Mathematics with Applications in Business and Social Sciences MABS_Front Matter.indd 1MABS_Front Matter.indd 111/10/2021 5:47:26 PM11/10/2021 5:47:26 PMEditors: Danielle C. Bess, Marvin Glover, Claudia Vance Assistant Editor: Daniel Breuer Creative Services Manager: Trudy Tronco Designers: Lizbeth Mendoza, Patrick Thompson, Joel Travis Cover Design: Trudy Tronco Composition and Answer Key Assistance: Quant Systems India Pvt. Ltd. Courseware Developers: Douglas Chappell, Jolie Even, Adam Flaherty, Kyle Gilstrap Manager of Math Content Development: Blair Dunivan A division of Quant Systems, Inc. 546 Long Point Road Mount Pleasant, SC 29464 Copyright © 2022 by Hawkes Learning / Quant Systems, Inc. All rights reserved. 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. Library of Congress Control Number 2021917837 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-484-9 MABS_Front Matter.indd 2MABS_Front Matter.indd 211/10/2021 5:47:27 PM11/10/2021 5:47:27 PMTable of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 0 Fundamental Concepts of Algebra 0.1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 0.2 The Arithmetic of Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 0.3 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 0.4 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 0.5 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 0.6 Polynomials and Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 1 Equations and Inequalities in One Variable 1.1 Linear Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 1.2 Applications of Linear Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 1.3 Linear Inequalities in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 1.4 Quadratic Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 1.5 Higher Degree Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 1.6 Rational and Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 2 Linear Equations in Two Variables 2.1 The Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 2.2 Linear Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 2.3 Forms of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 2.4 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138 2.5 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146 3 Functions and Their Graphs 3.1 Introduction to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162 3.2 Functions and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171 3.3 Linear and Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185 3.4 Applications of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .196 3.5 Other Common Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .200 3.6 Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .210 3.7 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226 3.8 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .239 3.9 Rational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 4 Exponential and Logarithmic Functions 4.1 Exponential Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258 4.2 Applications of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267 4.3 Logarithmic Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280 4.4 Applications of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291 MABS_Front Matter.indd 3MABS_Front Matter.indd 311/10/2021 5:47:29 PM11/10/2021 5:47:29 PMiv Table of Contents 5 Mathematics of Finance 5.1 Basics of Personal Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304 5.2 Simple and Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315 5.3 Annuities: Present and Future Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 5.4 Borrowing Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341 6 Systems of Linear Equations; Matrices 6.1 Solving Systems of Linear Equations by Substitution and Elimination . . . . . . . . . . . . .360 6.2 Matrix Notation and Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374 6.3 Determinants and Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .388 6.4 Basic Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .400 6.5 Inverses of Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412 6.6 Leontief Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423 7 Inequalities and Linear Programming 7.1 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .440 7.2 Linear Programming: The Graphical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452 7.3 The Simplex Method: Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .460 7.4 The Simplex Method: Duality and Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .482 7.5 The Simplex