A New Note On Cauchy-Schwarz
|
|
- Reynold Osborne
- 5 years ago
- Views:
Transcription
1 A New Note On Cauchy-Schwarz Hong Ge Chen January 30, student in Guangzhou university,guangzhou c- ity,china 1
2 As we can see,the Cauchy-Schwarz inequality is a very important inequality in proving inequalities no matter in mathematic contest or our daily use.there has many ways to apply the Cauchy-Schwarz method.now,i will introduce a nice way to use Cauchy-Schwarz that is very useful in prove three variables and fourth degrees inequality,also work on fifth degrees inequality. 1 A Simple Introduce Of pqr method In 2007,the inequality genius Võ Quôc Bá Cân published his article On a class of three-variable inequalities 1 in Mathematical Reflections.that gave the following conclusions. Let a, b, c are real numbers and set p = a + b + c, ab + bc + ca = p2 q 2 3, r = abcwe have a powerful estimate of r, Which is: (p + q) 2 (p 2q) r, (p ] q)2 (p + 2q) (1) it s a beautiful form of r. but,in fact,we usually set p = a + b + c, q = ab + bc + ca, r = abc that will change the form into: r 2p 3 + 9pq 2(p 2 3q) p 2 3q ], 2p 3 + 9pq + 2(p 2 3q) p 2 3q (2) I admit this is a ugly form of r and needs many computes.but it s obviously powerful. but,we can see that many lic inequality has something like a 2 b + b 2 c + c 2 a and a 3 b + b 3 c + c 3 a,that cause the inequality lic,without these things,the inequality should be a symmetric,a reason is: each three variables inequality can be rewrite as F (a, b, c)+g(a, b, c) (a k b m +b k c m +c k a m ) I wouldn t prove this fact here,you should remember all inequalities you have met before to see if it has this form. Now,Let us back to our topic,due to these reason.estimate y = a k b m +b k c m + c k a m become a important issue.but deal with unknown k, m is no double a difficult things,because different k, m get different ranges of y with different equality occurs.also with much complicated compute when k is a great value.here,we discuss about k = 2. When k = 2,we have to estimate a 2 b + b 2 c + c 2 a,we set p = a + b + c, q = ab + bc + ca, r = abc: also y 1 = a 2 b + b 2 c + c 2 a, y 2 = ab 2 + bc 2 + ca 2 1 1Vo Quoc Ba Can-On a class of three-variable inequalities,2007,mathematical Reflections ] 2
3 by Vieta Theorem { y1 + y 2 = pq 3r, y 1 y 2 = q 3 + p 3 r 6pqr + 9r 2 it s easy see that y 1, y 2 are the roots of following equation: When p, q are fixed.note: y 2 (pq 3r)y + q 3 + p 3 r 6pqr + 9r 2 = 0 (3) F (r, y) = y 2 (pq 3r)y + q 3 + p 3 r 6pqr + 9r 2 = 0 dy dr = F r(r, y) F y (r, y) = 3y + p3 6pqr + 9r 2 2y pq + 3r { 3y + p 3 6pq + 18r = 0 y 2 (pq 3r)y + q 3 + p 3 r 6pqr + 9r 2 = 0 Solve this equation,gives two roots: r 1 = 1 2p 3 + 9pq + (p 2 3q) ] p 2 3q y 1 = 1 9 p 3 2(p 2 3q) ] p 2 3q and r 2 = 1 y 2 = 1 9 2p 3 + 9pq (p 2 3q) ] p 2 3q p 3 + 2(p 2 3q) p 2 3q ] Having a simple look at r 1, r 2,we can see that both r 1, r 2 are in the range which we mentioned before.2 Hence the range of y is: p 3 2(p 2 3q) ] p 2 3q p 3 + 2(p 2 3q) ] p 2 3q, (4) 9 9 I believe you will feel very boring at solving equation which has such complicated coefficient.of course,the coefficient will be more ugly if the value of k, m is much large,although we can use Cauchy-Schwarz in the estimation(see kuing s new article).if you don t think so,please try k = 3, m = 1 I will introduce a simple way in next section. 3
