Mathematics Review for MS Finance Students


 Warren Joseph
 5 years ago
 Views:
Transcription
1 Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions, Ordered Tuples and Product Sets Common Functions Appendix to Lecture 1: Notes on Logical Reasoning 1
2 Sets: Basics A set is a list or collection of objects. The objects which h compose a set are termed the elements or members of the set. Tabular versus set builder notation A = {1, 2, 3} A = {x x is a positive integer, 1 x 3} may be interchanged with : Sets: Basics The symbol " " reads "is an element of". In our example, 2 A. If every element of a set S 1 is also an element of a set S 2, then S 1 is a subset of S 2 and we write S 1 S 2 or S 2 S 1. The set S 2 is said to be a superset of S 1. Def 1: Two sets S 1 and S 2 are said to be equal if and only if S 1 S 2 and S 2 S 1. 2
3 Examples: Sets: Basics #1. If S 1 = {1, 2}, S 2 = {1, 2, 3}, then S 1 S 2 or S 2 S 1. #2 The set of all positive integers is a subset of the set of real numbers. Sets: Basics The largest subset of a set S is the set S itself. The smallest subset of a set S is the set which contains no elements. The set containing no elements is called the null set, denoted. For all sets S, we have S. If all sets are subsets of a given set, we call that set the universal set, denoted U. 3
4 Sets: Basics Def 2: Two sets S 1 and S 2 are disjoint if and only if there does not exist an x such that x S 1 and x S 2. Example: If S = {0} and A = {1, 2, 4}, then S and A are disjoint. Operations Def 3: The operations of union, intersection, ti difference (relative complement), and complement are defined for two sets A and B as follows: (i) A B {x : x A or x B}, (ii) A B {x : x A and x B}, (iii) A  B {x : x A, x B}, (iv) A {x : x A}. 4
5 Operational Laws 1. Idempotent laws 1. a. A A = A 1. b. A A = A 2. Associative laws 2. a. (A B) C = A ( B C) 2. b. (A B) C = A ( B C) 3. Commutative laws 3. a. A B = B A 3. b. A B = B A Operational Laws 4. Distributive laws 4. a. A (B C) = (A B) (A C) 4. b. A (B C) = (A B) (A C) 5. Identity laws 5. a. A = A 5. b. A U=U U 5. c. A U = A 5. d. A = 5
6 Operational Laws 6. Complement laws 6. a. A A = U 6. b. (A ) = A 6. c. A A = 6. d. U =, = U II. The Real Number System The real numbers can be geometrically represented by points on a straight line. Numbers to the right of zero are the positive numbers and those to the left of zero are the negative numbers. Zero is neither positive nor negative. 11/2 0 +1/2 +1 6
7 Real Numbers The integers are the whole real numbers. Let I be the set of integers, so that I = {..., 2, 1, 0, 1, 2,...}. The positive integers are called the natural numbers. The rational numbers, Q, are those real numbers which can be expressed as the ratio of two integers. Hence, Def: Q = {x x = p/q, p I, q I, q 0}. Note that I Q. Real Numbers The irrational numbers, Q', are those real numbers which h cannot be expressed as the ratio of two integers. They are the nonrepeating infinite decimals. The set of irrationals is just the complement of the set of rationales Q in the set of reals Examples: 5, 3 and 2. 7
8 Real Numbers The prime numbers are those natural numbers say p, excluding 1, which h are divisible only by 1 and p itself. A few examples are 2, 3, 5, 7, 11, 13, 17, 19, and 23. Illustration Real Rational Irrational Integers Negative Integers Zero Natural Prime 8
9 The extended real number system. The set of real numbers R may be extended to include  and +. These notions mean to become negatively infinite or positively infinite, respectively. The result would be the extended real number system or the augmented real line, Rˆ Rules: Extended Real Numbers The following operational rules apply. (i) If "a" is a real number, then  < a <+ (ii) a + = + a =, if a  (iii) a + ( ) = ( ) + a = , if a + (iv) If 0 < a +, then a = a = a ( ) = ( ) a =  (v) If  a < 0, then a = a =  a ( ) = ( ) a = + (vi) If a is a real number, then a/ = a/+ = 0 9
10 Absolute Value of a Real Number Def 1: The absolute value of any real number x, denoted x, is defined as follows: xif x 0 x xifx 0 Properties of x If x is a real number, its absolute value x geometrically ti represents the distance between the point x and the point 0 on the real line. If a, b are real numbers, then ab = ba would represent the distance between a and b on the real line. 10
11 Properties of x We have, for a, b R (i) a 0 (ii) a + b a+b (iii) a b = ab (iv) a / b = a/b Intervals on R Let a, b R where a < b, then we have the following terminology: (i) The set A = {x a x b}, denoted A = [a, b], is termed a closed interval on R. (note that a, b A) (ii) The set B = {x a < x b}, denoted B = (a, b], is termed an openclosed interval on R. (note a B, b B) (iii) The set C = {x a x < b}, denoted C = [a, b), is termed a closedopen interval on R. (note a C, b C) (iv) The set D = {x a < x < b}, denoted D = (a, b), is termed an openinterval on the real line. (note a, b D) 11
