Mathematics with Applications in Business and Social Sciences Content PreviewEditors: Ʉ Danielle C. Bess, Marvin Glover, Lisa Hinton Project Manager: Ʉ Claudia Vance Lead Developers: Ʉ Robert Alexander, Doug Chappell, Kenneth Hanson Designers: Ʉ Trudy Gove, Patrick Thompson Cover Design: Ʉ Trudy Gove VP Research & Development: Ʉ0DUFHO3UHYX]QDN Director of Content: Ʉ.DUD5RFK« A division of Quant Systems, Inc. 546 Long Point Road Mount Pleasant, SC 29464 &RS\ULJKWkE\+DZNHV/HDUQLQJɈɈ4XDQW6\VWHPVΖQF$OOULJKWVUHVHUYHG No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the publisher. Printed in the United States of America Ʉ ɄɄɄɄɄɄɄɄɄTable of Contents CHAPTER 0 Fundamental Concepts of Algebra Ʉ Real Numbers Ʉ The Arithmetic of Algebraic Expressions ɄΖQWHJHU([SRQHQWV Ʉ Radicals Ʉ Rational Exponents Ʉ Polynomials and Factoring CHAPTER 1 Equations and Inequalities in One Variable Ʉ Linear Equations in One Variable Ʉ Applications of Linear Equations in One Variable Ʉ/LQHDUΖQHTXDOLWLHVLQ2QH9DULDEOH Ʉ4XDGUDWLF(TXDWLRQVLQ2QH9DULDEOH Ʉ Higher Degree Polynomial Equations Ʉ Rational and Radical Equations CHAPTER Linear Equations in Two Variables Ʉ The Cartesian Coordinate System Ʉ Linear Equations in Two Variables Ʉ Forms of Linear Equations Ʉ Parallel and Perpendicular Lines Ʉ Linear Regression CHAPTER Functions and Their Graphs ɄΖQWURGXFWLRQWR)XQFWLRQV Ʉ Functions and Models Ʉ/LQHDUDQG4XDGUDWLF)XQFWLRQV Ʉ$SSOLFDWLRQVRI4XDGUDWLF)XQFWLRQV Ʉ Other Common Functions Ʉ Transformations of Functions Ʉ Polynomial Functions Ʉ Rational Functions Ʉ5DWLRQDOΖQHTXDOLWLHV CHAPTER Exponential and Logarithmic Functions Ʉ Exponential Functions and Their Graphs Ʉ Applications of Exponential Functions Ʉ Logarithmic Functions and Their Graphs Ʉ Applications of Logarithmic Functions CHAPTER Mathematics of Finance Ʉ Basics of Personal Finance Ʉ6LPSOHDQG&RPSRXQGΖQWHUHVW Ʉ Annuities: Present and Future Value Ʉ Borrowing MoneyCHAPTER Systems of Linear Equations; Matrices Ʉ Solving Systems of Linear Equations by Substitution and Elimination Ʉ Matrix Notation and Gauss- Jordan Elimination Ʉ Determinants and Cramer’s Rule Ʉ Basic Matrix Operations ɄΖQYHUVHVRI6TXDUH0DWULFHV Ʉ/HRQWLHIΖQSXW2XWSXW$QDO\VLV CHAPTER Inequalities and Linear Programming Ʉ/LQHDUΖQHTXDOLWLHVLQ7ZR9DULDEOHV Ʉ Linear Programming: The Graphical Approach Ʉ7KH6LPSOH[0HWKRG0D[LPL]DWLRQ Ʉ The Simplex Method: Duality DQG0LQLPL]DWLRQ Ʉ The Simplex Method: Mixed Constraints CHAPTER Probability Ʉ Set Notation Ʉ Operations with Sets ɄΖQWURGXFWLRQWR3UREDELOLW\ Ʉ Counting Principles: Combinations and Permutations Ʉ Counting Principles and Probability Ʉ Probability Rules and Bayes’ Theorem Ʉ Expected Value CHAPTER Statistics Ʉ Collecting Data Ʉ Displaying Data Ʉ'HVFULELQJDQG$QDO\]LQJ'DWD Ʉ The Binomial Distribution Ʉ The Normal Distribution Ʉ Normal Approximation to the Binomial Distribution CHAPTER 10 Limits and the Derivative Ʉ One-Sided Limits Ʉ Limits Ʉ More about Limits Ʉ Continuity Ʉ Average Rate of Change ɄΖQVWDQWDQHRXV5DWHRI&KDQJH Ʉ'HȴQLWLRQRIWKH'HULYDWLYH and the Power Rule Ʉ Techniques for Finding Derivatives Ʉ Applications: Marginal Analysis CHAPTER 11 More about the Derivative Ʉ7KH3URGXFWDQG4XRWLHQW5XOHV Ʉ The Chain Rule and the General Power Rule ɄΖPSOLFLW'LHUHQWLDWLRQDQG5HODWHG5DWHV ɄΖQFUHDVLQJDQG'HFUHDVLQJΖQWHUYDOV Ʉ Critical Points and the First Derivative Test Ʉ Absolute Maximum and MinimumCHAPTER Applications of the Derivative Ʉ&RQFDYLW\DQG3RLQWVRIΖQȵHFWLRQ Ʉ The Second Derivative Test Ʉ Curve Sketching: Polynomial Functions Ʉ Curve Sketching: Rational Functions Ʉ Business Applications Ʉ2WKHU$SSOLFDWLRQV2SWLPL]DWLRQ Distance, and Velocity CHAPTER Additional Applications of the Derivative Ʉ Derivatives of Logarithmic Functions Ʉ Derivatives of Exponential Functions Ʉ Growth and Decay Ʉ Elasticity of Demand Ʉ L’Hôpital’s Rule Ʉ'LHUHQWLDOV CHAPTER Integration with Applications Ʉ7KHΖQGHȴQLWHΖQWHJUDO ɄΖQWHJUDWLRQE\6XEVWLWXWLRQ Ʉ Area and Riemann Sums Ʉ7KH'HȴQLWHΖQWHJUDODQGWKH Fundamental Theorem of Calculus Ʉ Area under a Curve (with Applications) Ʉ Area between Two Curves (with Applications) Ʉ'LHUHQWLDO(TXDWLRQV CHAPTER Additional Integration Topics ɄΖQWHJUDWLRQE\3DUWV Ʉ$QQXLWLHVDQGΖQFRPH6WUHDPV Ʉ7DEOHVRIΖQWHJUDOV Ʉ1XPHULFDOΖQWHJUDWLRQ ɄΖPSURSHUΖQWHJUDOV Ʉ Volume CHAPTER Multivariable Calculus Ʉ Functions of Several Variables Ʉ Partial Derivatives Ʉ Local Extrema for