4 2 Choose a suitable Polynomial to applying Cauchy- Schwarz Why we have to estimate so many times? as you can see,estimate a 2 b + b 2 c + c 2 a and a 3 b + b 3 c + c 3 a is not a easy job,the question come in natural.can we find some general polynomial that can estimate some value of k and m in once time? Let s analysis this problem in detail.the reasone we need to estimate for so many times is that: Although we have: a 3 b + b 3 c + c 3 a = (a 2 b + b 2 c + c 2 a)(a + b + c) abc(a + b + c) a 2 b 2 We still can t use the range of (a 2 b+b 2 c+c 2 a) to get the range of a 3 b+b 3 c+c 3 a. Why? The reasone is: if p, q are fixed.we can t sure the (a 2 b + b 2 c + c 2 a) and abc(a + b + c) a 2 b 2 = rp q 2 will obtain the max or min value at the same time in the define range of r. I hope I have explained clearly,let s give a simple example to this principle: Example: set f(x), g(x) C1, 4], f(x) obtain its min value when x = 1 and max value when x = 4,g(x) obtain its min value in x = 2 and max value in x = 3,Note H(x) = f(x) + g(x).do you know when H(x) obtain the max and min value? certainly not! it s unknown. Now,we find a regular pattern,if the rest things don t have r,only have p, q then we can use the result we found before,therefore,choose the following polynomial is suitable. S k = a 2 b + b 2 c + c 2 a + kabc (5) Here we will use another way to estimate this polynomial. f(a, b, c) = a 2 b + b 2 c + c 2 a + kabc + λ 1 (a + b + c) + λ 2 (ab + bc + ca) by Lagrange multiplier method: 2ab + c 2 + kbc + λ 1 + λ 2 (b + c) = 0 2bc + a + kca + λ 1 + λ 2 (a + c) = 0 2ca + b 2 + kab + λ 1 + λ 2 (b + a) = 0 a + b + c ab + bc + ca = p = q (a + b + c) 2 + k(ab + bc + ca) + 3λ 1 + 2λ 2 (a + b + c) = 0 4
5 2a b c 6ab + 3c 2 + 3kbc p 2 kq = 3λ 1 = (p 2 + kq + 2pλ 2 ) 2b a c 6bc + 3a 2 + 3kac p 2 kq = this system which guide us to use Cauchy-Schwarz : 2c b a 6ca + 3b 2 + 3kab p 2 kq 2 ] (2a b c)(6ab + 3c 2 + 3kbc p 2 kq)] (2a b c) ] 2 (6ab + 3c 2 + 3kbc p 2 kq) 2 there are some identities available: ( (2a b c) 2 = 6 a 2 ) ab (6) (6ab + 3c 2 + 3kbc p 2 kq) 2 = 9 a k ab 3 + (9k 2 + 3k) a 2 b a 2 bc (6p 2 + 6kq) a 2 (12kq + 6kp 2 + 6k 2 q + 12p 2 ) ab + 3p 4 + 3k 2 q 2 + 6kq 2 The second one is a little bit ugly.but if we transform in to p, q, r,it s much simpler and beautiful. (6ab + 3c 2 + 3kbc p 2 kq) 2 = (12k 36)p 2 q + (6k k)q 2 + 6p 4 18pS k k (8) (2a b c) 2 = 6( a 2 ab) = 6(p 2 2q) (9) (2a b c)(6ab + 3c 2 + 3kbc p 2 kq) = S k (9 + 3k)pq 3p(p 2 3q) Therefore, Sk (9 + 3k)pq 3p(p 2 3q) ] 2 6(p 2 2q) (12k 36)p 2 q + (6k k)q 2 + 6p 4 18pkS k ] it gives the range of S k is: (3 2k)p 3 + 9kpq 2 ] (k 2 3k + 9)(p 2 3q) 3, (7) (10) (3 2k)p 3 + 9kpq + 2 ] (k 2 3k + 9)(p 2 3q) 3 5
6 3 Application In the last section,we have gotten the range of S k is (3 2k)p 3 + 9kpq 2 ] (k 2 3k + 9)(p 2 3q) (3 3 2k)p 3 + 9kpq + 2 ] (k 2 3k + 9)(p 2 3q) 3 Now,Let s see the application of this powerful tool.in order to be more convince,let s build a lemma at first. lemma: a 3 b + b 3 c + c 3 a p4 + 9p 2 q q 2 2(p 2 3q) 7p 2 (p 2 3q) and a 3 b + b 3 c + c 3 a p4 + 9p 2 q q 2 + 2(p 2 3q) 7p 2 (p 2 3q) Võ Quôc Bá Cân called this as pqr lemma,where p = a+b+c, q = ab + bc + ca Proof of the lemma: Just notice that: (a 3 b + b 3 c + c 3 a) = (a 2 b + b 2 c + c 2 a + abc)(a + b + c) (ab + bc + ca) 2 = S 1 p q 2 and using the range of S k for k = 1,The result follows immediately. Example 1 Let a, b, c R,Prove that: 1 a 3 b + b 3 c + c 3 a 7 3 (a 2 + b 2 + c 2 ) 2 8, (Vasile Cirtoaje,Dan Chen) For the left side of this inequality,it s really famous since Vasile Cirtoaje found it in 1992,there also has many nice proofs 2 of it.also we can use Vo Quoc Ba Can s sum of square technique 3 to solve it.but now,i will use the theory I talked before to prove this problem. proof:(by kuing) For the left side (a 2 + b 2 + c 2 ) 2 3(a 3 b + b 3 c + c 3 a) 2 Prove Vasc s inequality by Cauchy-Schwarz 3 The Sum of Square technique,vo Quoc Ba Can-Pham Thi Hung 6
