Basi c T E C H N I C A L M A T H E M A T I C S wit h C A L C U L U S Allyn J. Washington Richard S. Evans Eleventh Edition ELEVENTH EDITION Basic Technical Mathematics with Calculus Allyn J.Washington Dutchess Community College Richard S. Evans Corning Community College Director, Courseware Portfolio Management: Deirdre Lynch Executive Editor: Jeff Weidenaar Editorial Assistant: Jennifer Snyder Managing Producer: Karen Wernholm Content Producer: Tamela Ambush Producer: Jean Choe Manager, Content Development: Kristina Evans Math Content Developer: Megan M. Burns Field Marketing Manager: Jennifer Crum Product Marketing Manager: Alicia Frankel Marketing Assistant: Hanna Lafferty Senior Author Support/Technology Specialist: Joe Vetere Manager, Rights Management: Gina M. Cheselka Manufacturing Buyer: Carol Melville, LSC Communications Composition: SPi Global Associate Director of Design: Blair Brown Cover Design: Barbara Atkinson Cover Image: Daniel Schoenen/Image Broker/Alamy Stock Photo Copyright © 2018, 2014, 2009 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Rights & Permissions Department, please visit www.pearsoned.com/permissions/. PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the United States and/or other countries. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees or distributors. Library of Congress Cataloging-in-Publication Data Names: Washington, Allyn J. | Evans, Richard (Mathematics teacher) Title: Basic technical mathematics with calculus / Allyn J. Washington, Dutchess Community College, Richard Evans, Corning Community College. Description: 11th edition. | Boston : Pearson, [2018] | Includes indexes. Identifiers: LCCN 2016020426| ISBN 9780134437736 (hardcover) | ISBN 013443773X (hardcover) Subjects: LCSH: Mathematics–Textbooks. | Calculus–Textbooks. Classification: LCC QA37.3 .W38 2018 | DDC 510–dc23 LC record available at https://lccn.loc.gov/2016020426 1 16 Student Edition: ISBN 10: 0-13-443773-X ISBN 13: 978-0-13-443773-6 Contents Preface VII 4.4 The Right Triangle 124 1 Basic Algebraic Operations 1 4.5 Applications of Right Triangles 129 1.1 Numbers 2 Key Formulas and Equations, Review Exercises, 1.2 Fundamental Operations of Algebra 6 and Practice Test 134 1.3 Calculators and Approximate Numbers 12 1.4 Exponents and Unit Conversions 17 5 Systems of Linear Equations; 1.5 Scientific Notation 24 Determinants 140 1.6 Roots and Radicals 27 1.7 Addition and Subtraction of Algebraic 5.1 Linear Equations and Graphs of Linear Expressions 29 Functions 141 1.8 Multiplication of Algebraic Expressions 33 5.2 Systems of Equations and Graphical Solutions 147 1.9 Division of Algebraic Expressions 36 5.3 Solving Systems of Two Linear Equations 1.10 Solving Equations 39 in Two Unknowns Algebraically 152 1.11 Formulas and Literal Equations 43 5.4 Solving Systems of Two Linear Equations 1.12 Applied Word Problems 46 in Two Unknowns by Determinants 159 Key Formulas and Equations, Review Exercises, 5.5 Solving Systems of Three Linear Equations in and Practice Test 50 Three Unknowns Algebraically 164 5.6 Solving Systems of Three Linear Equations in 2 Geometry 54 Three Unknowns by Determinants 169 2.1 Lines and Angles 55 Key Formulas and Equations, Review Exercises, and Practice Test 174 2.2 Triangles 58 2.3 Quadrilaterals 65 2.4 Circles 68 6 Factoring and Fractions 180 2.5 Measurement of Irregular Areas 72 6.1 Factoring: Greatest Common Factor and 2.6 Solid Geometric Figures 76 Difference of Squares 181 Key Formulas and Equations, Review Exercises, 6.2 Factoring Trinomials 186 and Practice Test 80 6.3 The Sum and Difference of Cubes 193 6.4 Equivalent Fractions 195 3 Functions and Graphs 85 6.5 Multiplication and Division of Fractions 200 6.6 Addition and Subtraction of Fractions 204 3.1 Introduction to Functions 86 6.7 Equations Involving Fractions 210 3.2 More about Functions 89 Key Formulas and Equations, Review Exercises, 3.3 Rectangular Coordinates 