...MA1200 Basic Calculus and Linear Algebra I Lecture Note 1 Coordinate Geometry and Conic Sections υ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Topic Covered • Two representations of coordinate systems: Cartesian coordinates [ሺݕ ,ݔሻcoordinates] and Polar coordinates [ሺߠ ,ݎሻ-coordinates]. • Conic Sections: Circle, Ellipse, Parabola and Hyperbola. • Classify the conic section in 2-D plane General equation of conic section Identify the conic section in 2-D plane - Useful technique: Rotation of Axes - General results φ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Representations of coordinate systems in 2-D There are two different types of coordinate systems used in locating the position of a point in 2-D. First representation: Cartesian coordinates We describe the position of a given point by considering the (directed) distance between the point and -ݔaxis and the distance between the point and -ݕaxis. ݕ 0 ܽ ܲ ൌ ሺܽ, ܾሻ ܾ ݔ Here, ܽ is called “-ݔcoordinate” of ܲ and ܾ is called “-ݕcoordinate” of ܲ. χ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections ܲଶ ൌ ሺݔଶ , ݕଶ ሻ ܲଵ ൌ ሺݔଵ , ݕଵ ሻ Given two points ܲଵ ൌ ሺݔଵ , ݕଵ ሻ and ܲଶ ൌ ሺݔଶ , ݕଶ ሻ, we learned that • the distance between ܲଵ and ܲଶ : ܲଵ ܲଶ ൌ ඥሺݔଶ െ ݔଵ ሻଶ ሺݕଶ െ ݕଵ...
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...Discount: Reduction from the full amount of a price. The following are the five types of discounts which we see are: Simple Discount: Offer a price reduction on a product by a percentage. For example, buy a shirt and receive 25 % off the original price. Minimum Purchase Discount: Offer a price reduction on a minimum quantity purchase. For example, buy two shirts and receive 20 % off each shirt. Buy N, Get one Free. Offer a free gift with a minimum quantity purchase. For example, buy two shirts and receive a third shirt for free• Offer a price reduction on a product if another product is purchased. For example, buy a shirt and receive Rs.10 off a pair of jeans• Paired Set Discount: Offer a price reduction on an item if a certain quantity of another item is purchased. For example, buy three shirts and receive 30 % off a pair of jeans• Order Discount: Offer a price reduction or free shipping on the order total, if a certain amount is purchased. For example, buy Rs. 5000worth of merchandise, and receive 10 % off the total order. Banking: A system of trading in money which involved safeguarding deposits and making funds available for borrowers.* what is the use of mathematics in Banking •Bank is full of transactions. In turn the transaction is nothing but mathematics •Banks are also involved in stocks and bonds. Bond calculations are mathematical. Stock options are also quite mathematical Foreign...
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...August 2013 * Class Schedules (Activities and events) Date and Time | Event | Description | JulyTime to be placed | Issue Diagnostic tests | 1 - 2 hrs.This is used to assess the levels of the students and to highlight where improvements are need versus where they will need to be fully taught. Also to monitor their progress in the Factory Files/Records will be created and maintained for each student involved in the math factory.(Time assessment for each level may differ) | JulyTime to be placed | Consultation with parent(s) and child(ren) | 30 – 60 mins. A consultation will be held, in order for the parents and child come in. We discuss their current progress, where they need to improve and how the parents can help in their development. We also discuss their strengths and how they can harness or fine tune it.This is also where we wish to gather parent and student information in these sessions also | JulyTime to be placed | Arranging of the Classes | 60 – 90 mins. Students will be sorted in their respective grade levels and competencies: * Basics * Primary * High (split between 7,8 and 9,10,11) | JulyTime to be placed | Teaching begins | Introduction of students, register is taken and lesson begins.Class Days: * Tuesday (Basic) * Wednesday (Primary) * Thursday (High)Each group will be taught on different days and each day is two hrs. each | - Time between - | - Teaching - | - Any other activities will be done on a by weekly basis and be presented...
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...filled with notes on light and the human eye. It helped understand the laws of refraction and reflection. “Descartes’ geometry also helped to make possible discoveries in optics, relating to the law of refraction, or the amount that light is bent” (Badertscher 3). Descartes work in mathematics and cartesian system inspired him to work in optics. With the cartesian system, Descartes was able to predict the path of light, after he figured out how to measure light. “Thus all the parts of the subtle matter which are touched by the side of the sun which faces us, tend in a straight line towards our eyes at the very moment that they are opened, without impeding each other and even without being impeded by the heavier parts of the transparent bodies which are between the two: whether these bodies move in other ways, as does air, which is almost always agitated by some wind, or if they be without motion as are perhaps glass and crystal” (Descartes, Dioptrics 5). Descartes is saying that all rays from the sun traveling in a straight line towards our open eyes and they don’t get in each other’s way. Only glass or crystal might distort the straight line of light. Descartes goes on to provide examples of how to bend light. “...these balls had earlier had only a simple rectilinear motion, they will lose a part of it and acquire instead a circular motion, which can have a different proportion with that that they retain of their rectilinear motion, accordingly as the surface of the body which they...