12 III. Functions, Ordered Tuples and Product Sets Ordered Pairs and Ordered Tuples: An ordered d pair is a set consisting of two elements with a designated first element and a designated second element. If a,b are the two elements, we write (a, b) An ordered ntuple is the generalization of this idea to n elements (x 1,,x n ) Product Set Let X and Y be two sets. The product set of fx and dy or the Cartesian product of fx and Y consists of all of the possible ordered pairs (x, y), where x X and y Y. Def 1: The product set of two sets X and Y is defined as follows: X Y = {(x, y) x X, y Y} 12
13 Generalization The product set of the n sets X i, i = 1,,n, is given by X 1 x x X n = {(x 1,,x n ) : x i X i, i = 1,,n} (nterms) Product Set: Examples If A = {a, b}, B = {c, d, e}, then A B = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)}. The Cartesian plane or Euclidean twospace, R 2, is formed by R R = R 2. The nfold Cartesian product of R is Euclidean nspace R R R = R n. (nterms) 13
14 Functions Def 2: A function from a set X into a set Y is a rule f which assigns to every member x of the set X a single member y = f(x) of the set Y. The set X is said to be the domain of the function f and the set Y will be referred to as the codomain of the function f. If f is a function from X into Y, we write f : X Y. Functions Def. 2. a: The element in Y assigned by f to an x Xi is the value of f at x or the image of x under f. We write y = f(x). Def. 2. b: The graph Gr(f) of the function f:x Y is defined as follows: Gr(f) {(x, f(x)) : x X}, where Gr(f) X Y. 14
15 Functions Def 2. c: The range f [X] of the function f : X Yi is the set of fimages of x X under f or f[x] { f(x): x X}. Note that the range of a function f is a subset of the codomain of f, that is f[x] Y. Functions Def 3: A function f: X Y is said to be onto if and only if f[x] = Y. Def 4: A function f: X Y is said to be onetoone if and only if images of distinct members of the domain of f are always distinct; t in other words, if and only if, for any two members x, x X, f(x) = f(x ) implies x = x. 15
16 Inverse Function A onetoone function can be inverted. That is there exists a reverse functional relationship wherein each y maps into a unique x. Such a mapping is written f 1 : f[x] X. We have x = f 1 (y). For example, the inverse of the function y = 2 + 3x is x = y/3 2/3. The former function is onetoone. Common Functions When we write y = f(x), we mean that a functional relationship between y and x exits (each x maps into one y), however, we have not made the rule of the mapping explicit. In this section, we consider several specific functional types. Each is used in business applications. 16
17 Common Functions The first example is a specific linear function. Let the function f assign to every real number its double: y = 2x Both the domain and the codomain of f are the set of real numbers, R. Hence, f: R R. The image of the real number 2 is f(2) = 4. The range of f is given by f[r] = {2x : x R}. The Gr(f) is given by Gr(f) = {(x, 2x): x R}. Common Functions f(x) 2 1 x Clearly, the function f(x) = 2x is onetoone. Moreover, f(x) = 2x is onto, that is, f[r] = R. 17
18 Common Functions Example #1 is a specific example of a linear function. More generally, let a and b be two real numbers. The function y = f(x) = a + bx is a linear function of x. The graph of this function is shown below for the case where both a and b are positive. The constant a is called the intercept of the function, because, for x = 0, y = a. Common Functions The constant b is called the slope coefficient, i because the slope of the graph of this function is b at each point. By slope, we mean the rate of change of y per change in x. Let x change by some arbitrary amount x = x  x. This change in x generates a change in y through the functional relationship. 18
19 Common Functions The induced change in y is given by y = (a + bx )  (a + bx ) = b(x  x ). Thus, the rate of change of y per change in x is just y b(x'' x') = = b. x (x'' x' ) Illustration of Linear Function f(x) = a + bx slope = b y y a x x y 0 x x x x x 19
20 Illustration of Linear Function If in the above example, b = 0, then f is said to be a constant t function. For every value of x, y is equal to the constant a. In this case, the graph of f is a flat line (i.e., the slope is zero). y a x Polynomial Functions Next, we consider a general class of functions termed polynomial. l The term polynomial means multiterm. A polynomial function has the general form y = a o + a 1 x + a 2 x a n x n. If n = 0, then y = a o and we have a constant function. 20