Method: Mixed Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .494 8 Probability 8.1 Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .512 8.2 Operations with Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .522 8.3 Introduction to Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .532 8.4 Counting Principles: Combinations and Permutations . . . . . . . . . . . . . . . . . . . . . . . . . .542 8.5 Counting Principles and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .555 8.6 Probability Rules and Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .564 8.7 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582 9 Statistics 9.1 Collecting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594 9.2 Displaying Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605 9.3 Describing and Analyzing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631 9.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654 9.5 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .665 9.6 Normal Approximation to the Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .675 MABS_Front Matter.indd 4MABS_Front Matter.indd 411/10/2021 5:47:31 PM11/10/2021 5:47:31 PM Table of Contents v 10 Limits and the Derivative 10.1 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .686 10.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .698 10.3 More about Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .705 10.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714 10.5 Average Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .723 10.6 Instantaneous Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .726 10.7 Definition of the Derivative and the Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .740 10.8 Techniques for Finding Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .749 10.9 Applications: Marginal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .758 11 More about the Derivative 11.1 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .770 11.2 The Chain Rule and the General Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .779 11.3 Implicit Differentiation and Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .790 11.4 Increasing and Decreasing Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .802 11.5 Critical Points and the First Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .814 11.6 Absolute Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .823 12 Applications of the Derivative 12.1 Concavity and Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .836 12.2 The Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .850 12.3 Curve Sketching: Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .857 12.4 Curve Sketching: Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .867 12.5 Business Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .875 12.6 Other Applications: Optimization, Distance, and Velocity . . . . . . . . . . . . . . . . . . . . . .884 13 Additional Applications of the Derivative 13.1 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .896 13.2 Derivatives of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .906 13.3 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .915 13.4 Elasticity of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .926 13.5 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .932 13.6 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .943 14 Integration with Applications 14.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .952 14.2 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .966 14.3 Area and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .974 14.4 The Definite Integral and the Fundamental Theorem of Calculus . . . . . . . . . . . . . . . .980 14.5 Area under a Curve (with Applications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .989 14.6 Area between Two Curves (with Applications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .996 14.7 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1007 MABS_Front Matter.indd 5MABS_Front Matter.indd 511/10/2021 5:47:31 PM11/10/2021 5:47:31 PMvi Table of Contents 15 Additional Integration Topics 15.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1018 15.2 Annuities and Income Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1024 15.3 Tables of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1030 15.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1035 15.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1043 15.6 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1050 16 Multivariable Calculus 16.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1056 16.