94 and Practice Test 215 3.4 The Graph of a Function 96 3.5 Graphs on the Graphing Calculator 102 3.6 Graphs of Functions Defined by 7 Quadratic Equations 219 Tables of Data 107 7.1 Quadratic Equations; Solution by Factoring 220 Review Exercises and Practice Test 110 7.2 Completing the Square 225 7.3 The Quadratic Formula 227 4 The Trigonometric Functions 113 7.4 The Graph of the Quadratic Function 232 Key Formulas and Equations, Review Exercises, 4.1 Angles 114 and Practice Test 237 4.2 Defining the Trigonometric Functions 117 4.3 Values of the Trigonometric Functions 120 iii iv ConTEnTs 12.4 Polar Form of a Complex Number 354 8 Trigonometric Functions 12.5 Exponential Form of a Complex Number 356 of Any Angle 240 12.6 Products, Quotients, Powers, and Roots of 8.1 Signs of the Trigonometric Functions 241 Complex Numbers 358 8.2 Trigonometric Functions of Any Angle 243 12.7 An Application to Alternating-current (ac) 8.3 Radians 249 Circuits 364 8.4 Applications of Radian Measure 253 Key Formulas and Equations, Review Exercises, Key Formulas and Equations, Review Exercises, and Practice Test 370 and Practice Test 259 13 Exponential and Logarithmic 9 Vectors and Oblique Triangles 263 Functions 373 9.1 Introduction to Vectors 264 13.1 Exponential Functions 374 9.2 Components of Vectors 268 13.2 Logarithmic Functions 376 9.3 Vector Addition by Components 272 13.3 Properties of Logarithms 380 9.4 Applications of Vectors 277 13.4 Logarithms to the Base 10 385 9.5 Oblique Triangles, the Law of Sines 283 13.5 Natural Logarithms 388 9.6 The Law of Cosines 290 13.6 Exponential and Logarithmic Equations 391 Key Formulas and Equations, Review Exercises, 13.7 Graphs on Logarithmic and and Practice Test 295 Semilogarithmic Paper 395 Key Formulas and Equations, Review Exercises, and Practice Test 400 10 Graphs of the Trigonometric Functions 299 14 Additional Types of Equations 10.1 Graphs of y 5 a sin x and y 5 a cos x 300 and Systems of Equations 403 10.2 Graphs of y 5 a sin bx and y 5 a cos bx 303 10.3 Graphs of y 5 a sin (bx 1 c) and 14.1 Graphical Solution of Systems of Equations 404 y 5 a cos (bx 1 c) 306 14.2 Algebraic Solution of Systems of Equations 407 10.4 Graphs of y 5 tan x, y 5 cot x, y 5 sec x, 14.3 Equations in Quadratic Form 411 y 5 csc x 310 14.4 Equations with Radicals 414 10.5 Applications of the Trigonometric Graphs 312 Review Exercises and Practice Test 418 10.6 Composite Trigonometric Curves 315 Key Formulas and Equations, Review Exercises, 15 Equations of Higher Degree 420 and Practice Test 320 15.1 The Remainder and Factor Theorems; Synthetic Division 421 11 Exponents and Radicals 323 15.2 The Roots of an Equation 426 11.1 Simplifying Expressions with Integer 15.3 Rational and Irrational Roots 431 Exponents 324 Key Formulas and Equations, Review Exercises, 11.2 Fractional Exponents 328 and Practice Test 436 11.3 Simplest Radical Form 332 11.4 Addition and Subtraction of Radicals 336 16 Matrices; Systems of Linear 11.5 Multiplication and Division of Radicals 338 Equations 439 Key Formulas and Equations, Review Exercises, and Practice Test 342 16.1 Matrices: Definitions and Basic Operations 440 16.2 Multiplication of Matrices 444 16.3 Finding the Inverse of a Matrix 449 12 Complex Numbers 345 16.4 Matrices and Linear Equations 453 12.1 Basic Definitions 346 16.5 Gaussian Elimination 457 12.2 Basic Operations with Complex Numbers 349 16.6 Higher-order Determinants 461 12.3 Graphical Representation of Complex Key Formulas and Equations, Review Exercises, Numbers 352 and Practice Test 466 ConTEnTs v 21.10 Polar Coordinates 609 17 Inequalities 470 21.11 Curves in Polar Coordinates 612 17.1 Properties of Inequalities 471 Key Formulas and Equations, Review Exercises, 17.2 Solving Linear Inequalities 475 and Practice Test 616 17.3 Solving Nonlinear Inequalities 480 17.4 Inequalities Involving Absolute Values 486 22 Introduction to Statistics 621 17.5 Graphical Solution