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...the most appealing problems in the classic Euclidean plane geometry. The name of Butterfly Theorem is named very straightforward that the figure of Theorem just likes a butterfly. Over the last two hundreds, there are lots of research achievements about Butterfly Theorem that arouses many different mathematicians’ interests. Until now, there are more than sixty proofs of the Butterfly Theorem, including the synthetical proof, area proof, trigonometric proof, analytic proof and so on. And based on the extension and evolution of the Butterfly Theorem, people can get various interesting and beautiful results. The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and from the point of intersection A to the centre of the circle AO...
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...REPUBLIC OF TRINIDAD AND TOBAGO MINISTRY OF EDUCATION SECONDARY EDUCATION MODERNIZATION PROGRAMME DRAFT SECONDARY SCHOOL CURRICULUM Form Three Mathematics Curriculum Development Division October 2003 TABLE OF CONTENTS About this Draft i Foreword – A Note to Teachers iii Acknowledgements v PART ONE Introduction 1- 1 The Curriculum Underpinnings 1- 2 Philosophy of Education 1- 3 The Goals of Education 1- 5 The Essential Learning Outcomes 1- 6 The Curriculum Design and Development Process 1-11 PART TWO - CURRICULUM CONTENT Vision Statement 2- 2 Rationale for the Teaching and Learning of Mathematics 2- 3 Goals of the Mathematics Curriculum 2- 4 General Intended Outcomes For Forms I, II, and III. 2- 5 Connections to Other Core Curriculum Areas 2- 6 Framework for Mathematics for Forms I, II and III 2- 9 A General Curriculum Framework 2-11 Course Outline for Form III 2-12 PART THREE - STRATEGIES/METHODOLOGIES Teaching and Learning Strategies 3- 2 Suggested Activities 3- 6 Suggested Resources 3-15 PART FOUR - EVALUATION Elaboration of Assessment and Evaluation 4- 2 Evaluation Tools and Strategies 4- 5 Cross-referencing to Teachers’ Guide 4- 7 BIBLIOGRAPHY 4- 9 ___________________________________ i ABOUT THIS DRAFT Under the umbrella of the Secondary Education Modernization Programme (SEMP), since the latter...
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...integers from n to 1or 1 through n can be expressed by the symbol n! (read as n factorial). FACTORING -is it the process of finding the factor of a product. this is the inverse operation of special product. FACTORS -they are the numbers multiplied i.e, both the multiplicand and the multiplier. FINAL AMOUNT -it is the sum of the principal and the interest as computed. it is also called maturity value. FINITE SERIES -It is a series containing a fixed number of terms. FINITE SET -it is a set whose element can be counted or has a limited number of elements. FOOT -a measure of lenth. FOOT-POUND -it is a unit of work. it is the work done in raising 1 lb, a height of 1 ft., ot it is the pressure of 1 lb. xerted over a distance of 1 ft. in any direction. Face of a Polyhedron One of the flat surfaces making up a polyhedron. Note: The faces of a polyhedron are all polygons. Factor of a Polynomial Factorization of a Polynomial A factor of polynomial P(x) is any polynomial which divides evenly into P(x). For example, x + 2 is a factor of the polynomial x2 – 4. The factorization of a polynomial is its representation as a product its factors. For example, the factorization of x2 – 4 is (x – 2)(x + 2). Factor Theorem The theorem that establishes the connection between the zeros and factors of apolynomial. Factor Tree A structure used to find the prime factorization of a positive integer. Factoring Rules Algebra formulas for factoring. 1. x2 – (r...
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...Mathematics Proficiency Of Selected Bachelor of Science in Hospitality Management Students of PSU-NCCRD The Researchers: Bacabac, Jidiahlyn Caboteja, Jerrylyn Dela Pena, Kristine Joy Licay, Merlyn Grace Lucena, Jomarie Magbanua, Carmelle L. Tiosin, Allain B. Zeta, Eldon A. In partial fulfillment of the requirements for the degree Bachelor of Science in Entrepreneurship SY 2014-2015 i Biographical Sketch The first researcher was born on January 5, 1993 and was named Eldon A. Zeta. He studied in San Isidro Elementary School in Princess Urduja, Narra, Palawan. He pursued his high school life in Princess Urduja National High School. He is now currently taking up Bachelor of Science in Entrepreneurship at PSU-Narra CCRD. He is a second year student and currently a Student Government Organization Senator. The second researcher’s born on January 7, 1994 and was named Carmelle Magbanua. She finished her elementary years in Gregorio Oquendo Memorial School and pursued her high school life in Narra National High School. She is now a second year student of Bachelor of Science in Entrepreneurship at PSU-Narra CCRD. The third researcher was born on May 19,1996 and was named Allain B. Tiosin. He finished his elementary years in Narra Pilot School. He pursued his high school years in Narra National High School. He is now a second year student of Bachelor of Science in Entrepreneurship at PSU-Narra CCRD. The fourth researcher’s born on April 21...