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1066 16.3 Local Extrema for Functions of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1076 16.4 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1083 16.5 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1089 16.6 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1096 Appendix: Critical Values of the Pearson Correlation Coefficient . . . . . . . . . . . . . . . . . . . . .1104 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .AK-1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I-1 MABS_Front Matter.indd 6MABS_Front Matter.indd 611/10/2021 5:47:31 PM11/10/2021 5:47:31 PM Preface vii PREFACE Features Definitions, Theorems, Formulas, Properties, and Procedures Definitions, theorems, and formulas are clearly set apart in highly visible green boxes for easy reference, and properties and procedures are similarly set apart in distinctive blue boxes. All formally identified terms appear in bold print when first defined, and other useful terms appear in italic font. Cautions Many common errors are pointed out, along with how to correct them. These are set apart in red boxes. Notes and Helpful Hints These green and red boxes in the margins help clarify subtle details and provide problem-solving tips. Examples Examples are presented in a step-by-step manner that is easy for students to follow. Each example has a title indicating the problem-solving skill being presented. Examples make use of tables, diagrams, and graphs for additional clarity where applicable. 186 Chapter 3 w Functions and Their Graphsxf x () − − 18 05 12 21 x y−4 −2 2 4 8106 − 6 − 4 − 2462 FIGURE 1: Graph of f (x) = −3x + 5 Again, note that every point on the graph of the function in Figure 1 is an ordered pair of the form (x, f (x)); we have simply highlighted four of them with dots. We could have graphed the function f (x) = −3x + 5 by noting that it is a straight line with a slope of −3 and a y-intercept of 5. We use this approach in the following example. Example 1: Graphing Linear Functions Graph the following linear functions. a. f (x) = 3x + 2 b. g(x) = 3 Solution a. x y − 4 − 224 −2 2 4 6 The function f is a line with a slope of 3 and a y-intercept of 2. To graph the function, plot the ordered pair (0, 2) and locate another point on the line by moving up 3 units and over to the right 1 unit, giving the ordered pair (1, 5). Once these two points have been plotted, connecting them with a straight line completes the process. { NOTE A function cannot represent a vertical line (since it fails the vertical line test). Vertical lines can represent the graphs of equations, but not functions. 336 Chapter 5 w Mathematics of Finance PV 150 000 1 004 12 73 100 31 12 18 , . ,. This means that if Amber could deposit $73,100.31 into the account now, then after 18 years there will be $150,000 in the account. Ɨ TECH TRAINING To perform the calculation from Example 1 on a TI-30XIIS/B or TI-83/84 Plus calculator, use the following keystrokes. 150000 ÷ ( 1 + ( 0.04 ÷ 12 ) ) ^ ( 12 x 18 ) = To perform the calculation with Wolfram|Alpha, go to www.wolframalpha.com and type “150000/(1+(0.04/12))^(12*18)” into the input bar. Then, click the = button. Often it is preferable to make regular payments into a savings program rather than depositing a lump sum of money at once. Although these payments might be considerably smaller than a lump sum amount, as we discussed in the previous example, compound interest can work in our favor in just the same manner. The significant point with this type of savings plan is that consistent payments are ongoing. For instance, suppose you deposit $50 each month into an account. The first month, interest is calculated on the initial $50 deposit. However, in the next period, after you’ve made your second deposit of $50, interest is calculated on both $50 payments as well as the interest you earned in the first period. So, although you personally are building up your savings by contributing $50 a month, interest continues to add to your savings as well. In broad terms, whenever fixed regular payments are made we refer to them as annuity payments. Annuity An annuity is a sequence of regular payments made into an account, or taken out of an account, over time. We can calculate the future value of an annuity payment plan by using the following formula. It takes into account not only the additional compound interest each period, but also the regular payment added each time. Annuity Formula for Finding Future Value The future value (FV) of an annuity savings account is calculated with the formula FV PMT r n r n nt 11 where PMT is the payment amount that is deposited on a regular basis, r is the APR, n is the number of regular payments made each year, and FV is the future value after t years. ̻ HELPFUL HINT If calculating large formulas in pieces on a calculator, round each step to at least six decimal places to avoid rounding errors in the subsequent calculations. Skill Check 1 How much should you deposit now in an account with an APR of 7% compounded monthly if you wish for the account to have a balance of $200,000 in 25 years? 