Classroom Tips and Techniques: Real Distinct Roots of a Cubic. Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
|
|
- Amie Wilson
- 7 years ago
- Views:
Transcription
1 Introduction Classroom Tips and Techniques: Real Distinct Roots of a Cubic Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft The real distinct roots of the cubic equation can be expressed compactly in terms of trig functions by the algorithm in Table 1. Maple's solve command does not use this algorithm, so we explore how we can interpret and compact Maple's solution of this equation. If then three real and distinct roots. Table 1 In trigonometric form, the real distinct roots of the cubic Example
2 The cubic function has three distinct real zeros, as suggested by its graph in Figure 1. Figure 1 Graph of the cubic Indeed, approximate values for these three zeros are given in Table 2. Table 2 Approximate zeros of the cubic Trigonometric Form of the Zeros
3 Application of the algorithm in Table 1 results in the exact zeros shown in Table 3. Table 3 Trigonometric form of the zeros of The zeros in Table 3 are not ordered according to Table 2. In fact, the approximate values of the zeros in Table 3 are given in Table 4. Table 4 in Table 3 Approximate values of the zeros Notice that both and in Table 3 are expressed as sines, contrary to the algorithm in Table 1. To see why, we first determine that, then observe that, where. When, Maple immediately converts to. When, Maple immediately converts to.
4 When, Maple immediately converts to. These transformations are automatic, and cannot be prevented. But they do not cause the trigonometric expressions for the zeros to be any "larger" than they would have been if these transformations did not take place. Maple's Form for the Zeros Maple's solution to the equation is given in the form shown in Table 5, where,, and and are as defined in Table 1. Table 5 Maple's solution to the equation For the equation, Maple's solve command produces the solutions
5 Table 6 Maple's solutions of the equation Even though these three solutions are known to be real, they are expressed with the imaginary unit. At first glance, it wouldn't be clear that these three solutions are real! The three solutions in Table 6 are the members of a list. To this list we apply the evalf command in an attempt to see these solutions as floating-point numbers. Because of the small imaginary parts computed in floating-point form, even these floats do not immediately declare themselves as real numbers. It takes both the fnormal command (to set to zero floats below the threshold of ) and the simplify command to remove the " " remnant of the small imaginary parts. Hence, we get in agreement with Table 4. Table 7 shows the result of applying the simplify command to Maple's analytic solutions.
6 Table 7 Application of the simplify command to Maple's solutions The root retains the imaginary unit, but the other two roots are now expressed in terms of trig functions. Surprisingly, the argument of these trig functions are in terms of the arctangent function, not the arccosine function. Moreover, each such root is the sum of two sine functions, so the form of these solutions is clearly not as compact as those in Table 3. To separate into its real and imaginary parts, the evalc command can be applied. If this command is applied to the solutions in Table 6, an expression of length 1939 is obtained. If this 14-line result is simplified, and are given as in Table 7, and is given as in Table 8. Table 8 Application of simplify and evalc to in Table 7 The form of in Table 8 is the most compact we've been able to obtain for this root. However, because is given in terms of two sine functions, it is possible to make it more compact by applying the expand command that applies the addition formula for the sine of a sum to the first sine function in the expression. But because is expressed in terms of two sine functions whose arguments are sums, little compression is obtained by applying the expand and even the combine commands. These results are given in Table 9.
7 Table 9 The most compact form of Maple's solutions to the cubic The final step in our journey is to investigate why Maple's combine command does not reverse the expansion of when is numeric (and not a name) and is an angle for which the sine and cosine functions evaluate to a radical. A Maple Trig Simplification In Maple, the identity is implemented with the expand and combine commands. The expand command converts the left-hand side to the right-hand side, and the combine command converts the right-hand side to the left-hand side. However, the combine command is programmed to search for products of sines and cosines, in which case the following transformations are applied. Of course, the sum collapses to. This is why Maple's combine command fails to re-combine an expansion such as However, in this case, the relatively new (Maple 13) convert option "phaseamp" effects the re-combination, as we see with = Unfortunately, even this approach fails for something like
8 which is precisely the case, as described after Table 9. Working with pencil and paper, one would resort to a direct application of the elementary trigonometry inherent in where and. For this example, we have so and programmatically for in Table 9. Apply Maple's pattern-matching command to from Table 9.. Table 10 shows how this calculation could be implemented Examine parameters extracted by the patmatch command. = Apply the algorithm. Table 10 The transformation applied to from Table 9 A Final Observation
9 From Table 3 we have while from Table 8 or 9 we have To show directly that set, where and write. From Figure 2, we then have and Hence, does? Using the evaluation template from the expression palette and then the Context Menu, we have =
10 Notational Devices Used There are two notational devices used in this worksheet. First, tinted cells of tables contain hidden Maple input. To see how the displayed output is generated, use the Table menu and select Properties. In this dialog box, there is a checkbox for hiding input. The second device is even more subtle. In Table 9, is assigned its appropriate expression, but " " is converted to the status of an "Atomic Identifier." In plain English, this means that collectively, all the characters that make up the symbol have been frozen together as a single name. Unfortunately, there is no obvious way to detect this, and each time such an Atomic Identifier is used, this status must be re-applied to the symbol. About the only way to tell if something is an Atomic Identifier is to try to change it to an Atomic Identifier. If it already is an Atomic Identifier, the change can't be made. Legal Notice: Maplesoft, a division of Waterloo Maple Inc Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.
Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors. Content of the Precalculus Subpackage
Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This article provides a systematic exposition
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More informationMaple Quick Start. Introduction. Talking to Maple. Using [ENTER] 3 (2.1)
Introduction Maple Quick Start In this introductory course, you will become familiar with and comfortable in the Maple environment. You will learn how to use context menus, task assistants, and palettes
More informationFACTORING ANGLE EQUATIONS:
FACTORING ANGLE EQUATIONS: For convenience, algebraic names are assigned to the angles comprising the Standard Hip kernel. The names are completely arbitrary, and can vary from kernel to kernel. On the
More informationTrigonometry for AC circuits
Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
More informationSection 10.4 Vectors
Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationCurve Fitting with Maple
Curve Fitting with Maple Maplesoft, a division of Waterloo Maple Inc., 2007 Introduction Maple includes a number of assistants that allows a user to experiment and easily perform key tasks. This Tips and
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationBasic Use of the TI-84 Plus
Basic Use of the TI-84 Plus Topics: Key Board Sections Key Functions Screen Contrast Numerical Calculations Order of Operations Built-In Templates MATH menu Scientific Notation The key VS the (-) Key Navigation
More informationCreating Basic Excel Formulas
Creating Basic Excel Formulas Formulas are equations that perform calculations on values in your worksheet. Depending on how you build a formula in Excel will determine if the answer to your formula automatically
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationLIES MY CALCULATOR AND COMPUTER TOLD ME
LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocket-size calculating devices are currently
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationThe Center for Teaching, Learning, & Technology
The Center for Teaching, Learning, & Technology Instructional Technology Workshops Microsoft Excel 2010 Formulas and Charts Albert Robinson / Delwar Sayeed Faculty and Staff Development Programs Colston
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationX On record with the USOE.
Textbook Alignment to the Utah Core Algebra 2 Name of Company and Individual Conducting Alignment: Chris McHugh, McHugh Inc. A Credential Sheet has been completed on the above company/evaluator and is
More informationUsing Excel to Execute Trigonometric Functions
In this activity, you will learn how Microsoft Excel can compute the basic trigonometric functions (sine, cosine, and tangent) using both radians and degrees. 1. Open Microsoft Excel if it s not already
More informationIntroduction to Complex Fourier Series
Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationMATH 2 Course Syllabus Spring Semester 2007 Instructor: Brian Rodas
MATH 2 Course Syllabus Spring Semester 2007 Instructor: Brian Rodas Class Room and Time: MC83 MTWTh 2:15pm-3:20pm Office Room: MC38 Office Phone: (310)434-8673 E-mail: rodas brian@smc.edu Office Hours:
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationSYLLABUS. OFFICE AND HOURS: Karnoutsos 536 (Access through K506) M 12, T 1, R 10, 12, 2 or by appointment. I am available by e-mail at all times.
SYLLABUS COURSE TITLE: PreCalculus COURSE NUMBER: MATH0165 REFERENCE NUMBER: 1980 PREREQUISITE: MATH0112 Intermediate Algebra or equivalent INSTRUCTOR: Dr. Riggs OFFICE AND HOURS: Karnoutsos 536 (Access
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More informationANGELO STATE UNIVERSITY/GLEN ROSE HIGH SCHOOL TRIGONOMETRY WITH ANALYTIC GEOMETRY MATH 1353 SPRING OF 2016
ANGELO STATE UNIVERSITY/GLEN ROSE HIGH SCHOOL TRIGONOMETRY WITH ANALYTIC GEOMETRY MATH 1353 SPRING OF 2016 I. INSTRUCTOR MRS. JAMI LOVELADY Office: 504 Phone: 254-898-3824 Conference/Planning Period -
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More informationLAYOUT OF THE KEYBOARD
Dr. Charles Hofmann, LaSalle hofmann@lasalle.edu Dr. Roseanne Hofmann, MCCC rhofman@mc3.edu ------------------------------------------------------------------------------------------------- DISPLAY CONTRAST
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationIntroduction Assignment
PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationMath 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction
Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationAMSCO S Ann Xavier Gantert
AMSCO S Integrated ALGEBRA 1 Ann Xavier Gantert AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 Dedication This book is dedicated to Edward Keenan who left a profound influence
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationMAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:
MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex
More informationCOURSE OUTLINE FOR MATH 115. Instructor: Rich Tschritter, Ewing 268. Text: Precalculus, Sixth Edition, by Larson & Hostetler CHAPTER A: APPENDIX A
COURSE OUTLINE FOR MATH 115 Instructor: Rich Tschritter, Ewing 268 Text: Precalculus, Sixth Edition, by Larson & Hostetler CHAPTER A: APPENDIX A 1 A.4 2 Rational Expressions 2 A.5 1 Solving Equations 3
