams to facilitate the formula- tion of mathematical expressions; however, with a mathematical sketch, users can also leverage their physical intuition by watch- ing their hand-drawn diagrams animate in response to continuous or discrete parameter changes in their written formulas. Diagram animation is driven by implicit associations that are inferred, either automatically or with gestural guidance, from mathematical expres- sions, diagram labels, and drawing elements. The modeless nature of mathematical sketching enables users to switch freely between modifying diagrams or expressions and viewing animations. Math- ematical sketching can also support computational tools for graph- ing, manipulating and solving equations; initial feedback from a small user group of our mathematical sketching prototype applica- tion, MathPad 2 , suggests that it has the potential to be a powerful tool for mathematical problem solving and visualization. CR Categories: H.5.2 [Information Interfaces and Presentation]: User Interfaces—Interaction Styles G.4 [Mathematics of Comput- ing]: Mathematical Software—User Interfaces; Keywords: pen-based interfaces, mathematical sketching, ges- tures 1 Introduction Diagrams and illustrations are often used to help explain math- ematical concepts. They are commonplace in math and physics textbooks and provide a form of physical intuition about abstract principles [Hecht 2000; Varberg and Purcell 1992; Young 1992]. Similarly, students often draw pencil-and-paper diagrams for math problems to help in visualizing relationships among variables, con- stants, and functions. With the drawing as a guide, they can write the appropriate math to solve the problem. However, such static diagrams generally assist only in the initial formulation of mathe- matical expressions, not in the "debugging" or analysis of those ex- pressions. This can be a severe limitation, even for simple problems with natural mappings to the temporal dimension, or for problems with complex spatial relationships. By animating sketched diagrams from changes in associated math- ematical expressions, users can evaluate different formulations with * Email: { jjl,bcz } @cs.brown.edu Figure 1: A mathematical sketch used to explore damped harmonic oscillation. It shows a spring and mass drawing and the necessary equations for animating the sketch. The label inside the mass asso- ciates the mathematics with the drawing....
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This note was uploaded on 06/12/2011 for the course CAP 6105 taught by Professor Lavoila during the Spring '09 term at University of Central Florida. | 677.169 | 1 |
There are 3 key elements to pay attention to: Text, Figures, and Calculations Figures Calculations Text What are the MAIN points of the chapter? Find out by reading the SUMMARY first. Find the logical relationship between section topics. What is the Topic of the chapter? What does the title mean? For Each Paragraph: Write down the MAIN idea in your own words. Write down short definitions of new vocabulary. For Each Section: When examples are given, write down the point that the example is about, followed by a list of examples and their significance. What does the figure show? Be able to describe and explain each part of the
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Unformatted text preview: figure to a 12 year old. Write down what each equation is used for. What can be calculated from the equation? What information is needed to use the equation? When is each equation valid? Look at the section where this equation is introduced. Can the equation be applied to topics in other sections as well? Be able to read the equation and explain each part. If there are many steps to a solution or procedure, write down a general RECIPE. As you read, write down any questions you have and note places where you get stuck. Bring your questions to class....
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Unformatted text preview: Matrices and Determinants Advanced Level Pure Mathematics Advanced Level Pure Mathematics Chapter 8 Matrices and Determinants 8.1 INTRODUCTION : MATRIX / MATRICES 2 8.2 SOME SPECIAL MATRIX 3 8.3 ARITHMETRICS OF MATRICES 4 8.4 INVERSE OF A SQUARE MATRIX 16 8.5 DETERMINANTS 19 8.6 PROPERTIES OF DETERMINANTS 21 8.7 INVERSE OF SQUARE MATRIX BY DETERMINANTS 27 Prepared by K. F. Ngai Page 1 8 Matrices and Determinants Advanced Level Pure Mathematics 8.1 INTRODUCTION : MATRIX / MATRICES 1. A rectangular array of m × n numbers arranged in the form a a a a a a a a a n n m m mn 11 12 1 21 22 2 1 2 is called an m × n matrix . e.g. 2 3 4 1 8 5- is a 2 × 3 matrix. e.g. 2 7 3- is a 3 × 1 matrix. 2. If a matrix has m rows and n columns , it is said to be order m × n. e.g. 2 3 6 3 4 7 1 9 2 5 is a matrix of order 3 × 4. e.g. 1 2 2 1 5 1 3-- is a matrix of order 3. 3. [ ] a a a n 1 2 is called a row matrix or row vector . 4. b b b n 1 2 is called a column matrix or column vector . e.g. 2 7 3- is a column vector of order 3 × 1. e.g. [ ]--- 2 3 4 is a row vector of order 1 × 3. 5. If all elements are real, the matrix is called a real matrix. Prepared by K. F. Ngai Page 2 Matrices and Determinants Advanced Level Pure Mathematics 6. a a a a a a a a a n n n n nn 11 12 1 21 22 2 1 2 is called a square matrix of order n. And a a a nn 11 22 , , , is called the principal diagonal. e.g. 3 9 2- is a square matrix of order 2. 7. Notation : [ ] ( 29 a a A ij m n ij m n × × , , , ... 8.2 SOME SPECIAL MATRIX. Def.8.1 If all the elements are zero, the matrix is called a zero matrix or null matrix, denoted by O m n × . e.g. is a 2 × 2 zero matrix, and denoted by O 2 . Def.8.2 Let [ ] A a ij n n = × be a square matrix. (i) If a ij = for all i, j, then A is called a zero matrix. (ii) If a ij = for all i<j, then A is called a lower triangular matrix . (iii) If a ij = for all i>j, then A is called a upper triangular matrix . a a a a a a n n nn 11 21 22 1 2 a a a a a n nn 11 12 1 22 i.e. Lower triangular matrix Upper triangular matrix e.g. 1 2 1 1 4- is a lower triangular matrix. e.g. 2 3 5- is an upper triangular matrix....
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This note was uploaded on 11/26/2011 for the course COMPUTER S 1003 taught by Professor Angelosstavrou during the Spring '11 term at King Saud University. | 677.169 | 1 |
Intermediate Algebra
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Read More contains additional modeling and real-data coverage. A conceptual approach to functions is introduced early in the book and revisited in Ch. 5, 6, 7, 8, and 10--readers are exposed to a variety of realistic situations where functions are used to explain and record the changes we observe in the world. A discussion of solving linear equations in Chapter 2 now includes coverage of equations with no solution and equations with infinitely many solutions. The sections on determinants and Cramer's rule have been moved out of Chapter 4 into an appendix. This material can be covered with ease after Section 4.3 This is a loose-leaf edition textbook (same content, just cheaper! ! ). May contain wear throughout the pages. May not contain access codes or supplementary material. 2nd day shipping available, ships same or next business day. This is the U.S. student edition as pictured unless otherwise stated.
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Summary
Topics in Contemporary Mathematics is uniquely designed to help students see math at work in the contemporary world by presenting problem solving in purposeful and meaningful contexts. This Expanded Eighth Edition contains two additional chapters on Voting and Apportionment and Graph Theory. Strong technology focus encourages students to learn and apply their knowledge using the most up-to-date web links maintained by the author on a companion web site. Instructors may also use this site to access PowerPoint slides for convenient class presentations. In addition to these web resources, lecture and practice test videos have been developed to provide extra support and foster confidence outside of the classroom. For those students in Florida, a CLAST Test software package and video are available as well. A variety of pedagogical features reinforce ideas and motivate students to learn. Getting Started offers a motivating introduction for the techniques and ideas in each section. Through web references and Web It exercises, students utilize the Internet as an educational and creative tool to study mathematical concepts. Collaborative Learning encourages student interaction as they work together to solve problems. The Graph It feature found in the book margins provides step-by-step directions for solving specific examples using the TI-83 graphing calculator. Problem-solving approach throughout the text helps students learn techniques and methods that will benefit them throughout their lives and careers. These special examples use George Polya\'s problem-solving strategy (RSTUVRead, Select, Think of a plan, Use the techniques, Verify) and a unique two-column format for describing the general problem-solving method and demonstrating specific uses. Abundant applications and examples include more than 500 examples and 4100 carefully developed exercises that cover a wide range of topics and provide the instructor and student with flexibility in choosing computational, drill, or conceptual problems. Real-world applications motivate students and pique their interest. Other problems such as Using Your Knowledge, Discovery, Calculator Corner, and Research questions help reinforce concepts and further develop the students critical-thinking and problem-solving skills. Skill Checker helps students test their knowledge with a variety of problems to ensure they have a thorough grasp of the material before continuing on to new concepts. The Chapter Summary provides definitions and section references for key topics within a given chapter. A Practice Test after each chapter is followed by Answers to the Practice Test, with references to the appropriate section, page, and example for review, as needed.
Table of Contents
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Note: Each chapter concludes with a Summary, Research Questions, a Practice Test, and Answers to Practice Test | 677.169 | 1 |
Download Vhembe District Term Tasks Of Mathematics
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MATHEMATICS GRADES 10-12 ... Grade 11 Term: 1 . . around us, and, most of all, to teach us to think creatively. 2.2 Specific . the equation of a tangent to a circle at a given point on the circle. 10. st a. tistiC s . The order of topics is not prescriptive, but ensure that part of trigonometry is taught in the first term and more.
is to be taught and learnt on a term-by-term basis. ... LIFE ORIENTATION GRADES 10-12. 1. CAPS. CONTENTS. seCtion 1: introduCtion to 4.5.1 Written tasks . . teaching of Life Orientation in Grades 10 and 11, and 56 hours in Grade 12.
English Language Arts/Literacy and Mathematics, Grades 3-11. End-of-Year ... Extended tasks. Applications of Research Simulation Tasks students will analyze an PARCC assessment gives students the opportunity to gain partial credit if their . Quality: This is an example of a science passage from a third- grade .
ebook.dexcargas.com is a PDF Ebook search engine and unrelated to Adobe System Inc. No pdf files hosted in Our server. All trademarks and copyrights on this website are property of their respective owners. | 677.169 | 1 |
MATH Documents
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Chapter 6-Factoring Polynomials
Section 6.1-Greatest Common Factor & Factoring by Grouping
Question: What is a factor?