176 Chapter 3 w Functions and Their Graphs p = S(x) Supply function: price per unit at which the supplier will supply x units of an item. p = D(x) Demand function: price per unit at which consumers will buy x units of an item. The equilibrium point, denoted as (x E, p E), is the point where the price is such that the production level (supply) is equal to the purchase level (demand). (See Figure 1.) x 5 10 15 20 p 3 6 9 12 p = 0.2x + 3 x 5 10 15 20 p 3 6 9 12 p = −0.4x + 12 (15, 6) x 5 10 15 20 p 3 6 9 12 (x E, p E) p = S(x) Supply Function (A) p = D(x) Demand Function (B) S(x) = D(x) Equilibrium Point (C) FIGURE 1 In general, supply and demand functions are not linear functions as they are shown to be in Figure 1. However, supply functions are increasing functions, and demand functions are decreasing functions. Suppliers will happily provide more products as prices increase, and consumers will happily buy more products as prices decrease. Example 5: Equilibrium Point Suppose that the supply function for a particular product is p = S(x) = x2 + x + 3 and the demand function p = D(x) = (x − 5)2 where x represents thousands of units and p represents thousands of dollars. Find the equilibrium point (x E, p E). Solution We solve for x E by setting S(x) = D(x). Then we substitute this value for x into either S(x) or D(x) to find p E. SxDx xxx xxxx x x E EE EE EE EE E E () = () ++ =− () ++ =−+ = = 2 2 22 35 31025 1122 2 ppS xS EE = () = () =+ += 2 22 39 2 The equilibrium point occurs where x = 2000 units and p = $9000. Units (in thousands) 1 2 3 4 5 Price (in thousands of dollars) 2 4 6 8 10 12 14 16 (2, 9) Equilibrium point p x (x E, p E) p = S(x) = x2 + x + 3 p = D(x) = (x − 5)2 Linear Functions A linear function f in the variable x is any function that can be written in the form f (x) = mx + b, where m and b are real numbers. If m ≠ 0, f (x) = mx + b is also called a first-degree function. Solving Elementary Exponential Equations To solve an elementary exponential equation, complete the following steps. Step 1: Isolate the exponential. Move the exponential containing x to one side of the equation and any constants or other variables in the expression to the other side. Simplify, if necessary. Step 2: Find a base that can be used to rewrite both sides of the equation. Step 3: Equate the powers, and solve the resulting equation. 128 Chapter 2 w Linear Equations in Two Variables2.3 FORMS OF LINEAR EQUATIONS TOPICS The Slope of a Line Slope-Intercept Form of a Line Point-Slope Form of a Line The Slope of a Line There are several ways to characterize a given line in the plane. We have already used one way repeatedly: two distinct points in the Cartesian plane determine a line. Another, often more useful, approach is to identify just one point on the line and to indicate how “steeply” the line is rising or falling as we scan the plane from left to right. It turns out that a single number is sufficient to convey this notion of “steepness.” The Slope of a Line Let L stand for a given line in the Cartesian plane, and let (x 1, y 1) and (x 2, y 2) be the coordinates of any two distinct points on L. The slope of the line L is the ratio yy xx 21 21 − − which can be described in words as “change in y over change in x” or “rise over run.” x y Run = x2− x1 Rise = y2− y1 ( x2, y1) ( x1, y1) ( x2, y2) L FIGURE 1: Rise and Run between Two Points In the line drawn in Figure 1, the ratio yy xx 21 21 − − is positive, the line rises from the lower left to the upper right, and we say that the line has a positive slope. If the rise and run have opposite signs, the slope of the line is negative and the line under consideration would fall from the upper left to the lower right. ¡ CAUTION It doesn’t matter how you assign the labels (x 1, y 1) and (x 2, y 2) to the two points you are using to calculate slope, but it is important that you are consistent as you apply the formula. You cannot change the order in which you are subtracting as you determine the numerator and denominator in the slope formula. 