www.EngineeringBooksPDF.com Basic Technical Mathematics with Calculus www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com TENTH EDITION Basic Technical Mathematics with Calculus Allyn J Washington Dutchess Community College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.EngineeringBooksPDF.com EDITORIAL DIRECTOR: Vernon R Anthony ACQUISITIONS EDITOR: Sara Eilert EDITORIAL ASSISTANT: Doug Greive DIRECTOR OF MARKETING: David Gesell MARKETING MANAGER: Stacey Martinez SENIOR MARKETING COORDINATOR: Alicia Wozniak SENIOR MARKETING ASSISTANT: Les Roberts SENIOR MANAGING EDITOR: JoEllen Gohr SENIOR PROJECT MANAGER: Rex Davidson SENIOR OPERATIONS SUPERVISOR: Pat Tonneman CREATIVE DIRECTOR: Andrea Nix ART DIRECTOR: Diane Y Ernsberger COVER DESIGNER: Candace Rowley COVER IMAGE: Gencho Petkov/Shutterstock.com MEDIA PROJECT MANAGER: Christina Maestri FULL-SERVICE PROJECT MANAGEMENT: Nancy Kincade/ PreMediaGlobal COMPOSITION: PreMediaGlobal USA Inc PRINTER/BINDER: Edwards Brothers Malloy State COVER PRINTER: Lehigh/Phoenix Color TEXT FONT: Times Roman Copyright © 2014, 2009, and 2005 by Pearson Education, Inc All rights reserved Manufactured 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 likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290 Library of Congress Cataloging-in-Publication Data Washington, Allyn J Basic technical mathematics with calculus / Allyn J Washington, Dutchess Community College — Tenth edition pages cm Includes indexes ISBN-13: 978-0-13-311653-3 (hardcover) ISBN-10: 0-13-311653-0 (hardcover) Mathematics Calculus I Title QA37.3.W38 2014 510–dc23 2012039828 10 ISBN 10: 0-13-311653-0 ISBN 13: 978-0-13-311653-3 www.EngineeringBooksPDF.com Contents Basic Algebraic Operations 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Numbers Fundamental Operations of Algebra Calculators and Approximate Numbers Exponents Scientific Notation Roots and Radicals Addition and Subtraction of Algebraic Expressions 1.8 Multiplication of Algebraic Expressions 1.9 Division of Algebraic Expressions 1.10 Solving Equations 1.11 Formulas and Literal Equations 1.12 Applied Word Problems 11 16 21 24 Equations Quick Chapter Review Review Exercises Practice Test 45 46 46 48 Geometry 2.1 2.2 2.3 2.4 2.5 2.6 Lines and Angles Triangles Quadrilaterals Circles Measurement of Irregular Areas Solid Geometric Figures Equations Quick Chapter Review Review Exercises Practice Test Functions and Graphs 3.1 3.2 Introduction to Functions More about Functions 26 30 32 35 38 41 49 50 53 60 63 67 71 75 76 76 79 80 81 84 3.3 3.4 3.5 3.6 Rectangular Coordinates The Graph of a Function Graphs on the Graphing Calculator Graphs of Functions Defined by Tables of Data Quick Chapter Review Review Exercises Practice Test The Trigonometric Functions 4.1 4.2 4.3 4.4 4.5 Angles Defining the Trigonometric Functions Values of the Trigonometric Functions The Right Triangle Applications of Right Triangles Equations Quick Chapter Review Review Exercises Practice Test 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Systems of Linear Equations; Determinants Linear Equations Graphs of Linear Functions Solving Systems of Two Linear Equations in Two Unknowns Graphically Solving Systems of Two Linear Equations in Two Unknowns Algebraically Solving Systems of Two Linear Equations in Two Unknowns by Determinants Solving Systems of Three Linear Equations in Three Unknowns Algebraically Solving Systems of Three Linear Equations in Three Unknowns by Determinants 89 91 96 101 105 105 107 108 109 112 