of Inequalities with Two Variables 489 22.1 Graphical Displays of Data 622 17.6 Linear Programming 492 22.2 Measures of Central Tendency 626 22.3 Standard Deviation 630 Key Formulas and Equations, Review Exercises, and Practice Test 496 22.4 Normal Distributions 633 22.5 Statistical Process Control 637 22.6 Linear Regression 642 18 Variation 499 22.7 Nonlinear Regression 647 18.1 Ratio and Proportion 500 Key Formulas and Equations, Review Exercises, 18.2 Variation 504 and Practice Test 650 Key Formulas and Equations, Review Exercises, and Practice Test 510 23 The Derivative 655 23.1 Limits 656 19 Sequences and the Binomial 23.2 The Slope of a Tangent to a Curve 664 Theorem 514 23.3 The Derivative 667 19.1 Arithmetic Sequences 515 23.4 The Derivative as an Instantaneous Rate of 19.2 Geometric Sequences 519 Change 671 19.3 Infinite Geometric Series 522 23.5 Derivatives of Polynomials 675 19.4 The Binomial Theorem 526 23.6 Derivatives of Products and Quotients of Key Formulas and Equations, Review Exercises, Functions 680 and Practice Test 531 23.7 The Derivative of a Power of a Function 684 23.8 Differentiation of Implicit Functions 690 23.9 Higher Derivatives 693 20 Additional Topics in Trigonometry 535 Key Formulas and Equations, Review Exercises, 20.1 Fundamental Trigonometric Identities 536 Practice Test 696 20.2 The Sum and Difference Formulas 542 20.3 Double-Angle Formulas 547 24 Applications of the Derivative 700 20.4 Half-Angle Formulas 551 20.5 Solving Trigonometric Equations 554 24.1 Tangents and Normals 701 20.6 The Inverse Trigonometric Functions 558 24.2 Newton’s Method for Solving Equations 703 Key Formulas and Equations, Review Exercises, 24.3 Curvilinear Motion 706 and Practice Test 564 24.4 Related Rates 711 24.5 Using Derivatives in Curve Sketching 715 24.6 More on Curve Sketching 721 21 Plane Analytic Geometry 568 24.7 Applied Maximum and Minimum Problems 726 21.1 Basic Definitions 569 24.8 Differentials and Linear Approximations 733 21.2 The Straight Line 573 Key Formulas and Equations, Review Exercises, 21.3 The Circle 579 Practice Test 737 21.4 The Parabola 584 21.5 The Ellipse 588 25 Integration 742 21.6 The Hyperbola 593 21.7 Translation of Axes 599 25.1 Antiderivatives 743 21.8 The Second-degree Equation 602 25.2 The Indefinite Integral 745 21.9 Rotation of Axes 605 25.3 The Area Under a Curve 750 vi ConTEnTs 25.4 The Definite Integral 755 29 Partial Derivatives and Double 25.5 Numerical Integration: Integrals 884 The Trapezoidal Rule 758 25.6 Simpson's Rule 761 29.1 Functions of Two Variables 885 Key Formulas and Equations, Review Exercises, 29.2 Curves and Surfaces in Three Dimensions 888 Practice Test 765 29.3 Partial Derivatives 894 29.4 Double Integrals 898 26 Applications of Integration 768 Key Formulas and Equations, Review Exercises, Practice Test 902 26.1 Applications of the Indefinite Integral 769 26.2 Areas by Integration 773 30 Expansion of Functions in Series 904 26.3 Volumes by Integration 779 26.4 Centroids 784 30.1 Infinite Series 905 26.5 Moments of Inertia 790 30.2 Maclaurin Series 909 26.6 Other Applications 795 30.3 Operations with Series 913 Key Formulas and Equations, Review Exercises, 30.4 Computations by Use of Series Expansions 917 Practice Test 800 30.5 Taylor Series 920 30.6 Introduction to Fourier Series 923 30.7 More About Fourier Series 928 27 Differentiation of Transcendental Functions 805 Key Formulas and Equations, Review Exercises, Practice Test 933 27.1 Derivatives of the Sine and Cosine Functions 806 31 Differential Equations 937 27.2 Derivatives of the Other Trigonometric Functions 810 31.1 Solutions of Differential Equations 938 27.3 Derivatives of the Inverse Trigonometric 31.2 Separation of Variables 940 Functions 813 31.3 Integrating Combinations 943 27.4 Applications 816 31.4 The Linear Differential Equation 27.5 Derivative of the Logarithmic Function 821 of the First Order 946 27.6 Derivative of the