21 21 yy xx − − 12 21 y y xx − − 12 12 y y x x − − yy x x 21 12 − − Correct Incorrect MABS_Preface.indd 7MABS_Preface.indd 711/10/2021 5:41:46 PM11/10/2021 5:41:46 PMviii Preface 6.1 w Solving Systems of Linear Equations by Substitution and Elimination 369 Solving Systems of Equations Using Technology The solution to a consistent pair of linear equations is the point common to both equations. Graphically speaking, the solution is the point where the graphs of the two equations intersect. We can use a graphing calculator to find this point. Consider the following system of equations: 2313 6 xy xy . One way to solve this system using a calculator is to graph each equation. Remember to solve for y before entering the equation in Y= and selecting graph . Once the graph of the two lines is displayed, press 2nd trace to access the CALC menu and select intersect . The phrase “First curve? ” should appear. Use the arrows to move the cursor along the first line to where it appears to intersect the other line and press enter . When the phrase “Second curve? ” appears, press enter again (as the cursor should now be on the second line, still near the point of intersection). Now the word “Guess?” should appear. Press enter a final time and the x- and y-values of the point of intersection will appear at the bottom. So the point where the lines intersect, and thus the solution to this system of equations, is 51, . This method works with any system of equations that can be graphed on a calculator, not just linear ones. MABS_Chapter 6.indd 369MABS_Chapter 6.indd 36911/10/2021 1:51:52 PM11/10/2021 1:51:52 PM Technology Instructions Technology notes and screenshots are included throughout the text to highlight ways that graphing utilities and other forms of technology can help solve problems or explain concepts. Step-by-step instructions for using a TI-84 Plus are given in many cases. Applications Many examples and exercises illustrate practical applications, keeping students engaged. 358 Chapter 5 w Mathematics of F inance Example 5: How Much House Can You Afford? Suppose you have recently graduated from college and want to purchase a house. Your take-home pay is $3220 per month and you wish to stay within the recommended guidelines for mortgage amounts by only spending 1 4 of your take-home pay on a house payment. You have $15,300 saved for a down payment. With your good credit and the down payment you can get an APR from your bank of 3.37% compounded monthly. a. What is the total cost of a house you could afford with a 15-year mortgage? b. What is the most that you could afford with a traditional 30-year mortgage instead of a 15-year? Solution a. The first thing to do is to calculate the size of the monthly mortgage payment you are willing to spend. Since you have $3220 per month in take-home pay, multiply this by 25% to find your maximum monthly payment. advised monthly payment = $3220 ⋅ 0.25 = $805 We know that r = 0.0337 and that because this is a 15-year mortgage, n = 12 and t = 15. Substituting these values in the formula, we have the following. maximum purchase price =⋅ −+ − PMT r n r n nt 11 ==⋅ −+ ≈ −⋅ 805 11 0 0337 12 0 0337 12 113 6 12 15 . . ,117 8221. So, you could afford a 15-year mortgage of approximately $113,617.82. Remember that the amount of down payment you have available will add to the maximum amount you can spend on a house. Therefore, with a 15-year mortgage you can afford to buy a house with a maximum price of $113,617.82 + $15,300 = $128,917.82. Ɨ TECH TRAINING Use the following keystrokes on a TI-30XIIS/B or TI-83/84 Plus calculator for the calculation of the maximum purchase price for part a. of Example 5. 805 x ( 1 Þ ( 1 + 0.0337 ÷ 12 ) ^ ( (-) 12 x 15 ) ) ÷ ( 0.0337 ÷ 12 ) = Wolfram|Alpha can be used to find the maximum purchase price. Go to www. wolframalpha.com and type “805*(1-(1+0.0337/12)^(-12*15))/(0.0337/12)” into the input bar. Then, click the = button. ̻ HELPFUL HINT If you are using a TI-30XS/B Multiview calculator, the keystrokes will be slightly different for your calculator. 5.4 w Borrowing Money 355 ĭ APPLICATIONS Round your answer to the nearest cent, if necessary. 7. Given the chart below, solve the following problems. 1 62124186248 250 0 500 750 1000 Number of Monthly Payments Debt Balance (in Dollars) Minimum Payments vs . Fixed Payments Minimum Payments Fixed Payments a. Estimate the total amount paid when a debt balance was paid using a fixed monthly payment of $40. b. Estimate the total amount paid when a debt balance was paid using the minimum monthly payment of $18. 