# 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

Save this PDF as:

Size: px
Start display at page:

Download "5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0"

## Transcription

1 a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a<0 Example 1 5 =5 = ( ) = 0 =0 Since 5 > 0 Since < 0 Since 0 0 Note that the effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is nonnegative. Example Solve x 3 =. Solution. Depending on whether x 3 is positive or negative, the equation x 3 = can be written as x 3 = or x 3 = Solving these two equations gives x = and x = 1. Example 3 Solve 3x = 5x +. Solution. Because two numbers with the same absolute value are either equal or differ in sign, the given equation will be satisfied if either 3x = 5x + or 3x = (5x + ) Solving the first equation yields x = 3 and solving the second yields x = 1 ; thus, the given equation has the solutions x = 3and x = 1. RELATIONSHIP BETWEEN SQUARE ROOTS AND ABSOLUTE VALUES Recall from algebra that a number is called a square root of a if its square is a. Recall also that every positive real number has two square roots, one positive and one negative; the A1

2 A Appendix E: Absolute Value positive square root is denoted by a and the negative square root by a. For example, the positive square root of 9 is 9 = 3, and the negative square root of 9 is 9 = 3. Readers who may have been taught to write 9 as ±3 should stop doing so, since it is incorrect. It is a common error to replace a by a. Although this is correct when a is nonnegative, it is false for negative a. For example, if a =, then a = ( ) = 16 = = a A result that is correct for all a is given in the following theorem. E. theorem. For any real number a, a = a proof. Since a = (+a) = ( a), the numbers +a and a are square roots of a.if a 0, then +a is the nonnegative square root of a, and if a<0, then a is the nonnegative square root of a. Since a denotes the nonnegative square root of a, it follows that a =+a if a 0 a = a if a<0 That is, a = a. PROPERTIES OF ABSOLUTE VALUE E.3 theorem. If a and b are real numbers, then (a) a = a A number and its negative have the same absolute value. (b) ab = a b The absolute value of a product is the product of the absolute values. (c) a/b = a / b The absolute value of a ratio is the ratio of the absolute values. We will prove parts (a) and (b) only. In part (c) of Theorem E.3 we did not explicitly state that b = 0, but this must be so since division by zero is not allowed. Whenever divisions occur in this text, it will be assumed that the denominator is not zero, even if we do not mention it explicitly. proof (a). proof (b). From Theorem E., a = ( a) = a = a From Theorem E. and a basic property of square roots, ab = (ab) = a b = a b = a b The result in part (b) of Theorem E.3 can be extended to three or more factors. More precisely, for any n real numbers, a 1, a,...,a n, it follows that a 1 a a n = a 1 a a n (1)

3 Appendix E: Absolute Value A3 In the special case where a 1, a,...,a n have the same value, a, it follows from (1) that a n = a n () A a b a (a) B b GEOMETRIC INTERPRETATION OF ABSOLUTE VALUE The notion of absolute value arises naturally in distance problems. For example, suppose that A and B are points on a coordinate line that have coordinates a and b, respectively. Depending on the relative positions of the points, the distance d between them will be b a or a b (Figure E.1). In either case, the distance can be written as d = b a, so we have the following result. B A b a b (b) a E. theorem (Distance Formula). If A and B are points on a coordinate line with coordinates a and b, respectively, then the distance d between A and B is d = b a. Figure E.1 This theorem provides useful geometric interpretations of some common mathematical expressions: expression x a x + a x geometric interpretation on a coordinate line The distance between x and a The distance between x and a (since x + a = x ( a) ) The distance between x and the origin (since x = x 0 ) INEQUALITIES WITH ABSOLUTE VALUES Inequalities of the form x a <kand x a >karise so often that we have summarized the key facts about them in Table 1. Table 1 inequality (k > 0) geometric interpretation figure alternative forms of the inequality The statements in Table 1 remain true if < is replaced by and > by, and if the open dots are replaced by closed dots in the illustrations. x a < k x is within k k units k units k < x a < k units of a. a k a x a + k a k < x < a + k x a > k x is more than k units away k units k units from a. a k a a + k x x a < k or x a > k x < a k or x > a + k Example Solution (a). Solve Adding 3 throughout yields (a) x 3 < (b) x + (c) The inequality x 3 < can be rewritten as <x 3 < 1 <x< 1 x 3 > 5