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationEXCEL SOLVER TUTORIAL
ENGR62/MS&E111 Autumn 2003 2004 Prof. Ben Van Roy October 1, 2003 EXCEL SOLVER TUTORIAL This tutorial will introduce you to some essential features of Excel and its plug-in, Solver, that we will be using
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name
More informationProgramming with Mathcad Prime
Programming with Mathcad Prime PTC Academic Program Learn. Create. Collaborate. Succeed. Written By Chris Hartmann, Anji Seberino & Roger Yeh These materials are 2011, Parametric Technology Corporation
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationCHAPTER 5 Round-off errors
CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any
More informationIra Fine and Thomas J. Osler Department of Mathematics Rowan University Glassboro, NJ 08028. osler@rowan.edu. 1. Introduction
1 08/0/00 THE REMARKABLE INCIRCLE OF A TRIANGLE Ira Fine and Thomas J. Osler Department of Mathematics Rowan University Glassboro, NJ 0808 osler@rowan.edu 1. Introduction The incircle of a triangle is
More informationThis activity will guide you to create formulas and use some of the built-in math functions in EXCEL.
Purpose: This activity will guide you to create formulas and use some of the built-in math functions in EXCEL. The three goals of the spreadsheet are: Given a triangle with two out of three angles known,
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationQ&As: Microsoft Excel 2013: Chapter 2
Q&As: Microsoft Excel 2013: Chapter 2 In Step 5, why did the date that was entered change from 4/5/10 to 4/5/2010? When Excel recognizes that you entered a date in mm/dd/yy format, it automatically formats
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationhttp://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304
MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio
More informationPre-Calculus Semester 1 Course Syllabus
Pre-Calculus Semester 1 Course Syllabus The Plano ISD eschool Mission is to create a borderless classroom based on a positive student-teacher relationship that fosters independent, innovative critical
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationExcel 2003 PivotTables Summarizing, Analyzing, and Presenting Your Data
The Company Rocks Excel 2003 PivotTables Summarizing, Analyzing, and Presenting Step-by-step instructions to accompany video lessons Danny Rocks 5/19/2011 Creating PivotTables in Excel 2003 PivotTables
More informationPrecalculus Orientation and FAQ
Precalculus Orientation and FAQ MATH 1011 (Precalculus) is a four hour 3 credit course that prepares a student for Calculus. Topics covered include linear, quadratic, polynomial, rational, exponential,
More informationIntro to Excel spreadsheets
Intro to Excel spreadsheets What are the objectives of this document? The objectives of document are: 1. Familiarize you with what a spreadsheet is, how it works, and what its capabilities are; 2. Using
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationDATA 301 Introduction to Data Analytics Microsoft Excel VBA. Dr. Ramon Lawrence University of British Columbia Okanagan
DATA 301 Introduction to Data Analytics Microsoft Excel VBA Dr. Ramon Lawrence University of British Columbia Okanagan ramon.lawrence@ubc.ca DATA 301: Data Analytics (2) Why Microsoft Excel Visual Basic
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationMath 1B Syllabus. Course Description. Text. Course Assignments. Exams. Course Grade
Course Description Math 1B Syllabus This Pre-Calculus course is designed to prepare students for a Calculus course. This course is taught so that students will acquire a solid foundation in algebra and
More informationExpense Management. Configuration and Use of the Expense Management Module of Xpert.NET
Expense Management Configuration and Use of the Expense Management Module of Xpert.NET Table of Contents 1 Introduction 3 1.1 Purpose of the Document.............................. 3 1.2 Addressees of the
More informationDecimal Notations for Fractions Number and Operations Fractions /4.NF
Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More information14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
More informationGraphing Calculator Scientific Calculator Version 2.0
Graphing Calculator Scientific Calculator Version 2.0 2006-1012 Infinity Softworks, Inc. www.infinitysw.com/ets August 7, 2012 1! Table of Contents Table of Contents 1 Overview! 3 2 Navigation! 4 3 Using
More informationIntroduction to the TI-Nspire CX
Introduction to the TI-Nspire CX Activity Overview: In this activity, you will become familiar with the layout of the TI-Nspire CX. Step 1: Locate the Touchpad. The Touchpad is used to navigate the cursor
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationGeorgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
More informationMathematics, Basic Math and Algebra
NONRESIDENT TRAINING COURSE Mathematics, Basic Math and Algebra NAVEDTRA 14139 DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. PREFACE About this course: This is a self-study
More information