Question: Why do we want to know how to factor quickly and
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Question: What is a greatest common factor (GCF)? How does the G
1
Section 9.1 Square Roots
a is the square root of a and if
a b , this means b 2 a . In other words,
a
means we are looking for a number when squared gives us a . An example of this
is 100 10 since 10 2 100
Properties of Square Roots
1. Every positive rea
Calculus II Review for Test #2
Remember ~ absolutely no calculators of any kind on this test! You may bring a 3 inch by 5 inch
index card with notes in your own handwriting on the front and back readable without the aid
of any magnication devices other th
Caiculus H Study Guide for Test #3
As usual, you may bring a 3 inch by 5 inch index card with notes on the front and back in
your own handwriting, readable without the aid of any magnication other than your normal
reading glasses (if you normally wear the
Calculus II Study Guide for Test #1
You may bring a 3 inch by 5 inch index card with notes on the front and the back, in your own
handwriting, readable without the aid of magnification devices other than reading glasses (if you
normally wear them). You ma
Calculus II Study Guide Test #4
Things you should be able to do:
Find the radius and interval of convergence for a given power series
Given a function, find the Maclaurin or Taylor polynomial of a specified degree about a
specified point
Find the Macla
2-A
Lab Report
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Throughout this experiment we investigated to determine the Water Quality Index of certain
samples of water. We did this to get a full picture of how an ecosystem may change overtime. We
had certain outlined bases for these tests,
Mr. Gonzalez
English 1010
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Organizational Pattern: Emphatic/Simple to Complex
The Idea of Being Fearless
A spark being ignited in someone is like an awakening of a being rising inevitably. In today's
world/society there are many that want and w
Electrical Engineering: Positively Changing Society One Product at a Time
What would the world be without electrical engineering? Electrical engineering has so
many fields of interest that peoples everyday lives are positively affected by it. The world of
How Data Storage Works
How Data Storage Works
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The purpose of this paper is to highlight some best practices for data input and output. In
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Goraya 1
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30 May 2012
The Collision on Culture
The way people perceive Okonkwo in the novel Things Fall Apart changes many times
throughout the story. It is portrayed in many ways, that a person's perspective of culture can be changed
in a b
Name:_
Date:_
China Still Has Some Tarnish on Its Image Article Questions
Read the article from the link and answer the following questions below in red.
Save as: Chinas Olympics-Lastname to H: and Inbox.
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Assignment 1: Intercultural Sensitivity
1. B I would tell them how I feel about the situation but always continue to have the bond
we had. The only reason I would confront them about it is because I want to know what
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1-B
Dolphins
Franchise
Over the last several weeks, we have been learning how to start a new football
franchise from scratch. Also, we have learned many different concepts of this simulation
that will help us achieve our goal. Along with, making several k | 677.169 | 1 |
The text is for a two semester course in advanced calculus. It develops the basic ideas of calculus rigorously but with an eye to showing how mathematics connects with other areas of science and engineering. In particular, effective numerical computation is developed as an important aspect of mathematical analysis. | 677.169 | 1 |
Download Free As Level Maths For Edexcel Core 2 Student Book Book in PDF and EPUB Free Download. You can read online As Level Maths For Edexcel Core 2 Student Book and write the review.
This book is the leading title in a series targeted at the average A Level mathematics student which aims to tackle the basic ideas & misconceptions associated with this subject. The inclusion of stretch & challenge material caters for the most able students, & lots of regular exercises & exam questions provide plenty of practice.
Updated for the 2004 specification, these new Core books are in full colour to ease the transition from GCSE to A Level. Tailor-made for the new specification and written by members of an experienced Senior Examining Team, you can be sure they provide everything students need to succeed.
The clear route to A Level success - new Core titles for the new specification Written by the same authors as the textbooks for a complete match, so are ideal for use alongside the course books. Worked examination questions and examples with hints on answering questions successfully help students push for those top grades. A test-yourself section makes sure students are fully prepared for the exam. Key points help reinforce learning and help students reach their best potential. Answers to all the questions ensure students can check their work. Written by experienced Senior Examiners.
Drawing on over 10 years' experience of publishing for Edexcel maths, Heinemann Modular Maths for Edexcel AS and A Level brings you dedicated textbooks to help you give your students a clear route to success, now with new Core maths titles to match the new 2004 specification. Further Pure 3 replaces Pure 6 in the new specification.
"This book helps in raising and sustaining motivation for better grades. These books are the best possible match to the specification, motivating readers by making maths easier to learn. They include complete past exam papers and student-friendly worked solutions which build up to practice questions, for all round exam preparation. These books also feature real-life applications of maths through the 'Life-links' and 'Why...?' pages to show readers how this maths relates, presenting opportunities to stretch and challenge more apply students. Each book includes a Live Text CDROM which features: fully worked solutions examined step-by-step, animations for key learning points, and revision support through the Exam Cafe." -- Publisher's description.
Build your students' confidence in applying mathematical techniques to solving problems with resources developed with leading Assessment Consultant Keith Pledger and Mathematics in Education and Industry (MEI). Build reasoning and problem-solving skills with practice questions and well-structured exercises that build skills and mathematical techniques. Develop a fuller understanding of mathematical concepts with real world examples that help build connections between topics and develop mathematical modelling skills. Address misconceptions and develop problem-solving with annotated worked examples. Supports students at every stage of their learning with graduated exercises that build understanding and measure progress. Provide clear paths of progression that combine pure and applied maths into a coherent whole. Reinforce Year 1 content with short review chapters - Year 2 only. | 677.169 | 1 |
Tag Archives: learning trajectory
In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston's seven different ways to think of the derivative: Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function. Symbolic: the derivative of x^n is nx^n−1, the derivative of sin(x) is cos(x), the… Read More »
Here is one way of describing students levels of problem solving skills in mathematics. I call them levels of problem solving skills rather than process of reflective abstraction as described in the original paper. As math teachers it is important that we are aware of our students learning trajectory in problem solving so we can… Read More »
There are at least three representational systems used to study function: graphs, tables and equations. But unlike graphs and tables that are used to visually show the relationships between two varying quantities, students first experience with equation is not as a representation of function but a statement which state the condition on a single unknown quantity. Also,… Read More » | 677.169 | 1 |
Methods of Solving Complex Geometry Problems
Containing over 160 complex problems with hints and detailed solutions, this book can be used as a self-study guide for mathematics competitions and for improving problem-solving skills in courses on plane geometry or the history of mathematics.
Containing over 160 complex problems with hints and detailed solutions, this book can be used as a self-study guide for mathematics competitions and for improving problem-solving skills in courses on plane geometry or the history of mathematics. | 677.169 | 1 |
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The Mathematics of Various Entertaining Subjects: Research in Recreational Math
The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected.
Maths - grade 8 Say It With Symbols: Making Sense of Symbols, explores the topic that beginning algebra used to focus on almost exclusively: the use of symbols. This mathematics curriculum emphasizes the meaning behind the symbols. This helps students build their own understanding of the basics of algebra and its usefulness for solving problems. REUPLOAD for the TEACHERS'S GUIDE NEEDED
Here is a list of the main concepts and formulas you'll need to know for the GRE. Each of these concepts is addressed in the Magoosh's Complete Guide to Math Formulas eBook. In this resource, you'll also find recommended strategies for how to best use (or not use!) and remember these math formulas on the GRE. | 677.169 | 1 |
1001 Basic Math & Pre- Algebra Practice Problems For Dummies Practice makes perfect and helps deepen your understanding of basic math and pre-algebra by solving problems 1001 t only a solution but a step-by-step explanation. From the book, go online and find: * One year free subscription to all 1001 practice problems * On-the-go access any way you want it from your computer, smart phone, or tablet * Multiple choice questions on all you math course topics * Personalized reports that track your progress and help show you where you need to study the most * Customized practice sets for self-directed study * Practice problems categorized as easy, medium, or hard The practice problems in 1001 Basic Math & Pre-Algebra Practice Problems For Dummies give you a chance to practice and reinforce the skills you learn in class and help you refine your understanding of basic math & pre-algebra. NoteAuthor Biography
Mark Zegarelli is a math and test prep tutor and instructor in SanFrancisco and New Jersey. He is the author of Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, ACT Math For Dummies, Logic For Dummies, and Calculus II For Dummies . | 677.169 | 1 |
CAPS GRADE 12 TEACHER GUIDE. 1. INTRODUCTION. Assessment is a ... tasks, particularly the investigation and assignment; hence these exemplars were The sum of the first n terms of a sequence is given by: Sn = n(23 3n). 2.1 Write .
OVERVIEW OF GRADE 11 TOPICS AND SUBTOPICS IN MATHEMATICAL LITERACY. CAPS . .... This activity is taken from a Grade 11 Mathematical Literacy Topic. Teacher's Guide, which provides guidance and answers for Programme of.
ebook.dexcargas.com is a PDF Ebook search engine and unrelated to Adobe System Inc. No pdf files hosted in Our server. All trademarks and copyrights on this website are property of their respective owners. | 677.169 | 1 |
I'ts surprisingly difficult to visualize functions in 3D, making graphing a pain, at least for me. The actual math isn't all that much harder, we are finishing partial derivatives and implicit differentiation which were fairly simple. | 677.169 | 1 |
Top Math Software for Mac
Revolution in mathematics software with 2D, 3D, and time graphing with MobileCAS for algebra and calculus. Explore mathematical concepts in an innovative interface and write your own scripts in our new powerful programming language. With features only...
MacBreadboard is a Digital Electronics Breadboarding Simulator.It has a very unique interface in which you "move" chips & wires on a virtual breadboard. MacBreadboard and WinBreadboard (the PC version) are used in many colleges and universities around...
BasicFacts Maker II generates student worksheets with 25 different math problems per page. The program will supply addition, subtraction, multiplication, or division problems for practicing basic math skills. An unlimited number of worksheets can be made...
Hawk visualizes information structures in database systems and other data sources. By defining semantic variables (much like Fuzzy Logic) based on information from an EOF adaptor, sub-concept/super-concept relationships are being computed by using methods...