Exponential Function 825 31.5 Numerical Solutions of First-order 27.7 L’Hospital’s Rule 828 Equations 948 27.8 Applications 832 31.6 Elementary Applications 951 Key Formulas and Equations, Review Exercises, 31.7 Higher-order Homogeneous Equations 957 Practice Test 835 31.8 Auxiliary Equation with Repeated or Complex Roots 961 31.9 Solutions of Nonhomogeneous Equations 964 28 Methods of Integration 840 31.10 Applications of Higher-order Equations 969 28.1 The Power Rule for Integration 841 31.11 Laplace Transforms 976 28.2 The Basic Logarithmic Form 843 31.12 Solving Differential Equations by Laplace 28.3 The Exponential Form 847 Transforms 981 28.4 Basic Trigonometric Forms 850 Key Formulas and Equations, Review Exercises, 28.5 Other Trigonometric Forms 854 Practice Test 985 28.6 Inverse Trigonometric Forms 858 28.7 Integration by Parts 862 Appendix A Solving Word Problems A.1 28.8 Integration by Trigonometric Substitution 866 Appendix B Units of Measurement A.2 28.9 Integration by Partial Fractions: Appendix C Newton’s Method A.4 Nonrepeated Linear Factors 869 Appendix D A Table of Integrals A.5 28.10 Integration by Partial Fractions: Photo Credits A.8 Other Cases 872 Answers to Odd-Numbered Exercises 28.11 Integration by Use of Tables 877 and Chapter Review Exercises B.1 Key Formulas and Equations, Review Exercises, Solutions to Practice Test Problems C.1 Practice Test 880 Index of Applications D.1 Index E.1 Preface scope of the Book Basic Technical Mathematics with Calculus, Eleventh Edition, is intended primarily for students in technical and pre-engineering technical programs or other programs for which coverage of mathematics is required. Chapters 1 through 20 provide the necessary background for further study with an integrated treatment of algebra and trigonometry. Chapter 21 covers the basic topics of analytic geometry, and Chapter 22 gives an intro- duction to statistics. Chapters 23 through 31 cover fundamental concepts of calculus including limits, derivatives, integrals, series representation of functions, and differential equations. In the examples and exercises, numerous applications from the various fields of technology are included, primarily to indicate where and how mathematical tech- niques are used. However, it is not necessary that the student have a specific knowledge of the technical area from which any given problem is taken. Most students using this text will have a background that includes some algebra and geometry. However, the mate- rial is presented in adequate detail for those who may need more study in these areas. The material presented here is sufficient for two to three semesters. One of the principal reasons for the arrangement of topics in this text is to present material in an order that allows a student to take courses concurrently in allied technical areas, such as physics and electricity. These allied courses normally require a student to know certain mathematics topics by certain definite times; yet the traditional order of topics in mathematics courses makes it difficult to attain this coverage without loss of continuity. However, the material in this book can be rearranged to fit any appropriate sequence of topics. The approach used in this text is not unduly rigorous mathematically, although all appropriate terms and concepts are introduced as needed and given an intuitive or algebraic foundation. The aim is to help the student develop an understanding of mathematical methods without simply providing a collection of formulas. The text material is developed recognizing that it is essential for the student to have a sound background in algebra and trigonometry in order to understand and succeed in any subsequent work in mathematics. new to This Edition You may have noticed something new on the cover of this book. Another author! Yes, after 50 years as a “solo act,” Allyn Washington has a partner. New co-author Rich Evans is a