4 A Appendix E: Absolute Value -1 3 Figure E. Figure E which can be written in interval notation as ( 1, ). Observe that this solution set consists of all x that are within units of 3 on a number line (Figure E.), which is consistent with Table 1. Solution (b). The inequality x + will be satisfied if Solving for x in the two cases yields x + or x + x 6 or x which can be expressed in interval notation as (, 6] [, + ) Observe that the solution set consists of all x that are at least units away from ona number line (Figure E.3), which is consistent with Table 1 and the remark that follows it. Solution (c). Observe first that x = 3 results in a division by zero, so this value of x cannot be in the solution set. Putting this aside for the moment, we will begin by taking reciprocals on both sides and reversing the sense of the inequality in accordance with Theorem A.1(e ) of Appendix A; then we will use Theorem E.3 to rewrite the inequality 1/ x 3 > 5 in a more familiar form: x 3 < 1 5 x 3 < 1 5 x 3 < 1 10 Theorem E.3(b) We multiplied both sides by 1/ =1/. 5 Figure E. Figure E.5 d units 3 d units 8 5 a d a a + d 0 < x a < d 1 10 <x 3 < <x< 8 5 Table 1 We added 3/ throughout. As noted earlier, we must eliminate x = 3 to avoid a division by zero, so the solution set is 5 <x< 3 or 3 <x< 8 5 which can be expressed in interval notation as ( 5, 3 ) ( 3, 8 5). (See Figure E..) AN INEQUALITY FROM CALCULUS One of the most important inequalities in calculus is 0 < x a <δ (3) where δ (Greek delta ) is a positive real number. This is equivalent to the two inequalities 0 < x a and x a <δ the first of which is satisfied by all x except x = a, and the second of which is satisfied by all x that are within δ units of a on a coordinate line. Combining these two restrictions, we conclude that the solution set of (3) consists of all x in the interval (a δ, a + δ) except x = a (Figure E.5). Stated another way, the solution set of (3) is (a δ,a) (a, a + δ) () THE TRIANGLE INEQUALITY It is not generally true that a + b = a + b. For example, if a = 1 and b = 1, then a + b =0, whereas a + b =. It is true, however, that the absolute value of a sum

5 Appendix E: Absolute Value A5 is always less than or equal to the sum of the absolute values. This is the content of the following useful theorem, called the triangle inequality. The name triangle inequality arises from a geometric interpretation of the inequality that can be made when a and b are complex numbers. A more detailed explanation is outside the scope of this text. E.5 theorem (Triangle Inequality). If a and b are any real numbers, then a + b a + b (5) proof. Observe first that a satisfies the inequality a a a because either a = a or a = a, depending on the sign of a. The corresponding inequality for b is b b b Adding the two inequalities we obtain ( a + b ) a + b ( a + b ) (6) Let us now consider the cases a + b 0 and a + b<0separately. In the first case, a + b = a + b, so the right-hand inequality in (6) yields the triangle inequality (5). In the second case, a + b = a + b, so the left-hand inequality in (6) can be written as ( a + b ) a + b which yields the triangle inequality (5) on multiplying by 1. EXERCISE SET E 1. Compute x if (a) x = (b) x = (c) x = k (d) x = k.. Rewrite (x 6) without using a square root or absolute value sign Find all values of x for which the given statement is true. 3. x 3 =3 x. x + =x + 5. x + 9 =x x + 5x =x + 5x. 3x + x =x 3x x = x 3 9. (x + 5) = x (3x ) = 3x 11. Verify a = a for a = and a =. 1. Verify the inequalities a a a for a = and for a = Let A and B be points with coordinates a and b. In each part find the distance between A and B. (a) a = 9, b = (b) a =, b = 3 (c) a = 8, b = 6 (d) a =, b = 3 (e) a = 11, b = (f) a = 0, b = 5 1. Is the equality a = a valid for all values of a? Explain. 15. Let A and B be points with coordinates a and b. In each part, use the given information to find b. (a) a = 3, B is to the left of A, and b a =6. (b) a =, B is to the right of A, and b a =9. (c) a = 5, b a =, and b> Let E and F be points with coordinates e and f. In each part, determine whether E is to the left or to the right of F on a coordinate line. (a) f e = (b) e f = (c) f e = 6 (d) e f = 1 Solve for x. 1. 6x = x = x = 3 + x 0. x + 5 = 8x x 11 = x. x = x x + 5 x = 6. x 3 x + = Solve for x and express the solution in terms of intervals. 5. x + 6 < 3 6. x 5. x 3 6