YP Image is designed to assist teachers and students when they study imageformation by mirrors and thin lenses (geometrical optics). It computes thelocation and magnification of the image formed by a spherical mirror or athin lens, and draws a ray...
"Numerology..." is a powerful macintosh numerology software that has been around the
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Roger Brown, professionnal numerologer has written the interpretation texts for... | 677.169 | 1 |
Function Operations and Compositions Student Notes and Practice
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Operations on Functions and Compositions of Functions Student Notes and Practice to complement the Relations Unit. Student Notes work well as guided instruction with the teacher providing examples of how to perform each operation (addition, subtraction, multiplication, division and composition. Using the examples from their Notes, students will practice 10 problems to demonstrate understanding. Suitable for advanced Algebra 1 or Algebra 2 | 677.169 | 1 |
STEP 3 - SELECT a PROGRAM
This is the list of available programs in arithmetic and pre-algebra. The programs differ in sets of user options, in number of included math problems and in prices. The _short versions contain less problems. | 677.169 | 1 |
Problems of the Day Milestone Review
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This worksheet gives students a problem to do every day of the week, labeled Monday-Friday. The problems review constructing functions from tables, finding the vertex of a quadratic function, interpreting linear functions, combining like terms, and exponential growth and decay. | 677.169 | 1 |
Conic Sections Review Task Cards QR Plus HW Quiz
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Conic Sections End of Unit
Task Cards really do work! They get the students engaged and keep them motivated to go through all of the problems, more so than a simple worksheet. This end of unit engaging activity is designed for PreCalculus or Algebra 2 students to reinforce the concepts and skills taught in class.
All conics are included in the activity:Parabolas, Hyperbolas, Circles, and Ellipses.
Included:
There are twelve cards encompassing 20 questions. Questions include recognition of equations from graphs of conics, both centered at zero and not centered at zero, determining the conics from equations, finding focus points, directrices, vertices, centers, and asymptotes.
There are two sets of the 12 cards, one with QR codes, and one without. Students do not need an internet connection to use the QR codes, but do need a device with a QR reader installed. There are four cards to a standard 8 1/2'' x 11" sheet of paper. These print fine in B/W.
Also included are two 20 questions handouts similar to the task cards and can be used as HW, assessment, or enrichment | 677.169 | 1 |
Inserting content from GeoGebra
GeoGebra is an interactive mathematics software for all levels of education. When you insert the GeoGebra widget in SMART Notebook software, you can explore geometry, algebra, tables, graphing, statistics and calculus with your students.
You can also search for worksheets from GeoGebraTube using the keyword search. After searching for a worksheet, you can add it to your .notebook file. GeoGebraTube (geogebratube.org) contains thousands of worksheets created and shared by other teachers.
Note
For more information on how to use GeoGebra software, go to the GeoGebra Wiki (wiki.geogebra.org). | 677.169 | 1 |
Instructor Information
What is This Course All About?
The primary objective of this course is to develop skills necessary for effective proof writing. Students will improve their ability to read and write mathematics. Successful completion of MAT 320 provides students with the background necessary for upper division mathematics courses. Also, the purpose of any mathematics course is to challenge and train the mind. Learning mathematics enhances critical thinking and problem solving skills. The course description says:
The course trains students on methods and techniques of mathematical communication, focusing on proofs but also covering expository writing and problem-solving explanations.
This course will likely be different than any other math class that you have taken before for two main reasons. First, you are used to being asked to do things like: "solve for $x$," "take the derivative of this function," "integrate this function," etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to "solve" problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof.
Unlike previous courses, this course is all about "proof." Mathematicians are in the business of proving theorems and this is exactly our endeavor. For the first time, you will be exposed to what "doing" mathematics is really all about. This will most likely be a shock to your system. Considering the number of math courses that you have taken before you arrived here, one would think that you have some idea what mathematics is all about. You must be prepared to modify your paradigm. The second reason why this course will be different for you is that the method by which the class will run and the expectations I have of you will be different. In a typical course, math or otherwise, you sit and listen to a lecture. (Hopefully) These lectures are polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I absolutely love lecturing and I do believe there is value in it, but I also believe that in reality most students do not learn by simply listening. You must be active in the learning you are doing. I'm sure each of you have said to yourselves, "Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost." In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL).
An Inquiry-Based Approach
This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand, with the readily available help of me and your classmates. Many of the concepts you learn and problems you work will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.
In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?
Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word "produce" because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!
Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn't expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located below.
In this course, everyone will be required to
read and interact with course notes on your own;
write up quality solutions/proofs to assigned problems;
present solutions/proofs on the board to the rest of the class;
participate in discussions centered around a student's presented solution/proof;
call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar on first glance.
As the semester progresses, it should become clear to you what the expectations are. This will be new to many of you and there may be some growing pains associated with it. For more details, see the syllabus. | 677.169 | 1 |
Algebra - Assign value to items so students can see algebra being applied to real life. I also like how this shows examples with money and what things cost. A fun way to come up with equations and great for visual learners.
Elementary Algebra is generalized form of arithmetic. It provides a language to represent problems and functions. Algebraic thinking is also one of the first forms of abstract thinking that students develop in mathematics. Lets look at some of the common gotchas of algebraic learning. | This is Excellent!! I'm loving the explanations!!!
In Algebra, a monomial or a term is comprised of a combination of the following: numbers, variables, and exponents. In Algebraic expressions and equations, terms or monomials are separated by addition and subtraction signs. Monomial is an algebraic expression with only one term. For example, 7xy, - 5m, 3z2, 4 etc.
LOVE this idea for teaching translating expressions into words....post different expressions around room, give each group diff color marker and have them write unique translations for each. Genius way to make this lesson more fun!! | 677.169 | 1 |
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Unformatted text preview: the left-hand side of the :=. The function definition (using the arrow notation) is on the right-hand side. The following statement defines f as the squaring function.
> f := x -> x^2; f := x → x2 Evaluating f at an argument produces the square of the argument of f.
> f(5); 25
> f(y+1); (y + 1)2 Predefined and Reserved Names
Maple has some predefined and reserved names. If you try to assign to a name that is predefined or reserved, Maple displays a message, informing you that the name you have chosen is protected.
> Pi := 3.14; Error, attempting to assign to 'Pi' which is protected > set := {1, 2, 3}; Error, attempting to assign to 'set' which is protected 2.5 Basic Types of Maple Objects • 21 2.5 Basic Types of Maple Objects This section examines basic types of Maple objects, including expression sequences, lists, sets, arrays, tables, and strings. These ideas are essential to the discussion in the rest of this book. Also, the following concepts in Maple are introduced. • Concatenation operator • Square bracket usage • Curly braces usage • Mapping • Colon (:) for suppressing output • Double quotation mark usage Types Expressions belong to a class or group that share common properities. The classes and groups are known as types. For a complete list of types in Maple, refer to the ?type help page. Expression Sequences
The basic Maple data structure is the expression sequence . This is a group of Maple expressions separated by commas.
> 1, 2, 3, 4; 1, 2, 3, 4
> x, y, z, w; x, y, z, w Expression sequences are neither lists nor sets. They are a distinct data structure within Maple and have their own properties. • Expression sequences preserve the order and repetition of their elements. Items stay in the order in which you enter them. If you enter an element twice, both copies remain. • Sequences are often used to build more sophisticated objects through such operations as concatenation. 22 • Chapter 2: Mathematics with Maple: The Basics Other properties of sequences will become apparent as you progress through this manual. Sequences extend the capabilities of many basic Maple operations. For example, concatenation is a basic name-forming operation. The concatenation operator in Maple is "||". You can use the operator in the following manner.
> a||b; ab When applying concatenation to a sequence, the operation affects each element. For example, if S is a sequence, then you can prepend the name a to each element in S by concatenating a and S .
> S := 1, 2, 3, 4; S := 1, 2, 3, 4
> a||S; a1 , a2 , a3 , a4 You can also perform multiple assignments using expression sequences. For example
> f,g,h := 3, 6, 1; f, g, h := 3, 6, 1
> f; 3
> h; 1 2.5 Basic Types of Maple Objects • 23 Lists
You create a list by enclosing any number of Maple objects (separated by commas) in square brackets.
> data_list := [1, 2, 3, 4, 5]; data _list := [1, 2, 3, 4, 5]
> polynomials := [x^2+3, x^2+3*x-1, 2*x]; polynomials := [x2 + 3, x2 + 3 x − 1, 2 x]
> particip...
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This note was uploaded on 08/27/2012 for the course MATH 1100 taught by Professor Nil during the Spring '12 term at National University of Singapore. | 677.169 | 1 |
How do you get help with Holt math textbook questions?
A:
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The Holt, Rinehart and Winston publishing company posts homework help for Holt math textbooks on its website. To find help with particular homework problems, go to Holt's math-specific website go.hrw.com/gopages/ma-msm.html.
Keep Learning
Holt publishes math textbooks for middle school, pre-algebra, algebra and geometry. From the website, select your subject from the list on the left. Next, select the specific curriculum. A tab labeled Homework Help is one of the options, with help available for each chapter.
In addition to the help offered by Holt, online sources are available to help with math homework. For example, Mathway.com has an interactive tool to assist with specific math problems. To use the tool, enter the math problem in the text box, click the Answer button, and receive the answer. Homework help is available for basic math, pre-algebra, algebra and trigonometry. The site also offers help with pre-calculus, calculus, statistics, finite math and linear algebra. To view the steps for solving complicated problems, users must create an account with a password and agree to terms of service.
The website Math.com prompts users to first select the type of math problem they need help with, such as K-8 math or algebra. Next, they must select the areas of their questions, such as polynomials or quadratic equations. Users are directed to a pages with a short lesson about the topic and boxes to enter their specific questions. | 677.169 | 1 |
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The Essential Tool for MathematicsMaple is math software that combines the world's most powerful math engine with an interface that makes it extremely easy to analyze, explore, visualize, and solve mathematical problems.... see full description | 677.169 | 1 |
Acomdata 75 Gb - Bookshelf
Designed for the undergraduate student with a calculus background but no prior experience with complex analysis, this text discusses the theory of the most relevant mathematical topics in a student-friendly manner.