veteran faculty member at Corning Community College (NY) and has brought a wealth of positive contributions to the book and accompanying MyMathLab course. The new features of the eleventh edition include: • Refreshed design – The book has been redesigned in full color to help students better use it and to help motivate students as they put in the hard work to learn the mathematics (because let’s face it—a more modern looking book has more appeal). CAUTION When you enter URLs for the • Graphing calculator – We have replaced the older TI-84 screens with those from the Graphing Calculator Manual, take care to new TI-84 Plus-C (the color version). And Benjamin Rushing [Northwestern State distinguish the following characters: University] has added graphing calculator help for students, accessible online via l = lowercase l short URLs in the margins. If you’d like to see the complete listing of entries for the I = uppercase I online graphing calculator manual, use the URL goo.gl/eAUgW3. • Applications – The text features a wealth of new applications in the examples 1 = one and exercises (over 200 in all!). Here is a sampling of the contexts for these new O = uppercase O applications: 0 = zero ■ Power of a wind turbine (Section 3.4) Height of One World Trade Center (Section 4.4) GPS satellite velocity (Section 8.4) Google’s self-driving car laser distance (Section 9.6) Phase angle for current/voltage lead and lag (Section 10.3) Growth of computer processor transistor counts (Section 13.7) vii viii PREFaCE Bezier curve roof design (Section 15.3) Cardioid microphone polar pattern (Section 21.7) Social networks usage (Section 22.1) Video game system market share (Section 22.1) Bluetooth headphone maximum revenue (Section 24.7) Saddledome roof slopes (Section 29.3) Weight loss differential equation (Section 31.6) • Exercises – There are over 1000 new and updated exercises in the new edition. In creating new exercises, the authors analyzed aggregated student usage and perfor- mance data from MyMathLab for the previous edition of this text. The results of this analysis helped improve the quality and quantity of exercises that matter the most to instructors and students. There are a total of 14,000 exercises and 1400 examples in the eleventh edition. • Chapter Endmatter – The exercises formerly called “Quick Chapter Review” are now labeled “Concept Check Exercises” (to better communicate their function within the chapter endmatter). • MyMathLab – Features of the MyMathLab course for the new edition include: Hundreds of new assignable algorithmic exercises help you address the homework needs of students. Additionally, all exercises are in the new HTML5 player, so they are accessible via mobile devices. 223 new instructional videos (to augment the existing 203 videos) provide help for students as they do homework. These videos were created by Sue Glascoe (Mesa Community College) and Benjamin Rushing (Northwestern State University). A new Graphing Calculator Manual, created specifically for this text, features instructions for the TI-84 and TI-89 family of calculators. New PowerPoint® files feature animations that are designed to help you better teach key concepts. Study skills modules help students with the life skills (e.g., time management) that can make the difference between passing and failing. Content updates for the eleventh edition were informed by the extensive reviews of the text completed for this revision. These include: • Unit analysis, including operations with units and unit conversions, has been moved from Appendix B to Section 1.4. Appendix B has been streamlined, but still contains the essential reference materials on units. • In Section 1.3, more specific instructions have been provided for rounding combined operations with approximate numbers. • Engineering notation has been added to Section 1.5. • Finding the domain and range of a function graphically has been added