7 Using the lemma,it s suffice to check Or (p 2 2q) 2 3 p4 + 9p 2 q q p 2 (p 2 3q) 3 9(2p 2 7q) 2 (p 2 3q) 2 0 Which is obviously true.equality occurs when p 2 = 3q Or 2p 2 = 7q.For the right side.using the lemma again,it s enough to prove that: 8 p4 + 9p 2 q q 2 2 7p 2 (p 2 3q) 3 and after expand gives 7(p 2 qa) 2 (8 + 7)p 4 + ( )p 2 q + ( )q p 2 (p 2 3q) 3 Due to p 2 3q,we can get that: (8 + 7)p 4 + ( )p 2 q + ( )q 2 ] p 2 (p 2 3q) 3 factor it,gives (2+ 7)p 2 +2q] 2 ( )p 4 +( )p 2 q +( )q 2 ] 0 Or ( )p 4 + ( )p 2 q + ( )q 2 0 actually,this inequality is hold in the condition p 2 3q Hence we are done! Example 2 Let x, y, z are real numbers,prove that: x 4 + y 4 + z 4 + 2xyz(x + y + z) x 3 y + y 3 z + z 3 x proof:as we can see,the inequality can be strength as (Vasile Cirtoaje) x 4 + 2xyz(x + y + z) (x 3 y + y 3 z + z 3 x) 2 (xy + yz + xz)2 3 Or 3( x 2 ) 2 8( xy) 2 3 x 3 y 6xyz( x)] 0 Normalize p = 3 3(p 2 2q) 2 8q 2 + 3(S 5 p q 2 ) 9S 5 7q 2 108q
8 Using our theory,we have: S q (3 q) 3 9 so it s enough to prove that: q 2 9q (3 q) 3 Or Which is prefectly true! Hence we are done! q 2 (q 3) 2 0 8
On a class of three-variable inequalities. Vo Quoc Ba Can
On a class of three-variable inequalities Vo Quoc Ba Can 1 Theem Let a, b, c be real numbers satisfying a + b + c = 1 By the AM - GM inequality, we have ab + bc + ca 1, therefe setting ab + bc + ca = 1
More informationDISCRETE INEQUALITIES
Vasile Cîrtoaje DISCRETE INEQUALITIES VOLUME 1 SYMMETRIC POLYNOMIAL INEQUALITIES ART OF PROBLEM SOLVING 2015 About the author Vasile Cîrtoaje is a Professor at the Department of Automatic Control and
More informationThe uvw method - Tejs. The uvw method. by Mathias Bæk Tejs Knudsen
The uvw method - Tejs The uvw method by Mathias Bæk Tejs Knudsen The uvw method - Tejs BASIC CONCEPTS Basic Concepts The basic concept of the method is this: With an inequality in the numbers a, b, c R,
More informationFactorisation CHAPTER Introduction
FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write
More informationAlgebraic Expressions and Identities
ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 Algebraic Expressions and Identities CHAPTER 9 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More information0.1 Squares Analysis Method S.O.S
1 0.1 Squares Analysis Method S.O.S 0.1.1 The Begining Problems Generally, if we have an usual inequalities, the ways for us to solve them are neither trying to fumble from well-known inequalities nor
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
More informationSCTT The pqr-method august 2016
SCTT The pqr-method august 2016 A. Doledenok, M. Fadin, À. Menshchikov, A. Semchankau Almost all inequalities considered in our project are symmetric. Hence if plugging (a 0, b 0, c 0 ) into our inequality
More informationGIL Publishing House PO box 44, PO 3 ROMANIA
57 Title: Secrets in Inequalities volume - advanced inequalities - free chapter Author: Pham Kim Hung Publisher: GIL Publishing House ISBN: 978-606-500-00-5 Copyright c 008 GIL Publishing House. All rights
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Algebra Homework: Chapter (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework M / Review of Sections.-.