Publisher: Jones & Bartlett Publishers
About this book
Complex Analysis: A First Course with Applications is a truly accessible introduction to the fundamental principles and applications of complex analysis. Designed for the undergraduate student with a calculus background but no prior experience with complex analysis, this text discusses the theory of the most relevant mathematical topics in a student-friendly manner. With a clear and straightforward writing style, concepts are introduced through numerous examples, illustrations, and applications. Each section of the text contains an extensive exercise set containing a range of computational, conceptual, and geometric problems. In the text and exercises, students are guided and supported through numerous proofs providing them with a higher level of mathematical insight and maturity. Each chapter contains a separate section devoted exclusively to the applications of complex analysis to science and engineering, providing students with the opportunity to develop a practical and clear understanding of complex analysis. New and Key Features: Clarity of exposition supported by numerous examples Extensive exercise sets with a mix of computational and conceptual problems Applications to science and engineering throughout the text New and revised problems and exercise sets throughout Portions of the text and examples have been revised or rewritten to clarify or expand upon the topics at hand The Mathematica syntax from the second edition has been updated to coincide with version 8 of the software.
Providing practical advice and advanced strategy tips, and discussing specific hands from his victories at the World Series of Poker and high-stakes cash games in which millions of dollars were on the line, this book promises to turn ...
Publisher: Triumph Books
About this book
In "Farha on Omaha," Sam Farha, the world's greatest Omaha player, and Storms Reback, a noted poker writer, offer those new to the game of Omaha poker simple strategic tips that will help transform them into winning players. The authors provide strategies on how to beat the three most popular forms of Omaha--limit, eight-or-better, and pot-limit--in both cash games and tournaments. Providing practical advice and advanced strategy tips, and discussing specific hands from his victories at the World Series of Poker and high-stakes cash games in which millions of dollars were on the line, this book promises to turn beginners into winning players and winning players into champions.
It is almost as if the authors peered into the future as many of the techniques and scenarios in these books have come to pass. This book contains all of the material from each of the four books in the Stealing the Network series.
Publisher: Syngress
About this book into the creative minds of some of today's best hackers, and even the best hackers will tell you that the game is a mental one." – from the Foreword to the first Stealing the Network book, How to Own the Box, Jeff Moss, Founder & Director, Black Hat, Inc. and Founder of DEFCON For the very first time the complete Stealing the Network epic is available in an enormous, over 1000 page volume complete with the final chapter of the saga and a DVD filled with behind the scenes video footage! These groundbreaking books created a fictional world of hacker superheroes and villains based on real world technology, tools, and tactics. It is almost as if the authors peered into the future as many of the techniques and scenarios in these books have come to pass. This book contains all of the material from each of the four books in the Stealing the Network series. All of the stories and tech from: How to Own the Box How to Own a Continent How to Own an Identity How to Own a Shadow Plus: Finally - find out how the story ends! The final chapter is here! A DVD full of behind the scenes stories and insider info about the making of these cult classics! * Now for the first time the entire series is one 1000+ page book* The DVD contains 20 minutes of behind the scenes footage* Readers will finally learn the fate of "Knuth" in the much anticipated Final Chapter
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Note: This is a standalone book, if you want the book/access card please order the ISBN listed below: 0321837533 / 9780321837530 A Survey of Mathematics with Applications plus MyMathLab Student Access Kit59664 / 9780321759665 Survey of Mathematics with Applications, A
For courses covering general topics in math course, often called liberal arts math, contemporary math, or survey of math. Everyday math, everyday language. The Tenth Edition of A Survey of Mathematics with Applications continues the tradition of showing students how we use mathematics in our daily lives and why it's important, in a clear and accessible way. With straightforward language, detailed examples, and interesting applications, the authors ensure non-majors will relate to the math and understand the mathematical concepts that pervade their lives. With this revision, an expanded media program in MyMathLab, and a new workbook further build upon the tradition of motivating and supporting student learning. Also available with MyMathLab MyMathLab is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and engage with media resources to help them absorb course material and understand difficult concepts. NEW! This edition's MyMathLab course provides additional tools to help with understanding and preparedness. Note: You are purchasing a standalone product; MyLab™ & Mastering™the physical text and MyLab & Mastering, search for: 0134115767 / 9780134115764 * A Survey of Mathematics with Applications plus MyMathLab Student Access Card -- Access Code Card Package Package consists of: 0134112105 / 9780134112107 * A Survey of Mathematics with Applications 0321431308 / 9780321431301 * MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 * MyMathLab Inside Star Sticker
Students, if interested in purchasing this title with MyMathLab, ask your instructor for the correct package ISBN and Course ID. Instructors, contact your Pearson representative for more information. If you would like to purchase both the physical text and MyMathLab, search for: 013398107X / 9780133981070 Finite Mathematics and Calculus with Applications Plus1979400 / 9780321979407 Finite Mathematics and Calculus with Applications
This is the eBook of the printed book and may not include any media, website access codes, or print supplements that may come packaged with the bound book. SURVEYING: PRINCIPLES & APPLICATIONS, 9/e is the clearest, easiest to understand, and most useful introduction to surveying as it is practiced today. It brings together expert coverage of surveying principles, remote sensing and other new advances in technological instrumentation, and modern applications for everything from mapping to engineering. Designed for maximum simplicity, it also covers sophisticated topics typically discussed in advanced surveying courses. This edition has been reorganized and streamlined to align tightly with current surveying practice, and to teach more rapidly and efficiently. It adds broader and more valuable coverage of aerial, space and ground imaging, GIS, land surveying, and other key topics. An extensive set of appendices makes it a useful reference for students entering the workplace. | 677.169 | 1 |
Groups
The Algebra I course begins with a fast-paced review of the previous year. The beauty, clarity, and utility of algebraic reasoning are explored through practical and not so practical challenges. We will conclude with a study of the quadratic formula, and introduce formal logical reasoning.
The main goals of this class are, 1) a last-ditch effort to recover the sense of wonder in authentic, creative mathematical thinking after too many years of "school math" and, 2) an opening exploration of abstract, algebraic thinking and the awesome, multi-dimensional world it enables us to model with mathematical precision.
In eighth grade, students begin bulding functional furniture using
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in these projects, as well as a greatly expending set of other tools. Mortise
and tennon joints are used to join the legs to the top. | 677.169 | 1 |
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A former math nerd like me also remembers these stuff quite well!
Here you'll learn how to represent the standard deviation of a normal distribution on a bell curve. Then you'll use the Empirical Rule to solve normal distribution problems. #Math #Empirical Rule #CK-12 | 677.169 | 1 |
This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional... more...
From signed numbers to story problems — calculate equations with ease Practice is the key to improving your algebra skills, and that's what this workbook is all about. This hands-on guide focuses on helping you solve the many types of algebra problems you'll encounter in a focused, step-by-step manner. With just enough refresher explanations... more...
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the cl- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the... more... | 677.169 | 1 |
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This course on turbulence modeling begins with a careful discussion of turbulence physics in the context of modeling. The exact equations governing the Reynolds stresses, and the ways in which these equations can be closed, is outlined. The course begins with the simplest turbulence models and charts a course leading to some of the most complex models that have been applied to a nontrivial turbulent flow problem. The course stresses the need to achieve a balance amongst the physics of turbulence, mathematical tools required to solve turbulence-model equations, and common numerical problems attending use of such equations. | 677.169 | 1 |
We will use MATLAB extensively in this course, and you will learn how to use it as we proceed. A basic introduction will not be given, but here are a few resources that will help you learn the basics. First, here's a link to a nice book chapter Intro to MATLAB for Egineering Studens from a course at Northwestern; second, here's a link to some lectures in Power Point format Intro to MATLAB for a course at MIT; third, here's some materials presented in html format MATLAB tutorial from a professor at Michigan Tech; and last, here's a web tutorial MATLAB tutorial created by MATLAB that I haven't watched. There are many more introductions to MATLAB that can be found via a Google search. | 677.169 | 1 |
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reading | 677.169 | 1 |
The fourth edition of this classic introduction to analysis retains the freshness of the first edition as well as its charming conversational style. This augmented and updated edition was completed by Harold Boas after his father's death. He calls the book "an heirloom and a memorial."
As originally published this book had two parts — chapters called "Sets" and "Functions". The latter took the discussion up through the analysis of continuous and differentiable functions, but stopped short of any treatment of integration. In his preface to the third edition, the author notes that he left out integration "reluctantly, because of the many technical details that are needed before one gets to the interesting results." He did, however, produce a draft of a chapter on integration (without all the technical details) on the grounds that "one need not understand the inner workings of the motor to appreciate a drive in the country." Indeed, it turns out to be a very pleasant ride. Harold Boas reworked his father's draft and added notes, exercises and solutions for what became the third chapter in this fourth edition.
This is not in any way a traditional textbook. It is more like a series of informal lectures, wordy, chatty and not the least bit concise. The author's aim, and the book's great strength, is to bring back a sense of wonder to a subject that, in his opinion, had been lost. The intended reader should have had a course in calculus. Nothing more — other than perhaps a small dose of that elusive thing called mathematical maturity — is called for. Typically the author starts slowly but the level of difficulty rises steeply. The author advises the reader to skip forward if the going is too tough, and he makes this approach workable.
Here are some elements of the book that I found especially notable: the treatment of the Baire Category theorem and its use to prove the existence of a continuous, ever-oscillating function; singular functions and an example of two functions with the same derivative that do not differ by a constant; the universal chord theorem for periodic continuous functions; and the Riesz representation theorem for Stieltjes integrals.
The level of proof varies considerably throughout: sometimes detailed proofs, sometimes sketches, sometimes "it can be shown". Exercises abound; they range from items essential to the text to the illustrative. Solutions to all the exercises are provided. The notes at the end of many sections are a true highlight. They are full of stories, references, and connections.