21 Polynomial Functions If n = 1, then y = a o + a 1 x and we have a linear function. If n = 2, then y = a o + a 1 x + a 2 x 2 and we have a quadratic function. If n = 3, then y = a o + a 1 x + a 2 x 2 + a 3 x 3 and we have a cubic function. Polynomial Functions When a quadratic function is plotted, it appears as a parabola. This is a curve with a single bump. A examples are given in the diagram below. y a o, a 1 > 0 and a 2 < 0. y a o, a 2 > 0 and a 1 < 0. ao a o x y = ao + a1x + a2x 2 y = ao + a1x + a2x 2 x 21
22 Polynomial Functions When a cubic function is plotted, it exhibits two bumps as shown in the example below. y ao y = a o + a 1 x + a 2 x 2 + a 3 x 3 (ao, a1, a2 > 0 and a3 < 0) x Rational Functions Rational functions are functions in which y is expressed as the ratio of two polynomial l functions in x. An example is y = x 1. x 2 4x One common rational function encountered in business applications is the rectangular hyperbola. This function has the form y = a/x. 22
23 Rectangular Hyperbola: Illustration Given a > 0 and x 0, thegraphofthisf this function nctionisasinthediagrambelois in the below. y y = a/x, a > 0 x Exponential and Logarithmic The above functions are termed algebraic. Generally, such functions are expressed in terms of polynomials and/or roots of polynomials (e.g, square root of a polynomial.). For example, y = square root {(x 3 + x 2 )} is not rational, but it is algebraic. Nonalgebraic functions include two types that are used in business applications. 23
24 Exponential and Logarithmic The first is the exponential function, y = b x, where the independent d variable appears in the exponent. The second is the closely related logarithmic function, y = log b x. The latter function is the inverse of the former function, Digression on Exponents Above, in the introduction to polynomial functions, we considered d the exponent as the indicator of the power to which a variable or number is to be raised. For example, 3 2 means that the number 3 is to be raised to the second power or that 3 is to be multiplied by itself. We have that 3 2 = 3 3 = 9. 24
25 Digression on Exponents Generally, x x x x (nterms) = x n. As a special case, x 1 = x. Exponents obey the following rules. Rule 1. x n x m = x n+m. Rule 2. x n /x m = x nm. Digression on Exponents The proofs of Rules 1 and 2 are obvious. However, if n < m, then the power of x becomes negative from Rule 2. What does this mean? Actually, Rule 2 tells us the answer. If n = 4 and m = 6, then 25
26 Digression on Exponents Thus x 2 = 1/x 2 and this can be generalized into another rule: Rule 3. x n = 1/x n. Another special case of Rule 2 is where n = m, in which case x n /x n = x 0. This must be one. Thus, in accordance with Rule 2, any nonzero number raised to the zero power is one. Rule 4. x 0 = 1. ( x 0) Digression on Exponents How do we interpret fractional exponents? Using Rule 1, we can interpret t a number such as x 1/2 : x 1/2 x 1/2 = x 1 = x. That is, because x 1/2 multiplied by itself is x, x 1/2 is the square root of x. Likewise, x 1/3 is the cube root of x. Generally, Rule 5. x 1/n = nth root of n. 26
27 Digression on Exponents Two other rules obeyed by exponents are as follows: Rule 6. (x m ) n = x nm. Rule 7. x m y m = (xy) m. Exponential and Logarithmic Functions An exponential function is a function in which the independent variable appears as an exponent: y = b x, where b > 1. We take b > 0 to avoid complex numbers. b > 1 is not restrictive because we can take (b 1 ) x = b x for this case. A logarithmic function is the inverse function of b x. That is, x = log b y. 27
28 Exponential and Logarithmic Functions The exponential function is graphed below. b x y 1 x Rules of logarithms Rl Rule 1. If x, y >0 0, then log (yx) = log y +log x. Rule 2. If x, y > 0, then log (y/x) = log y  log x. Rule 3. If x > 0, then log x a = a log x. 28
29 Preferred Base A preferred base is the number e (More formally, e = lim [1 + 1/n] n.) e is called the n natural logarithmic base. The corresponding log function is written x = ln y, meaning log e y. a. conversion Conversion and inversion of bases log b u = (log b c)(log c u) (log c u is known) Proof: Let u = c p. Then log c u = p. We know that log b u = log b c p = plog b c. By definition, p = log c u, so that log b u = (log c u)(log b c). b. inversion log b c = 1/(log c b). Proof: Using the conversion rule, 1 = log b b = (log b c)(log c b). Thus, log b c = 1/(log c b). 29
30 Functions of many variables So far we have only considered functions of a single independent variable. However, the concept can be extended to the case of two or more independent variables. We write y = f(x 1, x 2, x n ). The graph of the function is a surface in (n+1) dimensional space. For n greater than 2, it is of course impossible to graph the function. Notes on Logical Reasoning 1. Notation for logical reasoning: a. means "for all" b. ~ means "not" c. means "there exists 2. A Conditional Let A and B be two statements. A B means all of the following: If A, then B A implies B A is sufficient for B B is necessary for A 30