6 A6 Appendix E: Absolute Value 8. 3x + 1 < 9. x + > x x 3. x + 1 > x 1 < x x x + 3 < 1 (x 3. For which values of x is 5x + 6 ) = x 5x + 6? 38. Solve 3 x for x. 39. Solve x 3 x 3 =1 for x. [Hint: Begin by letting u = x 3.] 0. Verify the triangle inequality a + b a + b (Theorem E.5) for (a) a = 3, b = (b) a =, b = 6 (c) a =, b = 8 (d) a =, b =. 1. Prove: a b a + b.. Prove: a b a b. 3. Prove: a b a b. [Hint: Use Exercise.]

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### MATH Fundamental Mathematics II.

MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

### Chapter 1 Section 5: Equations and Inequalities involving Absolute Value

Introduction The concept of absolute value is very strongly connected to the concept of distance. The absolute value of a number is that number s distance from 0 on the number line. Since distance is always

### SECTION 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

### Analysis MA131. University of Warwick. Term

Analysis MA131 University of Warwick Term 1 01 13 September 8, 01 Contents 1 Inequalities 5 1.1 What are Inequalities?........................ 5 1. Using Graphs............................. 6 1.3 Case

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### 2. INEQUALITIES AND ABSOLUTE VALUES

2. INEQUALITIES AND ABSOLUTE VALUES 2.1. The Ordering of the Real Numbers In addition to the arithmetic structure of the real numbers there is the order structure. The real numbers can be represented by

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Calculus and the centroid of a triangle

Calculus and the centroid of a triangle The goal of this document is to verify the following physical assertion, which is presented immediately after the proof of Theorem I I I.4.1: If X is the solid triangular

### v 1. v n R n we have for each 1 j n that v j v n max 1 j n v j. i=1

1. Limits and Continuity It is often the case that a non-linear function of n-variables x = (x 1,..., x n ) is not really defined on all of R n. For instance f(x 1, x 2 ) = x 1x 2 is not defined when x

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

### Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5

Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### 2.3 Subtraction of Integers

2.3 Subtraction of Integers The key to subtraction of real numbers is to forget what you might have learned in elementary school about taking away. With positive numbers you can reasonably say take 3 away

### 1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

### Appendix A: Numbers, Inequalities, and Absolute Values. Outline

Appendix A: Numbers, Inequalities, and Absolute Values Tom Lewis Fall Semester 2015 Outline Types of numbers Notation for intervals Inequalities Absolute value A hierarchy of numbers Whole numbers 1, 2,

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.

### 1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Arithmetic Series. or The sum of any finite, or limited, number of terms is called a partial sum. are shorthand ways of writing n 1

CONDENSED LESSON 9.1 Arithmetic Series In this lesson you will learn the terminology and notation associated with series discover a formula for the partial sum of an arithmetic series A series is the indicated

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Solving Quadratic Equations by Completing the Square

9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

### 5.1 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

### Mathematical Notation and Symbols

Mathematical Notation and Symbols 1.1 Introduction This introductory block reminds you of important notations and conventions used throughout engineering mathematics. We discuss the arithmetic of numbers,

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### Rational 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

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### Absolute Value Equations and Inequalities

. Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Computing with Signed Numbers and Combining Like Terms

Section. Pre-Activity Preparation Computing with Signed Numbers and Combining Like Terms One of the distinctions between arithmetic and algebra is that arithmetic problems use concrete knowledge; you know

### 1.1 THE REAL NUMBERS. section. The Integers. The Rational Numbers

2 (1 2) Chapter 1 Real Numbers and Their Properties 1.1 THE REAL NUMBERS In this section In arithmetic we use only positive numbers and zero, but in algebra we use negative numbers also. The numbers that

### LECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A

LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Solving Equations with One Variable Type - The Algebraic Approach. Solving Equations with a Variable in the Denominator - The Algebraic Approach

3 Solving Equations Concepts: Number Lines The Definitions of Absolute Value Equivalent Equations Solving Equations with One Variable Type - The Algebraic Approach Solving Equations with a Variable in

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Spring As discussed at the end of the last section, we begin our construction of the rational

MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 3 The Rational Numbers 1 3.1 The set Q................................. 1 3.2 Addition and multiplication of rational numbers............