The intended audience for this book is the neophyte. As such, the book could be used for independent study or as a supplement to a more standard analysis text. Much of the material would also be of interest to more advanced students. Indeed, this is a book for anyone interested in renewing their sense of wonder in analysis | 677.169 | 1 |
The Sine Function
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Algebra 2 Honors: The Sine Function
This lesson is intended for Algebra 2 students with a general knowledge of right triangle properties. This is the fourth lesson of a six-topic unit on PERIODIC FUNCTIONS & TRIGONOMETRY.
The file includes an 8-page Bound-Book Dinah Zike Foldable*, used with permission, a Smart Notebook 11 Lesson Presentation, directions for making the Foldable*, and a completed answer key.This lesson is designed for ALGEBRA 2 students.
Students will be able to identify properties of the sine function and graph sine curves | 677.169 | 1 |
Algebra Through Puzzles
We'll cover equations, ratios, sequences, exponents, and more, while showing how algebra works and why it matters. You'll also learn unique problem-solving approaches in Algebra that aren't typically covered in school, which will help hone the intuition and strategic thinking that you use when approaching difficult problems. | 677.169 | 1 |
Year 12 Mathematical Methods
Year 12 Mathematical Methods
For parents and students alike, choosing the appropriate SACE courses can be a challenging and confusing experience, especially when there are a number of options, as is the case with mathematics.
This guide, produced by our vastly experienced maths tutors, is designed to assist parents and pupils to understand the various SACE mathematics courses that are available in order to help you to make better informed choices and to ensure that you are choosing a course that is appropriate for your needs and capabilities.
Mathematical Methods v Mathematical Studies
There is a large degree of overlap in terms of content in these two subjects. Both include units on Matrices, Statistics, Exponents and Logarithms, Differential Calculus, and Modelling. The major difference is that the Mathematical Methods course includes a unit on Linear Modelling that features Linear Programming, whereas this unit is not covered in Mathematical Studies. Instead, Mathematical Studies includes a unit on Integral Calculus, which is not found in Mathematical Methods .
There are also some differences within the common topics. For example, Mathematical Methods students are expected to know and use residuals when discussing models. In addition, while both Mathematical Methods and Mathematical Studies students study Confidence Intervals in Statistics, Hypothesis Testing is only covered by Mathematical Studies students.
There are also more subtle differences between the subjects. Methods tends to concentrate on the applications of the mathematics and while this is also true for Studies, there is more of an expectation in the latter that students will be familiar with the theoretical aspects of the topics. The other difference is perhaps harder to quantify, but the general feeling is that Mathematical Methods is slightly "easier" than Mathematical Studies.
An important point which should be taken into account when deciding which of these two subjects you choose is how they are viewed by Universities. From 2016, all students who achieve at least a C- in Mathematical Studies will gain 2 Bonus Points from all three South Australian universities when calculating their ATAR. These Bonus Points are not gained by Mathematical Methods students. | 677.169 | 1 |
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THE PRINCETON REVIEW GETS RESULTS Ace the SAT math sections with 10 need-to-know essential topics for acing the exam. In our number-crunching world, basic math knowledge is a must--especially for acing tests like the SAT. For many people, though, math is confusing and often anxiety inducing. That's why we've created"e; SAT Power Math,"e; which uses a simple, straightforward approach to break down and explain complicated math concepts and common problems. This book is your powerful tool for building essential math skills for the SAT, school, and beyond. "e; Everything You Need to Help Achieve a High Math Score."e; - A comprehensive review of math topics like algebra, geometry, and statistics - Strategies for cracking the most common question types found on the SAT - A glossary of key math terms at the end of every chapter "e; Practice Your Way to Perfection."e; - Practice drills for every math topic covered in the book - Detailed step-by-step answer explanations - Targeted strategies to help you score high on the math section of the SAT | 677.169 | 1 |
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A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. In general, matrices can contain complex numbers but we won't see those here | 677.169 | 1 |
Portland Community College
Mathematics
Mathematics includes the study of numbers, patterns, graphs, and abstract models using analytic reasoning and systematic problem solving skills. Mathematics and mathematical reasoning are used in situations as diverse as household budgeting and space shuttle design, subjects as different as art and law, and occupations as varied as nursing and computer programming. Mathematics can be used by everyone to enhance their understanding of the world.
PCC offers developmental and pre-college math courses (numbered below 100) that focus on algebraic skills and prepare students for certificate programs, two year degree programs, and college level coursework. Math courses at PCC numbered 100 and above are equivalent and transferable to the similarly numbered courses at Oregon's public universities. All math classes at PCC are designed to challenge students to improve their analytic reasoning, problem solving, and communication skills.
Transferable Credits
Although PCC does not offer a degree in math, math courses are lower division collegiate courses that transfer to a four-year college or university. Math courses may transfer as:
elective credits
program requirement credits
and/or graduation requirements for the receiving institution
Students are always encouraged to check with the receiving institution, your PCC academic advisor and the University Transfer website for the most accurate and timely transfer requirement information. | 677.169 | 1 |
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Maths
The Mathematics Department aims to enable all pupils to acquire mathematical skills and knowledge and to provide opportunities for them to use these skills to undertake problem solving with confidence, enjoyment and success. Pupils are encouraged to think for themselves within a clear, rigorous mathematical framework, to be critical of their own thinking and prepared to test, justify and improve their conclusions or solutions. Students are assessed at the end of each term and they also complete an end of year test.
Year 7
In Year 7 the aim is to ensure that the students' experience of transitioning to secondary school is positive and challenging. Formal, whole class learning is introduced, especially of topics new to all. There is an intended overlap with the primary school syllabus on basic topics. Exercises are set on these basic topics to reassure, to build confidence and to challenge. Mathematical activities are planned which enable teachers to get to know students individually and which encourage students to work cooperatively. Emphasis, encouragement and support is given to the establishment of the standards of organisation (equipment, books, deadlines) and of presentation of written work that are expected at the Wellington Academy.
Year 8
In Year 8 basic understanding and competence are established in fundamental mathematical topics – algebraic simplification, solving equations, using percentages and ratio, finding areas and volumes, drawing graphs, finding and using averages and calculating probability. Problem solving tasks and investigations are based on recognising patterns, solving puzzles and appreciating strategy in mathematical games. Tasks are chosen in which it is possible for the pupil to relate their solution to the underlying mathematical structure. The curriculum provides the foundation from which students tackle their GCSE course in Years 9, 10 and 11.
GCSE
"Do not worry about your difficulties in mathematics, I assure you that mine are greater." - Albert Einstein
Mathematics is a core subject at GCSE and all students will take a full GCSE qualification or an Entry Level certificate. The Maths GCSE encourages students to develop confidence in, and a positive attitude towards mathematics and to recognise the importance of mathematics in their own lives and to society. Skills that are developed throughout the course include the management of money, key mathematical concepts for a range of professions (i.e. engineering, medicine, finance), and general problem solving and thinking skills.
The students will develop their knowledge, skills and understanding of mathematical methods and concepts in the following areas:
They will use their knowledge and understanding to make connections between mathematical concepts and apply the functional elements of mathematics in everyday and real-life situations. The content of the GCSE course is a natural progression from the Key Stage 3 syllabus. Topics covered in Year 9 will be revisited to ensure that progression through each unit of work is based on a solid foundation.
How will I be assessed?
The GCSE is available at Higher tier (grades A*-D) and Foundation tier (grades C-G). The assessment consists of two exam papers each lasting one hour forty-five minutes which are taken at the end of Year 11. One exam paper is non-calculator. There is no controlled assessment.
The Entry Level Certificate is assessed using a series of tasks which are completed during normal lessons.
A Level Mathematics
A Level mathematics is one of the most revered A Level qualifications that can be attained in KS5 and is sought after by top Universities and employers alike. The course is designed so that you will develop your ability to reason logically, extend your range of mathematical skills and use mathematics as an effective means of communication. Students are required to have achieved a minimum of a grade B in GCSE Mathematics to be considered for study at A Level.
Throughout the course you will study four Core Mathematics modules which will build upon the topics studied at GCSE. Your studies will be supplemented by applications units in Statistics and Decision Mathematics.
In Statistics we look at the process of data collection and how such data can be analysed.
Decision Maths is an aspect of mathematics that has developed in line with the growth in computer technology and looks at the use of algorithms to solve problems.
Course Content
In Year 12 students will study two units of pure mathematics (Core 1 and Core 2) and Statistics 1 (S1). In Year 13 a further two units of pure mathematics are studied (Core 3 and Core 4) along with a module in Decision Mathematics (D1)
Examinations
All modules are assessed with a written examination at the end of each year. All modules carry equal weighting.
A Level Further Mathematics
This course is designed for students who wish to study Mathematics at university or pursue a mathematics related subject beyond A Level. The course builds upon the Core Mathematics component of the A Level course with additional optional modules in Mechanics, Statistics or Decision Maths. The recommendation is that students need a minimum of a grade A at GCSE to succeed on this course.
Course Content
In Year 12 students currently study three units, Further Pure 1 (FP1) which is compulsory and Mechanics 1 and Mechanics 2 which are optional and currently dependent to the requirements of the students.
In Year 13 students take three more units; either one pure unit (Further Pure 2 (FP2) or Further Pure 3 (FP3)) and two applied units or two pure units FP2 and FP3 and one applied unit. The course is quite flexible and depending on students numbers can be tailored to their needs. Previous courses have opted for FP1, FP2, FP3 and M1, M2 and M3.
Assessment All modules are assessed with a written examination. All modules carry equal weighting.
Year 12
Further mathematics GCE-AS - Edexcel
Year 13
Further Mathematics GCE-A2- Edexcel
Click here for more information on the Edexcel GCE Maths Specification and course material. | 677.169 | 1 |
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College Uses Math Software to Interest Students in Calculus
02/01/96
Lafayette College is a small liberal-arts college in Easton, Pa., with an unusually large mathematics faculty. Over the past five years the math department has integrated a software program called Mathematica into its curriculum, beginning with its Scientific Calculus course. Several Lafayette faculty members teach a three-semester scientific and engineering calculus sequence, all using Mathematica-based teaching materials.