31 Notes on Logical Reasoning 3. Proving a Conditional: Methods of Proof a. Direct: Show that B follows from A. b. Indirect: Find a statement C where C B. Show that A C. c. Contrapositive: Show that (~B) (~ A). d. Contradiction: Show that (~ B and A) (false statement). Notes on Logical Reasoning 3. A Biconditional Let A and B be two statements. A B means all of the following: A if and only if B ( A iff B) A is necessary and sufficient for B A and B are equivalent A implies B and B implies A 31
32 Notes on Logical Reasoning 4. Proving a Biconditional Use any of the above methods for proving a conditional and show that A B and that B A. 32
Algebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
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 informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
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 informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More informationMath 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
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebra II. Weeks 13 TEKS
Algebra II Pacing Guide Weeks 13: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 46: Linear Equations and Functions:
More informationA 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
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationMBA Jump Start Program
MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
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 informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationDRAFT. Algebra 1 EOC Item Specifications
DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as
More informationSome Lecture Notes and InClass Examples for PreCalculus:
Some Lecture Notes and InClass Examples for PreCalculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring noncourse based remediation in developmental mathematics. This structure will
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 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 informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
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 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 informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
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 information1 The Concept of a Mapping
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationHow To Understand The Difference Between Economic Theory And Set Theory
[1] Introduction TOPIC I INTRODUCTION AND SET THEORY Economics Vs. Mathematical Economics. "Income positively affects consumption. Consumption level can never be negative. The marginal propensity consume
More informationAnswer 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.
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
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 informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More informationCrosswalk Directions:
Crosswalk Directions: UMS Standards for College Readiness to 2007 MLR 1. Use a (yes), an (no), or a (partially) to indicate the extent to which the standard, performance indicator, or descriptor of the
More information04 Mathematics COSGFLD00403. Program for Licensing Assessments for Colorado Educators
04 Mathematics COSGFLD00403 Program for Licensing Assessments for Colorado Educators Readers should be advised that this study guide, including many of the excerpts used herein, is protected by federal
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationTo define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions
Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationhttp://www.aleks.com Access Code: RVAE4EGKVN Financial Aid Code: 6A9DBDEE3B74F5157304
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 informationparent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL
parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
More informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More informationMathematics Curriculum
Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: AAPR.3: Identify zeros of polynomials
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationNorth Carolina Math 2
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.
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 249x + 6 x  6 A) 1, x 6 B) 8x  1, x 6 x 
More informationKEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 447) Surveys and
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationTim Kerins. Leaving Certificate Honours Maths  Algebra. Tim Kerins. the date
Leaving Certificate Honours Maths  Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many
More informationGRADES 7, 8, AND 9 BIG IDEAS
Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for
More information