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...mathematical ideas and questions. Talking and writing about their mathematical thinking helps ELLs build word knowledge and oral expression and clarify their thinking. Discussions with the teacher or peers are also useful monitoring tools for teachers. Through listening and recording student conversations and peer problem solving, teachers can monitor individual student progress. Mathematics is no longer viewed as isolated, individualistic, or competitive. Mathematics problems are ideally suited to cooperative group discussions because they have solutions that can be objectively demonstrated. Students can persuade one another by the logic of their arguments. Mathematics problems can often be solved by several different approaches, and students in groups can discuss the merits of different proposed solutions (Robertson, Davidson, & Dees, 1994). For this and several other reasons mentioned throughout this chapter, cooperative learning takes a central place in mathematics instruction. Teaching and Learning Meaningful Math The lesson template for ExC-ELL is the same as that described in previous...
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...Nikolai Ivanovich Lobachevsky “The Copernicus of Geometry” Part I – An Intro to the Life & Time of Nikolai Lobachevsky Nikolai Lobachevsky was born and lived in Russia from 1792 until 1856. During this historic time in Russia, one era of rulers ended and another began. In 1796, 7 decades of women rulers came to an end. Catherine the Great died in 1796 after thirty-four years as Empress of Russia. The throne then falls to her son Paul I, whose reign is cut short when he is murdered in his bed in 1801. After Paul’s demise, his son, Alexander I ascends the throne.[4] Alexander I was going to have his work cut out for him. Due to the Russians lack of trust in Western ideas at the end of the 18th century, advances in science and math in Russia where practically non-existent. In fact, the “modern” Saint Petersburg Academy was nearly abandoned. At this low point, the school had only 14 full-time staff members. Upon becoming Tsar, Alexander was determined to reform the suffering education system. He knew that advances in the areas of math & science would help to improve the strength of the military as well as make an impact on the economy of his nation. Just in the first three years after inheriting the throne, Alexander reopened the Dorpat University and opened 3 new universities, including Vilna in 1802, Kazan in 1804, and Karkov in 1804. With the opening of these new institutions, he still faced one major challenge: Who was going to teach the students all this math...
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...CARIBBEAN EXAMINATIONS COUNCIL Caribbean Secondary Education Certificate CSEC MATHEMATICS SYLLABUS Effective for examinations from May/June 2010 CXC 05/G/SYLL 08 Published in Jamaica © 2010, Caribbean Examinations Council All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means electronic, photocopying, recording or otherwise without prior permission of the author or publisher. Correspondence related to the syllabus should be addressed to: The Pro-Registrar Caribbean Examinations Council Caenwood Centre 37 Arnold Road, Kingston 5, Jamaica, W.I. Telephone: (876) 630-5200 Facsimile Number: (876) 967-4972 E-mail address: cxcwzo@cxc.org Website: www.cxc.org Copyright © 2008, by Caribbean Examinations Council The Garrison, St Michael BB11158, Barbados CXC 05/OSYLL 00 Contents RATIONALE. .......................................................................................................................................... 1 AIMS. ....................................................................................................................................................... 1 ORGANISATION OF THE SYLLABUS. ............................................................................................. 2 FORMAT OF THE EXAMINATIONS ................................................................................................ 2 CERTIFICATION AND PROFILE DIMENSIONS .....
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...however, that this review is not intended to be allinclusive—there may be some concepts on the test that are not explicitly presented in this review. Also, if any topics in this review seem especially unfamiliar or are covered too briefly, we encourage you to consult appropriate mathematics texts for a more detailed treatment. Copyright © 2012 by Educational Testing Service. All rights reserved. ETS, the ETS logo, LISTENING. LEARNING. LEADING. and GRE are registered trademarks of Educational Testing Service (ETS). Table of Contents ARITHMETIC .............................................................................................................................. 1 1.1 Integers.................................................................................................................................. 1 1.2 Fractions ................................................................................................................................ 3 1.3 Exponents and Roots............................................................................................................. 5 1.4 Decimals ............................................................................................................................... 6 1.5 Real...
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...are the five words that changed the world of learning in today’s era. But what is Common Core and what makes this education curriculum so common? The Common Core State Standards Initiative is an effort to establish a common set of standards for all public schools in all states. For example, “by the end of first grade, all kids should be able to count to X, add, multiply, divide, know fractions and be able to read a chapter book” (BladdyK). Besides education standards, it standardizes things like homework and tests. This adaptation in a new standard is a work-in-progress, thus giving common core a very vague and broad feeling to it. I've heard from numerous college students within my classrooms...
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...mathematical thinking and the development of mathematical concepts, and make a point to emphasize contributions from both Western and non-Western civilizations. This learning module teaches students about the history of the Pythagorean theorem, along with proofs and applications. Feel free to use your own motivational ideas and tailor it to your students! This lesson is geared toward high school Geometry student that have completed a year of Algebra. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups, I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers. Calculators are optional. This lesson addresses the national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning, specifically: • • • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships Use visualization, spatial reasoning, and geometric modeling to solve problems Understand and apply basic and advanced properties of the concepts of geometry; Use the Pythagorean theorem...
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...Calculus From Wikipedia, the free encyclopedia This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus [show]Integral calculus [show]Vector calculus [show]Multivariable calculus Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus...
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