From Champaign, Ill.-based Wolfram Research, Mathematica can be characterized by its ability to perform symbolic calculations and extensive graphics, as well as the type of numerical computation commonly associated with computers.
New capabilities of software systems like Mathematica have a profound impact on the nature of mathematical investigation. The very existence of such tools necessitates a careful rethinking of the undergraduate mathematics curriculum.
For example, the value of having skills in calculational proficiency diminishes with the arrival of such tools. Instead, it becomes important that students learn the concepts underlying the calculations, so that they recognize opportunities to apply this powerful tool.
A New Look at Math
When using Mathematica, students no longer slog through long, difficult calculations, losing interest. Students can think more deeply about the concepts involved, and develop a better understanding of the ideas that underlie these calculations, especially the links with the geometry used to visualize them.
"Mathematica's graphics capabilities also offer a tremendous advantage," comments Dr. Robert Root, assistant professor of mathematics at Lafayette. "By alleviating the wearisome chore of graphing, the software leaves students with energy to look at what they have. This is especially evident in multivariable Calculus, where the 3D graphics can be particularly tedious and difficult to sketch accurately. Mathematica plots, in contrast, are simple to create and easy to interpret."
"I am convinced that what separates most students from the mathematical maturity discussed above is an ability to visualize what is going on," notes Root. "If I can show them an insightful plot or animation and make a connection with the algebra, they see calculus in a new way. This revelation is especially rewarding for students who never realized they had a talent for mathematics."
Concepts, Not Complications
Lafayette calculus students attend three lectures and one lab session a week, each lasting 75 minutes. Root relies on Mathematica's cross-platform compatibility to create his courseware on a Macintosh, then ports it to Windows for the lab's PCs.
In the lab, students use interactive Mathematica files, called "notebooks," plus lab handouts, which they complete with an accompanying lab report. Because the software is also a programming language, Root's lab notebooks often combine standard Mathematica with a few programmed functions that illustrate particular points.
He has written one such package that animates Newton's method. Another illustrates the "Brachistochrone Problem" with a race between beads sliding down different curves. These visualizations help the students to focus on the concepts, and sustain their interest through the accompanying algebra and calculus.
Traditional vs. Progressive
Root and the other professors must consider carefully how the Mathematica labs affect the rest of the course and how students think about math. "It's quite a shock for many first-year students to discover their assignments exclude problems with long, involved computations. That is no longer an important part of the curriculum," says Root.
Even in the traditional areas of the course, the presence of Mathematica is pervasive. It affects every lecture by Root and his colleagues. The use is deliberate ¬ to change the students' outlook on mathematics -- on what they can do, and on what is reasonable for them to attempt.
"I want students who not only understand the math concepts involved, but who can also perform perfunctory calculations without Mathematica," Root says. "I expect them to turn to the program only for intricate problems, to avoid wasting time on extensive calculations."
"We cannot reach all of our students," admits Root. "I nonetheless believe that Mathematica helps reach more, and in particular helps us encourage students who have never been excited about mathematics to develop their analytical abilities."
This article originally appeared in the 02 | 677.169 | 1 |
Discrete Math Jeopardy- Digraphs & Scheduling
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Jeopardy Review on Digraphs and Scheduling. Has 30 different questions with answers. Lots of practice on identifying what points are adjacent to and incident to on a digraph, drawing digraphs, and figuring out what the in degree and the out degree are. Also there are questions related to scheduling, making a schedule, what is the optimal time and finding the critical path. | 677.169 | 1 |
Mathematics (MATH)
MDL is a discovery-based project course in mathematics. Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. No lecture component; in-class meetings reserved for student presentations, attendance mandatory. Admission is by application: Motivated students with any level of mathematical background are encouraged to apply. WIM.
MATH 104. Applied Matrix Theory. 3 Units.
Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. (MATH 113 offers a more theoretical treatment of linear algebra.) Prerequisites: MATH 51 and programming experience on par with CS106nnMath 104 and EE103/CME103 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of EE103 is on a few linear algebra concepts, and many applications.
Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM.
Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization. (MATH 104 offers a more application-oriented treatment.).
The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 51.
MATH 116. Complex Analysis. 3 Units.
Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. (MATH 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.
Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite: MATH 120. Also recommended: 113.
Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.
MATH 142. Hyperbolic Geometry. 3 Units.
An introductory course in hyperbolic geometry. Topics may include: different models of hyperbolic geometry, hyperbolic area and geodesics, Isometries and Mobius transformations, conformal maps, Fuchsian groups, Farey tessellation, hyperbolic structures on surfaces and three manifolds, limit sets. Prerequisites: some familiarity with the basic concepts of differential geometrynand the topology of surfaces and manifolds is strongly recommended.
Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
MATH 155. Analytic Number Theory. 3 Units.
Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.
Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with emphasis on applications from economics, physics and engineering, such as filtering and control. Prerequisites: exposure to basic probability.
Same as: CME 298
Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs.
MATH 162. Philosophy of Mathematics. 4 Units.
(Graduate students register for PHIL 262.) General survey of the philosophy of mathematics, focusing on epistemological issues. Includes survey of some basic concepts (proof, axiom, definition, number, set); mind-bending theorems about the limits of our current mathematical knowledge, such as Gödel's Incompleteness Theorems, and the independence of the continuum hypothesis from the current axioms of set theory; major philosophical accounts of mathematics: Logicism, Intuitionism, Hilbert's program, Quine's empiricism, Field's program, Structuralism; concluding with a discussion of Eugene Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. Students won't be expected to prove theorems or complete mathematical exercises. However, includes some material of a technical nature. Prerequisite: PHIL150 or consent of instructor.
Same as: PHIL 162, PHIL 262
MATH 163. The Greek Invention of Mathematics. 3-5 Units.
How was mathematics invented? A survey of the main creative ideas of ancient Greek mathematics. Among the issues explored are the axiomatic system of Euclid's Elements, the origins of the calculus in Greek measurements of solids and surfaces, and Archimedes' creation of mathematical physics. We will provide proofs of ancient theorems, and also learn how such theorems are even known today thanks to the recovery of ancient manuscripts.
Same as: CLASSICS 136
MATH 171. Fundamental Concepts of Analysis. 3 Units.
Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 61CM or 61DM or 115 or consent of the instructor. WIM.
MATH 172. Lebesgue Integration and Fourier Analysis. 3 Units.
Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
MATH 173. Theory of Partial Differential Equations. 3 Units.
A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. In years when MATH 173 is not offered, MATH 220 is a recommended alternative (with similar content but a different emphasis). Prerequisite: 171 or equivalent.
MATH 174. Calculus of Variations. 3 Units.
An introductory course emphasizing the historical development of the theory, its connections to physics and mechanics, its independent mathematical interest, and its contacts with daily life experience. Applications to minimal surfaces and to capillary surface interfaces. Prerequisites: MATH 171 or equivalent.
Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions (including exponentials and logarithms). You must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
MATH 193. Polya Problem Solving Seminar. 1 Unit.
Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.
MATH 197. Senior Honors Thesis. 1-6 Unit.
Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll.
MATH 198. Practical Training. 1 Unit.
Only for undergraduate students majoring in mathematicsMATH 199. Independent Work. 1-3 Unit.
For math majors only. Undergraduates pursue a reading program; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for math majors. Approval of Undergraduate Affairs Committee is required to use credit for honors majors area requirement. Contact department student services specialist to enroll.
MATH 20. Calculus. 3 Units.
The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus, and the Mean Value Theorem for integrals. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value problems, exponential and logistic models, direction fields, and parametric curves. Prerequisite: MATH 19 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
MATH 205A. Real Analysis. 3 Units.
Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.
Review of limit rules. Sequences, functions, limits at infinity, and comparison of growth of functions. Review of integration rules, integrating rational functions, and improper integrals. Infinite series, special examples, convergence and divergence tests (limit comparison and alternating series tests). Power series and interval of convergence, Taylor polynomials, Taylor series and applications. Prerequisite: MATH 20 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
This course will be an introduction to Riemannian Geometry. Topics will include the Levi-Civita connection, Riemann curvature tensor, Ricci and scalar curvature, geodesics, parallel transport, completeness, geodesics and Jacobi fields, and comparison techniques. Prerequisites 146 or 215B.
Image denoising and deblurring with optimization and partial differential equations methods. Imaging functionals based on total variation and l-1 minimization. Fast algorithms and their implementation.
Same as: CME 321A
MATH 221B. Mathematical Methods of Imaging. 3 Units.
Array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of array imaging algorithms.
Same as: CME 321B
Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: MATH 131 and MATH 136/STATS 219, or equivalents.
MATH 228. Stochastic Methods in Engineering. 3 Units.
The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.
Same as: CME 308, MS&E 324
Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.
Same as: STATS 351A
MATH 231C. Free Probability. 3 Units.
Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite: STATS 310B or equivalent.
MATH 232. Topics in Probability: Percolation Theory. 3 Units.
An introduction to first passage percolation and related general tools and models. Topics include early results on shape theorems and fluctuations, more modern development using hyper-contractivity, recent breakthrough regarding scaling exponents, and providing exposure to some fundamental long-standing open problems. Course prerequisite: graduate-level probability.
Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite: MATH 230A or STATS 310. Offered every 2-3 years.
Same as: STATS 374
MATH 235A. Topics in combinatorics. 3 Units.
This advanced course in extremal combinatorics covers several major themes in the area. These include extremal combinatorics and Ramsey theory, the graph regularity method, and algebraic methods.
Introduction to mathematical models of complex static and dynamic stochastic systems that undergo sudden regime change in response to small changes in parameters. Examples from materials science (phase transitions), power grid models, financial and banking systems. Special emphasis on mean field models and their large deviations, including computational issues. Dynamic network models of financial systems and their stability.
Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing.
Same as: CME 372
Covers a list of topics in mathematical physics. The specific topics may vary from year to year, depending on the instructor's discretion. Background in graduate level probability theory and analysis is desirable.
Same as: STATS 359
MATH 280. Evolution Equations in Differential Geometry. 3 Units.
.