to Section 3.4. • The terms input, output, piecewise defined functions, and practical domain and range have been added to Chapter 3. • In response to reviewer feedback, the beginning of Chapter 5 has been reorganized so that systems of equations has a strong introduction in Section 5.2. The prerequisite material needed for systems of equations (linear equations and graphs of linear func- tions) has been consolidated into Section 5.1. An example involving linear regression has also been added to Section 5.1. • Solving systems using reduced row echelon form (rref) on a calculator has been added to Chapter 5. • Several reviewers made the excellent suggestion to strengthen the focus on factor- ing in Chapter 6 by taking the contents of 6.1 (Special Products) and spreading it throughout the chapter. This change has been implemented. The terminology greatest common factor (GCF) has also been added to this chapter. PREFaCE ix • In Chapter 7, the square root property is explicitly stated and illustrated. • In Chapter 8, the unit circle definition of the trigonometric functions has been added. • In Chapter 9, more emphasis had been given to solving equilibrium problems, includ- ing those that have more than one unknown. • In Chapter 10, an example was added to show how the phase angle can be interpreted, and how it is different from the phase shift. • In Chapter 16, the terminology row echelon form is used. Also, solving a system using rref is again illustrated. The material on using properties to evaluate determinants was deleted. • The terminology binomial coefficients was added to Chapter 19. • Chapter 22 (Introduction to Statistics) has undergone significant changes. Section 22.1 now discusses common graphs used for both qualitative data (bar graphs and pie charts) and quantitative data (histograms, stem-and-leaf plots, and time series plots). In Section 22.2, what was previously called the arithmetic mean is now referred to as simply the mean. The empirical rule had been added to Section 22.4. The sampling distribution of x has been formalized including the statement of the central limit theorem. A discussion of interpolation and extrapolation has been added in the context of regression, as well as information on how to interpret the values of r and r2. The emphasis of Section 22.7 on nonlinear regression has been changed. Informa- tion on how to choose an appropriate type of model depending on the shape of the data has been added. However, a calculator is now used to obtain the actual regression equation. • In Chapter 23, the terminology direct substitution has been introduced in the context of limits. • Throughout the calculus chapters, many of the differentiation and integration rules have been given names so they can be easily referred to. These include, the constant rule, power rule, constant multiple rule, product rule, quotient rule, general power rule, power rule for integration, etc. • In Chapter 30, the proof of the Fourier coefficients has been moved online. Continuing Features PagE LayouT Special attention has been given to the page layout. We specifically tried to avoid break- ing examples or important discussions across pages. Also, all figures are shown imme- diately adjacent to the material in which they are discussed. Finally, we tried to avoid referring to equations or formulas by number when the referent is not on the same page spread. ChaPTER inTRoduCTions Each chapter introduction illustrates specific examples of how the development of tech- nology has been related to the development of mathematics. In these introductions, it is shown that these past discoveries in technology led to some of the methods in mathemat- ics, whereas in other cases mathematical topics already known were later very useful in bringing about advances in technology. Also, each chapter introduction contains a photo that refers to an example that is presented within that chapter.