8. Rachel is purchasing a new camera that costs $3800 for her photography business. Rachel uses a credit card that has an APR of 16.99%. a. How long will it take her to pay off the camera if she makes monthly payments of $75? b. How much will she pay in the long run for the camera if she makes monthly payments of $75? c. How long will it take her to pay off the camera if she makes monthly payments of $150? d. How much will she pay in the long run for the camera if she makes monthly payments of $150? 9. Tommy gets to choose from one of the new car incentives when he purchases his car next week. He can either choose 0.9% APR financing for 48 months or $1000 cash back with a 4.75% APR over 48 months. Compare the two incentives that Tommy has to choose from if the new car he wishes to buy is $32,457 and he has saved a down payment of $3500. 10. Mike bought a new car and financed $25,000 to make the purchase. He financed the car for 60 months with an APR of 6.5%. Determine each of the following. a. Mike’s monthly payment b. Total cost of Mike’s car c. Total interest Mike pays over the life of the loan 11. Omar wants to purchase three vans for his delivery business. Each van costs $38,000. He wishes to finance the purchase for 48 months and has acquired an APR of 4.5%. Determine each of the following. a. Omar’s monthly payment b. Total cost of Omar’s vans c. Total interest paid by Omar over the life of the loan 12. Jamal bought a new car for $32,000. He paid a 10% down payment and financed the remaining balance for 36 months with an APR of 4.5%. Determine each of the following. a. Jamal’s monthly payment b. Total cost of Jamal’s car c. Total interest Jamal pays over the life of the loan MABS_Chapter 5.indd 355MABS_Chapter 5.indd 35511/10/2021 1:37:23 PM11/10/2021 1:37:23 PM 876 Chapter 12 w Applications of the Derivative C(x) inventory costs = = (storage cost per item) storage costs x 2 + + (cost per order) ordering costs total ordered x Example 1: Minimizing Inventory Costs A furniture dealer sells 500 desks per year. The desks take up floor space and warehouse space, and the dealer estimates his storage costs at $6 per desk. The distributor charges the dealer a $60 fee for each order. How many times per year and in what lot size should the dealer order to minimize inventory costs? Solution Using the given information, we can determine a function for the inventory costs, C(x). Let x = lot size. Then 500 x is the number of orders per year, and x 2 is the average inventory. Cx x storage cost per itemcost per order 2 500 xx Cx x x 6 2 60 500 The number of desks ordered is between 1 and 500. At the extremes, one order for 500 desks would cost C 500 500 500 6 2 60 500 1560$, and 500 orders for one desk at a time would cost C 1 1 1 6 2 60 500 30 003$,. Now we need to differentiate C(x) so we can determine the local minima. Cx x x xx Cxx 6 2 60 500 330 000 330 000 1 2 , , 33 30 000 2 , x Rewrite C(x) using exponents. We set C ′(x) = 0 and solve for x. 3 30 000 0 330 000 10 000 100 2 2 2 , , , x x x x Ɨ TECH TRAINING To evaluate the function C(x) in Example 1 with a TI-83/84 Plus calculator, perform the following steps: 1. Enter the function C(x) that was found into Y1 . 2. Now, on the main screen, press VARS , scroll right to Y-VARS , select 1:Function , and then 1:Y1 to display Y1 on the main screen. 3. With your cursor after Y1 , type an opening parenthesis and then the x-value we are evaluating, 500. Type a closing parenthesis. 4. Press enter to calculate the corresponding y-value. 5. This process can be repeated for the next x-value, 1. MABS_Chapter 12.indd 876MABS_Chapter 12.indd 87611/8/2021 12:35:43 PM11/8/2021 12:35:43 PM MABS_Preface.indd 8MABS_Preface.indd 811/10/2021 5:42:04 PM11/10/2021 5:42:04 PM Preface ix 9.2 w Displaying Data 625 Misleading GraphsAs we said at the beginning of the section, itʼs important to display data so that it is organized clearly and it effectively conveys the intended message. Ideally, graphs should be able to stand alone without the need for additional information in order to be understood. However, sometimes graphs either intentionally or unintentionally convey the wrong message about data or are not quite clear enough to get their message across. It’s important to be aware that there are visually misleading and/or ambiguous graphs out there. For example, making the bars different widths on a bar chart might imply that one category is somehow larger than another. Similarly, a distorted piece of a pie graph might inaccurately lead the reader to assume that one section of data is larger than another. An example of this is the graph in Figure 10.In their graph, the freshmen wanted to emphasize the fact that they came in second place for spirit week—although they just grabbed second place by a tiny margin. If the exact numbers were not on the graph, the emphasis on the freshmen piece of pie might visually imply that the freshman wedge is considerably larger than the junior wedge. Watch out for these visual manipulations when interpreting graphs.Skill Check Answers 1. 