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

### ON THE FIBONACCI NUMBERS

ON THE FIBONACCI NUMBERS Prepared by Kei Nakamura The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties

### Rational Exponents. Given that extension, suppose that. Squaring both sides of the equation yields. a 2 (4 1/2 ) 2 a 2 4 (1/2)(2) a a 2 4 (2)

SECTION 0. Rational Exponents 0. OBJECTIVES. Define rational exponents. Simplify expressions with rational exponents. Estimate the value of an expression using a scientific calculator. Write expressions

### SECTION 1-4 Absolute Value in Equations and Inequalities

1-4 Absolute Value in Equations and Inequalities 37 SECTION 1-4 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and

### Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

2-8 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Solve. 1. y + 7 < 11 2. 4m 12 3. 5 2x 17 y < 18 m 3 x 6 Use interval notation to indicate the graphed numbers. 4. (-2, 3] 5. (-, 1] Objectives Solve

### Mathematical Induction with MS

Mathematical Induction 2008-14 with MS 1a. [4 marks] Using the definition of a derivative as, show that the derivative of. 1b. [9 marks] Prove by induction that the derivative of is. 2a. [3 marks] Consider

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### 1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Math 002 Unit 5 - Student Notes

Sections 7.1 Radicals and Radical Functions Math 002 Unit 5 - Student Notes Objectives: Find square roots, cube roots, nth roots. Find where a is a real number. Look at the graphs of square root and cube

### PRINCIPLES OF PROBLEM SOLVING

PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Math 1111 Journal Entries Unit I (Sections , )

Math 1111 Journal Entries Unit I (Sections 1.1-1.2, 1.4-1.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Skill Builders. (Extra Practice) Volume I

Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining

### Welcome to Analysis 1

Welcome to Analysis 1 Each week you will get one workbook with assignments to complete. Typically you will be able to get most of it done in class and will finish it off by yourselves. In week one there

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### CHAPTER I THE REAL AND COMPLEX NUMBERS DEFINITION OF THE NUMBERS 1, i, AND 2

CHAPTER I THE REAL AND COMPLEX NUMBERS DEFINITION OF THE NUMBERS 1, i, AND 2 In order to mae precise sense out of the concepts we study in mathematical analysis, we must first come to terms with what the

### Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### A BRIEF GUIDE TO ABSOLUTE VALUE. For High-School Students

1 A BRIEF GUIDE TO ABSOLUTE VALUE For High-School Students This material is based on: THINKING MATHEMATICS! Volume 4: Functions and Their Graphs Chapter 1 and Chapter 3 CONTENTS Absolute Value as Distance

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Lies 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

### TI-83 Plus Graphing Calculator Keystroke Guide

TI-83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this

### 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

### 10-3 Geometric Sequences and Series

1 2 Determine the common ratio, and find the next three terms of each geometric sequence = 2 = 2 The common ratio is 2 Multiply the third term by 2 to find the fourth term, and so on 1( 2) = 2 2( 2) =

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### PROVING STATEMENTS IN LINEAR ALGEBRA

Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### pp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64

Semester 1 Text: Chapter 1: Tools of Algebra Lesson 1-1: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 1-2: Algebraic Expressions

### Basic numerical skills: POWERS AND LOGARITHMS

1. Introduction (easy) Basic numerical skills: POWERS AND LOGARITHMS Powers and logarithms provide a powerful way of representing large and small quantities, and performing complex calculations. Understanding

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Math 1B, lecture 14: Taylor s Theorem

Math B, lecture 4: Taylor s Theorem Nathan Pflueger 7 October 20 Introduction Taylor polynomials give a convenient way to describe the local behavior of a function, by encapsulating its first several derivatives

### Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson

Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent

### 16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style

Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### North Carolina Math 1

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.