More information1 Functions of Several Variables 2019 v2
1 Functions of Several Variables 2019 v2 11 Notation The subject of this course is the study of functions f : R n R m The elements of R n, for n 2, will be called vectors so, if m > 1, f will be said to
More informationGraphing Square Roots - Class Work Graph the following equations by hand. State the domain and range of each using interval notation.
Graphing Square Roots - Class Work Graph the following equations by hand. State the domain and range of each using interval notation. 1. y = x + 2 2. f(x) = x 1. y = x +. g(x) = 2 x 1. y = x + 2 + 6. h(x)
More informationA New Method About Using Polynomial Roots and Arithmetic-Geometric Mean Inequality to Solve Olympiad Problems
Polynomial Roots and Arithmetic-Geometric Mean Inequality 1 A New Method About Using Polynomial Roots and Arithmetic-Geometric Mean Inequality to Solve Olympiad Problems The purpose of this article is
More information26. LECTURE 26. Objectives
6. LECTURE 6 Objectives I understand the idea behind the Method of Lagrange Multipliers. I can use the method of Lagrange Multipliers to maximize a multivariate function subject to a constraint. Suppose
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Notes #: Sections.. Section.: Review Simplify; leave all answers in positive exponents:.) m -.) y -.) m 0.) -.) -.) - -.) (m ) 0.) 0 x y Evaluate if a =
More informationUSA Mathematics Talent Search
ID#: 036 16 4 1 We begin by noting that a convex regular polygon has interior angle measures (in degrees) that are integers if and only if the exterior angle measures are also integers. Since the sum of
More informationAlgebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
More informationAn Useful Technique in Proving Inequalities
An Useful Technique in Proving Inequalities HSGS, Hanoi University of Science, Vietnam Abstract There are a lot of distinct ways to prove inequalities. This paper mentions a simple and useful technique
More informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationa. Define a function called an inner product on pairs of points x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in R n by
Real Analysis Homework 1 Solutions 1. Show that R n with the usual euclidean distance is a metric space. Items a-c will guide you through the proof. a. Define a function called an inner product on pairs
More informationreview To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17
1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient
More informationExpanding brackets and factorising
CHAPTER 8 Epanding brackets and factorising 8 CHAPTER Epanding brackets and factorising 8.1 Epanding brackets There are three rows. Each row has n students. The number of students is 3 n 3n. Two students
More informationChapter 1 Basic (Elementary) Inequalities and Their Application
Chapter 1 Basic (Elementary) Inequalities and Their Application There are many trivial facts which are the basis for proving inequalities. Some of them are as follows: 1. If x y and y z then x z, for any
More informationMath 3C Midterm 1 Study Guide
Math 3C Midterm 1 Study Guide October 23, 2014 Acknowledgement I want to say thanks to Mark Kempton for letting me update this study guide for my class. General Information: The test will be held Thursday,
More informationMATH 255 Applied Honors Calculus III Winter Homework 5 Solutions
MATH 255 Applied Honors Calculus III Winter 2011 Homework 5 Solutions Note: In what follows, numbers in parentheses indicate the problem numbers for users of the sixth edition. A * indicates that this
More informationLESSON 7.1 FACTORING POLYNOMIALS I
LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of
More informationSolution to Proof Questions from September 1st
Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L + W + L + W. Can this formula be written in a simpler way? If it is true, that we can
More informationECE 238L Boolean Algebra - Part I
ECE 238L Boolean Algebra - Part I August 29, 2008 Typeset by FoilTEX Understand basic Boolean Algebra Boolean Algebra Objectives Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand
More informationLecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
More informationMath Boot Camp Functions and Algebra
Fall 017 Math Boot Camp Functions and Algebra FUNCTIONS Much of mathematics relies on functions, the pairing (relation) of one object (typically a real number) with another object (typically a real number).