0 pets2. 6,488,203 people 9.2 EXERCISES L CONCEPT CHECK Fill in each blank with the correct term. 1. A is a graph that represents a frequency distribution. 2. When comparing parts of data to the whole, a visually shows this using sections of a circle. 3. A graph is best to use when showing data over a time period. 4. When comparing multiple categories from different populations, a graph or a graph can be used. 5. A is a literal count of each member of a data set and how often it occurs. Spirit Week ParticipationSenior14Junior30Sophomore75Freshmen31FIGURE 10MABS_Chapter 9.indd 625MABS_Chapter 9.indd 62511/10/2021 2:46:31 PM11/10/2021 2:46:31 PM 3.3 w Linear and Quadratic Functions 193 3.3 EXERCISES ô PRACTICE Graph the following linear functions. See Example 1. 1. fxx 52 2. gx x 32 4 3. hxx2 4. px2 5. gxx 32 6. rx x 2 5 7. fxx 21 8. axxx 31 1 3 9. fxx 24 10. gx x 28 4 11. hxx 510 12. kxx x 3 26 2 13. mx x 25 10 14. qxx 151. 15. wxxx22 Match the following functions with their graphs. 16. fxxx 814172 17. fxx x 3 78 3 18. fxx 6 2 2 8 19. fxxx 22 8 5 a. −2−4−6246 2 4 6 −2 −4 −6 y x b. −2−4−6246 2 4 6 −2 −4 −6 y x c. −2−4−6246 2 4 6 −2 − 4 −6 y x d. − 2 − 4 − 6246 2 4 6 −2 −4 −6 y x MABS_Chapter 3.indd 193MABS_Chapter 3.indd 19311/10/2021 11:58:05 AM11/10/2021 11:58:05 AM 880 Chapter 12 w Applications of the Derivative Pxx 1 500 7 Set P ′(x) equal to 0, and solve for x. 1 500 70 3500 3500 x x x Thus, a maximum profit occurs if the company produces and sells 3500 calculators, which is only half its production capabilities. The price for each calculator would be p 1 1000 350010650$.. The graph shown illustrates the relationship between profit, revenue, and cost. Note that maximum revenue and maximum profit do not necessarily occur at the same level of production and sales. 12.5 EXERCISES ĭAPPLICATIONS 1. Minimizing inventory costs: An appliance store owner estimates that he will sell 125 vacuum cleaners of a particular model. It costs $12 to store one vacuum cleaner for one year. There is a fixed cost of $30 for each order. Find the lot size and the number of orders per year that will minimize inventory costs. 2. Minimizing inventory costs: A hardware store sells 96 chainsaws per year. It costs $5 to store one chainsaw for one year. There is a fixed reordering cost of $15. Find the lot size and the number of orders per year that will minimize inventory costs. 3. Minimizing inventory costs: An art gallery owner expects to sell 90 copies of a limited-edition print during the next year. It costs $1.50 to store one copy for one year. For each order she places, there is a fixed cost of $7.50, plus $0.50 for each copy. Find the lot size and the number of times the gallery owner should order per year to minimize her inventory costs. 4. Minimizing inventory costs: The owner of Lamps-4-U expects to sell 180 brass lamps during the year. For each order he places, there is a fixed cost of $18, plus $2 for each lamp ordered. It costs $5 to store one lamp for one year. In what lot size and how many times per year should he reorder to minimize the inventory costs? 5. Minimizing inventory costs: A T-shirt company sells 4000 sweatshirts per year. To reorder, there is a fixed cost of $6 plus $0.80 for each sweatshirt. It costs $1.20 to store one sweatshirt for one year. In what lot size and how many times per year should an order be placed to minimize inventory costs? x500010,000 y 10,000 20,000 30,000 Maximum revenue (5000, 25,000) Loss when cost is greater than revenue Maximum profit Profit Loss R(x) = − 1 1000 x2 + 10x C(x) = 5000 + 3x MABS_Chapter 12.indd 880MABS_Chapter 12.indd 88011/10/2021 3:51:07 PM11/10/2021 3:51:07 PM Categorized Exercises Each section concludes with a selection of exercises designed to allow the student to practice skills and master concepts. Many levels of difficulty exist within each exercise set, providing instructors with flexibility in assigning exercises and allowing students to practice elementary skills or stretch themselves, as appropriate. Exercise sets are organized into categories such as Concept Check, Practice, Applications, Writing & Thinking, and Technology. Answer Key The Answer Key in the back of the book contains the answers for odd-numbered exercises. This allows students to check their work to ensure they are accurately applying the methods and skills that they have learned. Chapter 3: Functions and Their Graphs 3.1 EXERCISES 1. a. 3 b. − 11 c. 2a − 5 d. 2a − 6 3. a. 9 b. 4 c. a2 d. aa 222 5. a. 4 b. −8 c. a3 + 4a2 + 2a d. a3 + a2 − 3a+ 2 7. a. 35 b. 4a2 + 16a + 15 c. 4841 22 xxhh d. 12 9. a. 2 b. 7a c. 5x h d. 36 11. a. −5 b. −2 c. 0.25 d. 3 13. 