MATH 282A. Low Dimensional Topology. 3 Units.
The theory of surfaces and 3-manifolds. Curves on surfaces, the classification of diffeomorphisms of surfaces, and Teichmuller space. The mapping class group and the braid group. Knot theory, including knot invariants. Decomposition of 3-manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3-manifolds. May be repeated for credit up to 6 total units.
The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
MATH 284B. Geometry and Topology in Dimension 3. 3 Units.
The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
MATH 286. Topics in Differential Geometry. 3 Units.
May be repeated for credit.
MATH 298. Graduate Practical Training. 1 Unit.
Only for mathematics graduate studentsThis course is a series of case-studies in doing applied mathematics on surprising phenomena we notice in daily life. Almost every class will show demos of these phenomena (toys and magic) and suggest open projects. The topics range over a great variety and cut across areas traditionally pigeonholed as physics, biology, engineering, computer science, mathematics ¿ but, instead of developing sophisticated mathematics on simple material, our aim is to extract simple mathematical understanding from sophisticated material which, at first, we may not yet know how to pigeonhole. In each class I will try to make the discussion self-contained and to give everybody something to take home, regardless of the background.
MATH 355. Graduate Teaching Seminar. 1 Unit.
Required of and limited to first-year Mathematics graduate students.
MATH 360. Advanced Reading and Research. 1-10 Unit.
.
MATH 382. Qualifying Examination Seminar. 1-3 Unit.
.
MATH 391. Research Seminar in Logic. 1-3 Unit.
Contemporary work. May be repeated a total of three times for credit.
Same as: PHIL 391
MATH 41. Calculus. 5 Units.
Introduction to differential and integral calculus of functions of one variable. Topics: limits, rates of change, the derivative and applications, introduction to the definite integral and integration. MATH 41 and 42 cover the same material as MATH 19-20-21, but in two quarters rather than three. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions, including exponentials and logarithms. *If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Same as: Accelerated
Continuation of 41. Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations, infinite series. Prerequisite: 41 or equivalent. *If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Same as: Accelerated
Geometry and algebra of vectors, matrices and linear transformations, eigenvalues of symmetric matrices, vector-valued functions and functions of several variables, partial derivatives and gradients, derivative as a matrix, chain rule in several variables, critical points and Hessian, least-squares, , constrained and unconstrained optimization in several variables, Lagrange multipliers. Prerequisite: 21, 42, or the math placement diagnostic (offered through the Math Department website) in order to register for this course.
This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space, inverse and implicit function theorems, and many examples. Part of the linear algebra content is covered jointly with MATH 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.
MATH 61DM. Modern Mathematics: Discrete Methods. 5 Units.
This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. Part of the linear algebra content is covered jointly with MATH 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.nnThis sequence is not appropriate for students planning to major in natural sciences, economics, or engineering, but is suitable for majors in any other field (such as MCS ("data science"), computer science, and mathematics).
MATH 62CM. Modern Mathematics: Continuous Methods. 5 Units.
A continuation of themes from MATH 61CM, centered around: manifolds, multivariable integration, and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space and an introduction to general manifolds (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisite: MATH 61CM.
MATH 62DM. Modern Mathematics: Discrete Methods. 5 Units.
This is the second part of a proof-based sequence in discrete mathematics. This course covers topics in elementary number theory, group theory, and discrete Fourier analysis. For example, we'll discuss the basic examples of abelian groups arising from congruences in elementary number theory, as well as the non-abelian symmetric group of permutations. Prerequisites: 61DM or 61CM.
MATH 63CM. Modern Mathematics: Continuous Methods. 5 Units.
A proof-based course on ordinary differential equations, continuing themes from MATH 61CM and MATH 62CM. Topics include linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations with some applications to manifolds, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisites: MATH 61CM and MATH 62CM.
MATH 63DM. Modern Mathematics: Discrete Methods. 5 Units.
Third part of a proof-based sequence in discrete mathematics. This course covers several topics in probability (random variables, independence and correlation, concentration bounds, the central limit theorem) and topology (metric spaces, point-set topology, continuous maps, compactness, Brouwer's fixed point and the Borsuk-Ulam theorem), with some applications in combinatorics. Prerequisites: 61DM or 61CM.
MATH 70SI. The Game of Go: Strategy, Theory, and History. 1 Unit.
Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player.
Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.
MATH 87Q. Mathematics of Knots, Braids, Links, and Tangles. 3 Units.
Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics. | 677.169 | 1 |
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In this playlist, students explore standard HSF.IF.B.6. They learn how to calculate the rate of change of a function over a specified interval and estimate a rate of change. Students look at examples of graphs and are guided through calculating and estimating the rate of change of a function. Students also have the option to view instructional videos and complete practice quizzes or activities.
The playlist includes:
• Links to two practice quizzes or activities
• Links to four instructional videos or texts
• Definitions of key terms, such as slope and function
Accompanying Teaching Notes include:
• Review of key terminology
• Links to video tutorials for students struggling with certain parts of the standard, such as calculating slope | 677.169 | 1 |
able to:
Evaluate definite integral using Riemann Sum.
Evaluate definite integral using the
trapezoidal rule.
Evaluate definite integral using Simpsons rule.
Evaluate definite integral using Romberg
integration
the end of the period, you should be
able to:
Determine the roots of an equation
using bisection method.
Determine the roots of an equation
using Newton-Rhapson method.
Determine the roots of an equation
using secant method.
Determine the roots of
SECANT METHOD
In numerical analysis, the secant method is a rootfinding algorithm that uses a succession of roots
of secant lines to better approximate a root of a
function f.
The first two iterations
of the secant method.
The red curve shows the
funct
At the end of the period, you should be able to:
Distinguish numerical solution from analytical
solution.
Get an overview of the topics that will be
encountered in numerical methods.
Get an idea of the necessary prerequisite
topics that will be needed
At
the end of the period, you should be
able to:
Use Quadratic interpolation to find the
maximum of a function.
Use Newtons method to find the
maximum of a function.
Root location and optimization are related in the
sense that both involve guessing an
At the end of the period, you should be
able to:
Use linear regression to determine the
best fit curve that can approximate a
group of ordered data points.
Use nonlinear regression to determine
the best fit curve that can approximate
a group of ordered | 677.169 | 1 |
7th Grade Module 1: Rate of Change / Graphing Linear Relationships
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Students complete two tables:
1. Completed Homework Assignments/ Weeks in School
2. Temperature / Time
Students graph the two relationships on two separate graphs.
Then, they use these graphs to make conjectures about whether the relationships are linear and/or proportional. Students calculate the rate of change and must explain their reasoning. | 677.169 | 1 |
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra is one of the important theorems of the mathematics.
The statement of the fundamental theorem of algebra is that every polynomial which has only
one variable which is not constant and which has the coefficients which are complex
possesses at least one root which is complex because we already know that the coefficients
which are real and the roots which are real come inside the definition of numbers which are
complex.
According to the statement of the fundamental theorem of algebra given in the last paragraph,
the fundamental theorem of algebra can alternatively be stated as follows.
The fundamental theorem of algebra says that the field of the numbers which are complex is
shut down algebraically.
Other than the statements of the fundamental theorem of algebra given in the last 2
paragraphs, the fundamental theorem of algebra can also be stated in a different manner as
follows.
The fundamental theorem of algebra states that each single polynomial which has only single
variable and is other than zero and which has coefficients which are complex possesses the
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no. of roots which are complex equal to the degree of the polynomial with the condition that
every single root is seen up to the multiplicity of the root.
After reading this statement of the fundamental theorem of algebra it seems that this is the
more powerful statement of the fundamental theorem of algebra but it should be known that
this statement of the fundamental theorem of algebra is just derived from the other statements
of the theorem given in the earlier paragraphs by the way of the continuous division of the
polynomial by the factors which are linear.
Although the word algebra is used in the name of the theorem as the name 'fundamental
theorem of algebra ' suggests but it should be known that there exists no proof of the
fundamental theorem of algebra which is completely algebraic.
Also the name of the theorem says that it is the fundamental theorem but the fundamental
theorem of algebra is not at all fundamental for the modern type algebra because the name of
the theorem was proposed at some time during which the study of the algebra was just related
to the solutions containing polynomial having the coefficients which could be either complex or
real.
We have discussed enough about the various statements of the fundamental theorem of
algebra so let us now discuss something about the proofs of the fundamental theorem of
algebra.
Every proof of the fundamental theorem of algebra contain some kind of the analysis and if
not that then at least it will contain the concept of the continuity of the real functions or the
complex functions which will be topological.
Some of the proofs of this fundamental theorem of algebra also utilize such type of the
functions which can be analytic or differential. These are the facts which have proved that the
fundamental theorem of algebra is not at all fundamental and also that it is not even the
theorem of the algebra.
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This because the proofs which are algebraic use only 2 things about the real numbers that
each polynomial which has odd degree and contain coefficients which are real possesses real
root and each positive real number possesses a square root.
Example 1:- Factorise completely: f (x) = x4 – 1 using fundamental theorem of algebra
Solution:- We know that since n = 4, there are exactly 4 complex zeros, roots, and linear
factors for f. The factorization for f could be done in this way:
f (x) = x4 - 1
= (x2 - 1 ) (x2 + 1)
= (x + 1)(x - 1)(x + i )(x - i)
These are the four linear factors of f and the four zeros of f are x = Âą 1 and x = Âąi
Example 2 : - Factoring a polynomial completely: f (x) = x3 - x2 using fundamental theorem of
algebra.
Solution :- The factorization for f could be done in this way,
f (x) = x3 - x2
We can pull out common terms x2 :
x3 - x2 = x2 (x - 1).