More informationTAYLOR POLYNOMIALS DARYL DEFORD
TAYLOR POLYNOMIALS DARYL DEFORD 1. Introduction We have seen in class that Taylor polynomials provide us with a valuable tool for approximating many different types of functions. However, in order to really
More informationENGINEERING MATH 1 Fall 2009 VECTOR SPACES
ENGINEERING MATH 1 Fall 2009 VECTOR SPACES A vector space, more specifically, a real vector space (as opposed to a complex one or some even stranger ones) is any set that is closed under an operation of
More informationAlgebra in Problem Solving (Senior) Konrad Pilch
Algebra in Problem Solving (Senior) Konrad Pilch March 29, 2016 1 Polynomials Definition. A polynomial is an expression of the form P(x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0. n is the degree of the
More informationPractice Test III, Math 314, Spring 2016
Practice Test III, Math 314, Spring 2016 Dr. Holmes April 26, 2016 This is the 2014 test reorganized to be more readable. I like it as a review test. The students who took this test had to do four sections
More informationMath 313 Midterm I KEY Winter 2011 section 003 Instructor: Scott Glasgow
Math 33 Midterm I KEY Winter 0 section 003 Instructor: Scott Glasgow Write your name very clearly on this exam In this booklet write your mathematics clearly legibly in big fonts and most important have
More informationP-adic numbers. Rich Schwartz. October 24, 2014
P-adic numbers Rich Schwartz October 24, 2014 1 The Arithmetic of Remainders In class we have talked a fair amount about doing arithmetic with remainders and now I m going to explain what it means in a
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationExpanding brackets and factorising
Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a
More informationLINEAR SYSTEMS AND MATRICES
CHAPTER 3 LINEAR SYSTEMS AND MATRICES SECTION 3. INTRODUCTION TO LINEAR SYSTEMS This initial section takes account of the fact that some students remember only hazily the method of elimination for and
More informationMathmatics 239 solutions to Homework for Chapter 2
Mathmatics 239 solutions to Homework for Chapter 2 Old version of 8.5 My compact disc player has space for 5 CDs; there are five trays numbered 1 through 5 into which I load the CDs. I own 100 CDs. a)
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... 7 Introduction... 7 Integer Exponents... 8 Rational Exponents...5 Radicals... Polynomials...30 Factoring Polynomials...36
More informationSect Least Common Denominator
4 Sect.3 - Least Common Denominator Concept #1 Writing Equivalent Rational Expressions Two fractions are equivalent if they are equal. In other words, they are equivalent if they both reduce to the same
More informationSTEP Support Programme. Hints and Partial Solutions for Assignment 1
STEP Support Programme Hints and Partial Solutions for Assignment 1 Warm-up 1 You can check many of your answers to this question by using Wolfram Alpha. Only use this as a check though and if your answer
More informationGrade 8 Factorisation
ID : ae-8-factorisation [1] Grade 8 Factorisation For more such worksheets visit www.edugain.com Answer the questions (1) Find factors of following polynomial A) y 2-2xy + 3y - 6x B) 3y 2-12xy - 2y + 8x
More informationA Few Elementary Properties of Polynomials. Adeel Khan June 21, 2006
A Few Elementary Properties of Polynomials Adeel Khan June 21, 2006 Page i CONTENTS Contents 1 Introduction 1 2 Vieta s Formulas 2 3 Tools for Finding the Roots of a Polynomial 4 4 Transforming Polynomials
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationConstructing Approximations to Functions
Constructing Approximations to Functions Given a function, f, if is often useful to it is often useful to approximate it by nicer functions. For example give a continuous function, f, it can be useful
More informationLagrange Multipliers
Calculus 3 Lia Vas Lagrange Multipliers Constrained Optimization for functions of two variables. To find the maximum and minimum values of z = f(x, y), objective function, subject to a constraint g(x,
More informationLinear Independence Reading: Lay 1.7
Linear Independence Reading: Lay 17 September 11, 213 In this section, we discuss the concept of linear dependence and independence I am going to introduce the definitions and then work some examples and
More informationDistances in R 3. Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane:
Distances in R 3 Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane: Definition: The equation of a line through point P(x 0, y 0, z 0 ) with directional vector
More informationThe P/Q Mathematics Study Guide
The P/Q Mathematics Study Guide Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved Table of Contents Ch. Numerical Operations - Integers... - Fractions... - Proportion and Percent...
More informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationAlgebra Using letters to represent numbers
3 Algebra 1 Using letters to represent numbers 47 3.1 Using letters to represent numbers Algebra is the branch of mathematics in which letters are used to represent numbers. This can help solve some mathematical
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More informationVector calculus background
Vector calculus background Jiří Lebl January 18, 2017 This class is really the vector calculus that you haven t really gotten to in Calc III. Let us start with a very quick review of the concepts from
More informationAlgebra I. Book 2. Powered by...
Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More informationThe Art of Dumbassing Brian Hamrick July 2, 2010
The Art of Dumbassing Brian Hamrick July, Introduction The field of inequalities is far from trivial. Many techniques, including Cauchy, Hölder, isolated fudging, Jensen, and smoothing exist and lead to
More informationExpansion of Terms. f (x) = x 2 6x + 9 = (x 3) 2 = 0. x 3 = 0
Expansion of Terms So, let s say we have a factorized equation. Wait, what s a factorized equation? A factorized equation is an equation which has been simplified into brackets (or otherwise) to make analyzing
More informationTWO USEFUL SUBSTITUTIONS
Contents TWO USEFUL SUBSTITUTIONS 2 ALWAYS CAUCHY-SCHWARZ 11 EQUATIONS AND BEYOND 25 LOOK AT THE EXPONENT! 38 PRIMES AND SQUARES 53 T 2 S LEMMA 65 ONLY GRAPHS, NO SUBGRAPHS! 81 COMPLEX COMBINATORICS 90
More informationChapter 6, Factoring from Beginning and Intermediate Algebra by Tyler Wallace is available under a Creative Commons Attribution 3.0 Unported license.
Chapter 6, Factoring from Beginning and Intermediate Algebra by Tyler Wallace is available under a Creative Commons Attribution 3.0 Unported license. 2010. 6.1 Factoring - Greatest Common Factor Objective:
More informationChapter 9 Notes SN AA U2C9
Chapter 9 Notes SN AA U2C9 Name Period Section 2-3: Direct Variation Section 9-1: Inverse Variation Two variables x and y show direct variation if y = kx for some nonzero constant k. Another kind of variation
More informationPolynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms
Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the
More informationCHAPTER 12 Boolean Algebra
318 Chapter 12 Boolean Algebra CHAPTER 12 Boolean Algebra SECTION 12.1 Boolean Functions 2. a) Since x 1 = x, the only solution is x = 0. b) Since 0 + 0 = 0 and 1 + 1 = 1, the only solution is x = 0. c)
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationLesson 24: True and False Number Sentences
NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values
More informationPermutations and Polynomials Sarah Kitchen February 7, 2006
Permutations and Polynomials Sarah Kitchen February 7, 2006 Suppose you are given the equations x + y + z = a and 1 x + 1 y + 1 z = 1 a, and are asked to prove that one of x,y, and z is equal to a. We
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationDefinition 1.2. Let p R n be a point and v R n be a non-zero vector. The line through p in direction v is the set
Important definitions and results 1. Algebra and geometry of vectors Definition 1.1. A linear combination of vectors v 1,..., v k R n is a vector of the form c 1 v 1 + + c k v k where c 1,..., c k R are
More informationGeorgia Department of Education Common Core Georgia Performance Standards Framework CCGPS Advanced Algebra Unit 2
Polynomials Patterns Task 1. To get an idea of what polynomial functions look like, we can graph the first through fifth degree polynomials with leading coefficients of 1. For each polynomial function,
More informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More informationWorkshops: The heart of the MagiKats Programme
Workshops: The heart of the MagiKats Programme Every student is assigned to a Stage, based on their academic year and assessed study level. Stage 5 students have completed all Stage 4 materials. The sheets
More informationMATH 115, SUMMER 2012 LECTURE 12
MATH 115, SUMMER 2012 LECTURE 12 JAMES MCIVOR - last time - we used hensel s lemma to go from roots of polynomial equations mod p to roots mod p 2, mod p 3, etc. - from there we can use CRT to construct
More informationMATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input,
MATH 20550 The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, F(x 1,, x n ) F 1 (x 1,, x n ),, F m (x 1,, x n ) where each
More informationMath Lecture 3 Notes
Math 1010 - Lecture 3 Notes Dylan Zwick Fall 2009 1 Operations with Real Numbers In our last lecture we covered some basic operations with real numbers like addition, subtraction and multiplication. This
More informationLesson 6-1: Relations and Functions
I ll bet you think numbers are pretty boring, don t you? I ll bet you think numbers have no life. For instance, numbers don t have relationships do they? And if you had no relationships, life would be
More informationLagrange Murderpliers Done Correctly
Lagrange Murderpliers Done Correctly Evan Chen June 8, 2014 The aim of this handout is to provide a mathematically complete treatise on Lagrange Multipliers and how to apply them on optimization problems.