3h 15. 22hhx 17. 224 hhx 19. hhxh 2 2 21. 232h hhx 23. 243 2 hxhh 25. 32233hh xhx 27. 32233hh xhx 29. x ≥ −5 31. x > −10 33. x1 35. 37. x 5 2 39. 41. Is a function 43. Is a function 45. Is a function 47. Not a function 49. Not a function 51. Is a function 3.2 EXERCISES 1. x = 3 3. x = 30, 50 5. (4, 11) 7. (5, 30) 9. a 11. c 13. a 15. a 17. a 19. a. C(x) = 135 + 0.5x b. $385 21. a. P(x) = 4.5x − 135 b. $2115 23. P(x) = 100(4.5x − 135) 25. 0.75 atm 27. 56.03 atm 29. a. R(x) = 6.5x b. C(x) = 1.1x + 378 c. P(x) = 5.4x − 378 d. x = 70 pies 31. a. R(x) = 243x b. C(x) = 73x + 5780 c. P(x) = 170x − 5780 d. x = 34 sets of clubs 33. a. $0.15/pen b. C(x) = 0.15x + 260 c. $260 35. a. Rxxx 3105 2 . b. Pxxx 0520500 2 . 37. a. D(x) = − 0.02x + 400 b. R(x) = −0.02x2 + 400x c. P(x) = −0.01x2 + 150x − 3600 39. a. I = 0.0625P b. $375 41. a. 340 p x b. $10 43. The least integer y such that y ≥ 800 x 45. Cx x x x 06503 0650 153 3 .; ..; for for 47. P(x) = 0.2575x 49. Pxxx 2 72720 51. A(x) = 138x − x2 53. Px x x 576 2 55. Axxx 360 3 2 2 3.3 EXERCISES 1. 0 − 2 − 4 − 62 4 6 0 − 2 −4 −6 2 4 6 y x 3. 0−2−4−62 4 6 0 −2 −4 − 6 2 4 6 y x 5. 0−2−4−62 4 6 0 −2 −4 −6 2 4 6 y x 7. 0−2−4−62 4 6 0 −2 −4 −6 2 4 6 y x 9. 0−2−4−62 4 6 0 −2 − 4 −6 2 4 6 y x 11. 0−2−4−62 4 6 0 −2 −4 − 6 2 4 6 y x 13. 0 − 2 − 4 − 62 4 6 0 − 2 −4 −6 2 4 6 y x AK-7 14.5 w Area under a Curve (with Applications) 995 ĭAPPLICATIONS 19. Profit: The marginal profit for a certain style of sports jacket is given by P ′(x) = 56 − 0.8x dollars per jacket, where x is the number of jackets produced and sold weekly. Find the profit for the first 50 jackets that are produced and sold. (Ignore any fixed costs.) 20. Profit: The marginal profit of an important product is given by Pxe x 10 0 015 06 . . dollars per item, where x is the number of items produced and sold. Find the profit for the first 8 items. (Ignore any fixed costs.) 21. Cost: The marginal cost of a product is given by 15 4 + x dollars per unit, where x is the number of units produced. The current level of production is 100 units weekly. If the level of production is increased to 169 units weekly, find the increase in the total costs. 22. Revenue: The marginal revenue from the sale of x bottles of a wine is given by 84 03..−x dollars per bottle. Find the increase in total revenue if the number of bottles sold is increased from 225 to 350. 23. Wildlife management: The manager of a wildlife preserve has started a management program to control the population of the preserve’s bison herd. It is estimated that the population will continue to grow according to the function Ntt 15 6 1 2 bison per year, where t is the number of years after implementation of the plan and 05t. Find the increase in the population during the first 4 years of the program. 24. Bacterial population: It is estimated that t hours after some particular bacteria are introduced into a culture, the population will be increasing at a rate of Pt t 1200 12 05 1 2 . bacteria per hour. Find the increase in the population during the first 6 hours. wWRITING & THINKING In Exercises 25 and 26, explain the meaning of the shaded region in each graph. 25. 26. x y (marginal cost) a 0 (units produced) C (x) x y (marginal revenue) a0 (units sold) R (x) MABS_Chapter 14.indd 995MABS_Chapter 14.indd 99511/10/2021 4:22:13 PM11/10/2021 4:22:13 PM 6.3 w Determinants and Cramer’s Rule 399 ƗTECHNOLOGY Using a graphing utility, find the determinant of the matrix. 69. 010407 030102 050203 ... ... ... −− −− −− 70. 010301 020201 010405 ... ... ... −−−− −−−− 71. 220317 040201 020316 ... ... ... −− −− −− 72. 310611 125273 014165 ... ... ... −− −− −−−− 73. 132321 173214 151216 −− −− −− 74. 253217 13 1424 162636 −−−− Use a graphing utility and Cramer’s Rule to solve each system of equations. 75. xyz xy xyz 239 34 25517 76. 241 232 1 xy z xyz xy z 77. wx yz wxy wxyz wxyz 6 230 2344 20 MABS_Chapter 6.indd 399MABS_Chapter 6.indd 39911/10/2021 2:04:24 PM11/10/2021 2:04:24 PM MABS_Preface.indd 9MABS_Preface.indd 911/10/2021 5:42:25 PM11/10/2021 5:42:25 PMNext >