Functions of Two Variables Ʉ Lagrange Multipliers Ʉ The Method of Least Squares Ʉ'RXEOHΖQWHJUDOV1-800-426-9538 HAWKES LEARNING.COM0.1 Real Numbers Common Subsets of Real Numbers Some types of numbers occur so frequently in mathematics that they have been given special names and symbols. These names will be used throughout this course and in later math courses when referring to members of the following sets: Types of Real Numbers The Natural (or Counting) Numbers: This is the set of numbers . The set is infinite, so in list form we can write only the first few numbers. The Whole Numbers: This is the set of natural numbers and : . Again, we can list only the first few members of this set. No special symbol will be assigned to this set in this text. The Integers: This is the set of natural numbers, their negatives, and . As a list, this is the set . Note that the list continues indefinitely in both directions. The Rational Numbers: This is the set, with symbol (for quotient), of ratios of integers(hencethename).Thatis,anyrationalnumbercanbewrittenintheform, where and are both integers and . When written in decimal form, rational numbers either terminate or have a repeating pattern of digits past some point. The Irrational Numbers: Every real number that is not rational is, by definition, irrational. In decimal form, irrational numbers are nonterminating and nonrepeating. No special symbol will be assigned to this set in this text. The Real Numbers: Every set above is a subset of the set of real numbers, which is denoted . Every real number is either rational or irrational, and no real number is both. The following figure shows the relationships among the subsets of defined above. This figure indicates, for example, that every natural number is automatically a whole number, and also an integer, and also a rational number. 1 0.1 | Real NumbersReal Numbers ( Ⴆ ) Rational Numbers ( Ⴃ ) Decimal form either terminates or repeats Integers ( Ⴌ ) Whole Numbers Natural Numbers ( ႟ ) Irrational Numbers Decimal form is nonterminating and nonrepeating Figure 1: The Real Numbers The Real Number Line Mathematicians often depict the set of real numbers as a horizontal line, with each point on the line representing a unique real number (so each real number is associated with a unique point on the line). The real number corresponding to a given point is called the coordinate of that point. Thus one (and only one) point on the real number line represents the number , and this point is called the origin. Points to the right of the origin represent positive real numbers, while points to the left of the origin represent negative EXAMPLE 1: Types of Real Numbers Consider the set . a. The natural numbers in are and . is a natural number since . b. The whole numbers in are , , and . c. The integers in are , , , and . d. The rational numbers in are , , , , , , and . The numbers and are both rational numbers since and (the bar over the last digit indicates that the digit repeats indefinitely). Note that any integer is also a rational number, sinceitcanbewrittenas. e. The only irrational numbers in are and . Although well known now, the irrationality of came as a bit of a surprise to the early Greek mathematicians who discovered this fact. The irrationality of was not proven until 1767. 2Chapter 0real numbers. Figure 2 is an illustration of the real number line with several points plotted. Note that two irrational numbers are plotted, though their locations on the line are approximations. í í 10 ¥ 25 Figure 2: The Real Number Line Order on the Real Number Line Representing the real numbers as a line leads naturally to the idea of ordering the real numbers. We say that the real number is less than the real number (in symbols, ) if lies to the left of on the real number line. This is equivalent to saying that is greater than (in symbols, ). The following definition gives the meaning of these and two other symbols indicating order. EXAMPLE 2: Drawing the Real Number Line We choose which portion of the real number line to show and the physical length that represents one unit based on the numbers that we wish to plot. a. If we want to plot the numbers , , and , we might construct the graph below. 101106107 b. If we want to plot the numbers , , and , we might make the unit interval longer. í 1 í 3 4 í 1 2 01 4 1 3 0.1 | Real NumbersNext >