= x2 ( x - 1 )
We have a factored the polynomial into three linear factors, thus the factorization is complete
using fundamental theorem of algebra. | 677.169 | 1 |
Basic Modern Algebra.pdf
书籍描述
内容简介 Part of the Jones & Bartlett Learning International Series in Mathematics Student-friendly and accessible in its approach, Basic Modern Algebra provides an introduction to abstract algebra for junior and senior undergraduates. Familiarity with factoring and solving polynomial equations, and properties of the integers, rationals, reals, as well as complex numbers and matrices is helpful prerequisite knowledge for this course. In an effort to promote clarity and understanding, the author is careful to provide full reasoning behind the conclusions to the important mathematical examples at hand. The text opens with a review and refocus chapter on writing proofs and from there moves on to integers, including number theory. With its flexible design, Basic Modern Algebra can be taught following either a rings first or group first approach. Instructor resources include PowerPoint Lecture Outlines, solutions to all of the text's exercises, an image bank, and a Test Bank. Key Features of Basic Modern Algebra *Includes a wealth of examples throughout to clearly illustrate and introduce key concepts. *Simple terminology is used in an effort to promote full understanding of complex material. *Provides full detail on proofs and proof writing *Mathematical portraits and histroical notes discuss the people behind the mathematics *A full glossary allows students to quickly define algebraic terms. | 677.169 | 1 |
As we learned on the previous page, vectors alone have limited use other
than providing a simple, yet effective, means of displaying quantities possessing both a
magnitude and direction. The real power in vectors resides in the ability to perform
mathematical operations on them.
An algebra is a set of mathematical rules. And in order to use
vector algebra, you have to know the rules. Fortunately for life science majors,
there is only one rule you have to remember -- the rule for adding two vectors
together. (We will see that the operation of subtraction is essentially the same as
addition. And if you can add two vectors together, then adding three or more vectors
is straightforward.)
Those studying a calculus-based physics course also have to consider how to
multiply vectors, but we will not concern ourselves with this added burden. So
vector algebra is actually simpler than regular algebra because we only have to concern
ourselves with one operation -- addition.
Some Important Points about
Notation and Definitions
We need notation for labeling
vectors: A vector is denoted by a bold-face letter, and its length is denoted by the
same letter without the bold face.
B
B
A vector, with magnitude and direction
The magnitude of the vector B.
We now need to introduce a definition: The sum of two or
more vectors is another vector called the resultant vector.
Vector addition for two vectors A and B is
simply denoted A + B. Therefore, the equation C
= A + B simply means "C is the
resultant vector obtained by adding vector A to vector B."
Vector subtraction of vector B from vector A is simply
denoted A - B.
There are three ways to add vectors. Each method will be illustrated using
Java applets in the following pages. Naturally, all three methods must produce the
same result.
The head-to-tail method.
The parallelogram method.
The component method.
The head-to-tail and parallelogram methods are actually identical, as a Java
applet will later demonstrate. They only provide a rough description of the
resultant vector, but they are very easy to apply. The component method is used in
those situations where exact, numerical information about the resultant vector is
required. | 677.169 | 1 |
MATH Documents
Showing 1 to 30 of 115
Syllabus
Course Syllabus
Course This is MAT-121, College Algebra, section 03. Your instructor is
nomenclature Roy Kilgore.
You do not need to buy a hard copy text; the e-text is embedded in
this course.
This is a four credit hour course presented in a nom
Using the e-text
The e-text that is embedded in MyMathLab is an exact copy of the hard copy version. There are
several ways to access the e-text.
If you are enrolled in MAT110, skip to the section titled For MAT110.
For MAT121
The first paragraph of the d
Math 120 Chapter 1, Section 1 Handout
Angles and Degree Measure
Angles
A ray is a part of a line that has only one endpoint and extends forever in the opposite
direction.
An angle is formed by two rays that have a common endpoint. One ray is called the
in
Math 120 Chapter 1, Section 4 Handout
The Trigonometric Functions
r
y
x
It is helpful to interpret trigonometric functions in terms of right triangles for certain
kinds of problems.
The Six Trigonometric Functions
If (x, y) is any point other than the ori
Math 120 Chapter 1, Section 3 Handout
Angular and Linear Velocity
Angular Velocity
If a point is in motion on a circle through an angle of
angular velocity
is given by =
t
radians in time t, then its
.
Linear Velocity
If a point is in motion on a circle
Math 120 Chapter 1, Section 2 Handout
Radian Measure, Arc Length, and Area
Radians
Radians are another type of measurement of angles. We use a circle of radius, r, to
measure an angle in radians. A central angle is an angle whose vertex is at the center
o
Math 120 Chapter P, Section 1 Handout
The Cartesian Coordinate System
Cartesian coordinate system
Many applications of math involve linear equations and their graphs (straight lines).
We then need to begin with a review of the Cartesian coordinate system.
Math 120 Chapter 2, Section 4 Handout
Graphs of the Tangent and Cotangent Functions
The Graph of y = tan x
Graph y = tan x by listing some points on the graph. Since the period of tangent is ,
graph the function on [0, /2) and complete the graph on the in
Math 120 Chapter 3, Section 3 Handout
Sum and Difference Identities for Cosine
Cosine of a Sum
cos cos cos sin sin
The cosine of the sum of two angles equals the cosine of the first angle times the cosine
of the second angle minus the sine of the first a
Math 120 Chapter 3, Section 2 Handout
Verifying Identities
Strategy for Verifying Identities
1) Work on one side of the equation (usually the more complicated side), keeping in
mind the expression on the other side as your goal.
2) Some expressions can be
Math 120 Chapter 3, Section 6 Handout
Product and Sum Identities
Product-to-Sum Identities
How do we write the products of sines and/or cosines as sums or differences? We use
the following identities, which are product-to-sum formulas:
1
sin ( A) cos( B )
Math 120 Chapter 3, Section 4 Handout
Sum and Difference Identities for Sine and Tangent
Sine of a Sum or Difference
sin ( + ) = sin ( ) cos( ) + cos( ) sin ( )
The sine of the sum of two angles equals the sine of the first angle times the cosine of the
s
Math 120 Chapter 4, Section 2 Handout
Basic Sine, Cosine, and Tangent Equations
An identity is satisfied by all values of the variable for which both sides are defined. A
conditional equation is an equation that has at least one solution but isnt an ident
Math 120 Chapter 6, Section 1 Handout
Complex Numbers
The Imaginary Unit i
The imaginary unit i is defined as , where , i 3 = i , and i 4 = 1 .
Roots of Negative Number
For any positive real number b, the principal square root of the negative number is
de
Math 120 Chapter P, Section 2 Handout
Functions
Relations
A relation is any set of ordered pairs. The set of all first components of the ordered
pairs is called the domain of the relation and the set of all second components is
called the range of the rel
MyMathLab
Getting Started Guide for Students Spring 2015 in Math 120
MyMathLab within Pearsons My Lab and Mastering is an interactive website where you use customized
materials, do your homework and assessments online, and you can take advantage of tutori
Math 120 Chapter 5, Section 3 Handout
Area of a Triangle
The Area of an Oblique Triangle
The area of a triangle equals one-half the product of the lengths of two sides times the
1
2
sine of their included angle. This can be expressed by the formulas, Area
Math 120 Chapter 5, Section 1 Handout
The Law of Sines
The Law of Sines and Its Derivation
An oblique triangle is a triangle that does not contain a right angle. An oblique triangle
has either three acute angles or two acute angles and one obtuse angle. T
Math 115 Precalculus
Review Sheet for Test 3
Section 2.7
Solving polynomial and rational inequalities. Find the boundary points and build a sign table in order
to find the intervals where it works. The position function for a free-falling object near the | 677.169 | 1 |
Alumni of our math program have been very successful. Our alumni profiles feature some of their positions, including Actuary, Medical Doctor, Lawyer, High School Teacher and Vice President of Information Management.
The five professors in the our math department have a wide variety of mathematical interests, including game theory, mathematical modeling, statistics, chaos theory, geometry, knot theory, graph theory, and differential equations. They are also interested in interdisciplinary applications of math in fields such as political science, economics, education, biology, chemistry and physics.
Why Simpson?
Our innovative curriculum is designed to help students think critically, solve problems creatively and communicate effectively — exactly the kind of skills that employers and graduate schools want most. | 677.169 | 1 |
After being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. Both proofs involve long and complicated applications of algebraic geometry over free groups as well as an extension of methods to solve equations in free groups originally... more...
Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography,... more...
This volume covers many topics, including number theory, Boolean functions, combinatorial geometry, and algorithms over finite fields. It contains many new, theoretical and applicable results, as well as surveys that were presented by the top specialists in these areas. New results include an answer to one of Serre's questions, posted in a letter... more...
Algebraic geometry is one of the most classic subjects of university research in mathematics. It has a very complicated language that makes life very difficult for beginners. This book is a little dictionary of algebraic geometry: for every of the most common words in algebraic geometry, it contains its definition, several references and... more...
The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems and examples. The aim is to collect material that will help mathematicians already working or wishing to work... more...
Designed for a one-semester course at the junior undergraduate level, Transformational Plane Geometry takes a hands-on, interactive approach to teaching plane geometry. The book is self-contained, defining basic concepts from linear and abstract algebra gradually as needed. The text adheres to the National Council of Teachers of Mathematics ... more...
Analytical Quadrics focuses on the analytical geometry of three dimensions. The book first discusses the theory of the plane, sphere, cone, cylinder, straight line, and central quadrics in their standard forms. The idea of the plane at infinity is introduced through the homogenous Cartesian coordinates and applied to the nature of the intersection... more...
Analytical Geometry contains various topics in analytical geometry, which are required for the advanced and scholarship levels in mathematics of the various Examining Boards. This book is organized into nine chapters and begins with an examination of the coordinates, distance, ratio, area of a triangle, and the concept of a locus. These topics are... more...
THE PRESENT six-figure trigonometric tables complete the series of tables of the natural values of the trigonometric functions published by Fizmatgiz. Now that small computers have become very widely available, almost all computations are carried out by machine, and the majority of computational schemes arc suited to this purpose. The situation calls... more...
Ten-Decimal Tables of the Logarithms of Complex Numbers and for the Transformation from Cartesian to Polar Coordinates contains Tables of mathematical functions up to ten-decimal value. These tables are compiled in the Department for Approximate Computations of the Institute of Exact Mechanics and Computational Methods of the U.S.S.R. Academy of Sciences.... more... | 677.169 | 1 |
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