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More informationCONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION
CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4 - SUBSTITUTION - 5 - REMOVING BRACKETS - 7 - FACTORISING - 8 - COMMON FACTORS - 8 - DIFFERENCE
More informationMain topics for the First Midterm Exam
Main topics for the First Midterm Exam The final will cover Sections.-.0, 2.-2.5, and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday,
More informationCHAPTER 4 VECTORS. Before we go any further, we must talk about vectors. They are such a useful tool for
CHAPTER 4 VECTORS Before we go any further, we must talk about vectors. They are such a useful tool for the things to come. The concept of a vector is deeply rooted in the understanding of physical mechanics
More informationEcon Macroeconomics I Optimizing Consumption with Uncertainty
Econ 07 - Macroeconomics I Optimizing Consumption with Uncertainty Prof. David Weil September 16, 005 Overall, this is one of the most important sections of this course. The key to not getting lost is
More informationDiagnostic quiz for M303: Further Pure Mathematics
Diagnostic quiz for M303: Further Pure Mathematics Am I ready to start M303? M303, Further pure mathematics is a fascinating third level module, that builds on topics introduced in our second level module
More informationMathematics Higher Tier, November /2H (Paper 2, calculator)
Link to past paper on AQA website: www.aqa.org.uk This question paper is available to download freely from the AQA website. To navigate around the website, you want QUALIFICATIONS, GCSE, MATHS, MATHEMATICS,
More informationComplex Numbers in Trigonometry
Complex Numbers in Trigonometry Page 1 Complex Numbers in Trigonometry Author Vincent Huang The final version- with better LaTeX, more contest problems, and some new topics. Credit to Binomial-Theorem
More informationOptimization and Calculus
Optimization and Calculus To begin, there is a close relationship between finding the roots to a function and optimizing a function. In the former case, we solve for x. In the latter, we solve: g(x) =
More informationChapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
More information1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
More information2.2 Graphs of Functions
2.2 Graphs of Functions Introduction DEFINITION domain of f, D(f) Associated with every function is a set called the domain of the function. This set influences what the graph of the function looks like.
More informationChapter 7. Extremal Problems. 7.1 Extrema and Local Extrema
Chapter 7 Extremal Problems No matter in theoretical context or in applications many problems can be formulated as problems of finding the maximum or minimum of a function. Whenever this is the case, advanced
More informationTwo useful substitutions...
Two useful substitutions... We know that in most inequalities with a constraint such as abc = 1 the substitution a = x y, b = y z, c = z simplifies the solution (don t kid yourself, not x all problems
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More informationProblem Set 6: Inequalities
Math 8, Winter 27. Problem Set 6: Inequalities I will start with a list of famous elementary inequalities. The emphasis is on algebraic methods rather than calculus. Although I ve included a few words
More informationTheFourierTransformAndItsApplications-Lecture28
TheFourierTransformAndItsApplications-Lecture28 Instructor (Brad Osgood):All right. Let me remind you of the exam information as I said last time. I also sent out an announcement to the class this morning
More information8. Prime Factorization and Primary Decompositions
70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings
More informationINTRODUCTION TO SIGMA NOTATION
INTRODUCTION TO SIGMA NOTATION The notation itself Sigma notation is a way of writing a sum of many terms, in a concise form A sum in sigma notation looks something like this: 3k The Σ sigma) indicates
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationCHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
More information