115 119 124 129 130 130 134 135 136 139 143 147 154 159 164 v www.EngineeringBooksPDF.com vi Contents Equations Quick chapter Review Review Exercises Practice Test 169 170 171 174 Factoring and Fractions 6.1 6.2 Special Products Factoring: Common Factor and Difference of Squares Factoring Trinomials The Sum and Difference of Cubes Equivalent Fractions Multiplication and Division of Fractions Addition and Subtraction of Fractions Equations Involving Fractions 6.3 6.4 6.5 6.6 6.7 6.8 175 Equations Quick Chapter Review Review Exercises Practice Test Quadratic Equations 7.1 Quadratic Equations; Solution by Factoring Completing the Square The Quadratic Formula The Graph of the Quadratic Function 7.2 7.3 7.4 8.1 8.2 8.3 8.4 179 184 190 191 196 200 206 214 215 220 222 226 231 231 231 233 Trigonometric Functions of Any Angle 234 Signs of the Trigonometric Functions Trigonometric Functions of Any Angle Radians Applications of Radian Measure Equations Quick Chapter Review Review Exercises Practice Test 235 237 243 247 253 254 254 256 Vectors and Oblique Triangles 9.1 9.2 9.3 9.4 Introduction to Vectors Components of Vectors Vector Addition by Components Applications of Vectors Oblique Triangles, the Law of Sines The Law of Cosines Equations Quick Chapter Review Review Exercises Practice Test 276 283 287 288 288 290 176 211 211 211 213 Equations Quick Chapter Review Review Exercises Practice Test 9.5 9.6 257 258 262 266 271 10 Graphs of the Trigonometric Functions 291 10.1 Graphs of y = a sin x and y = a cos x 10.2 Graphs of y = a sin bx and y = a cos bx 10.3 Graphs of y = a sin1bx + c2 and y = a cos1bx + c2 10.4 Graphs of y = tan x, y = cot x, y = sec x, y = csc x 10.5 Applications of the Trigonometric Graphs 10.6 Composite Trigonometric Curves 292 Equations Quick Chapter Review Review Exercises Practice Test 312 312 312 314 11 Exponents and Radicals 295 298 302 304 307 315 11.1 Simplifying Expressions with Integral Exponents 11.2 Fractional Exponents 11.3 Simplest Radical Form 11.4 Addition and Subtraction of Radicals 11.5 Multiplication and Division of Radicals 316 320 324 328 330 Equations Quick Chapter Review Review Exercises Practice Test 334 334 334 336 12 Complex Numbers 12.1 Basic Definitions 12.2 Basic Operations with Complex Numbers 12.3 Graphical Representation of Complex Numbers 12.4 Polar Form of a Complex Number 12.5 Exponential Form of a Complex Number 12.6 Products, Quotients, Powers, and Roots of Complex Numbers 12.7 An Application to Alternating-current (ac) Circuits Equations Quick Chapter Review www.EngineeringBooksPDF.com 337 338 341 344 346 348 351 357 363 364 Contents Review Exercises Practice Test 364 365 13 Exponential and Logarithmic Functions 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Exponential Functions Logarithmic Functions Properties of Logarithms Logarithms to the Base 10 Natural Logarithms Exponential and Logarithmic Equations Graphs on Logarithmic and Semilogarithmic Paper Equations Quick Chapter Review Review Exercises Practice Test 14 Additional Types of Equations and Systems of Equations 14.1 Graphical Solution of Systems of Equations 14.2 Algebraic Solution of Systems of Equations 14.3 Equations in Quadratic Form 14.4 Equations with Radicals 367 369 373 378 381 384 388 392 392 392 394 Equations Quick Chapter Review Review Exercises Practice Test 491 491 491 493 395 18 Variation 484 487 494 18.1 Ratio and Proportion 18.2 Variation 495 499 399 403 406 Equations Quick Chapter Review Review Exercises Practice Test 505 505 505 508 410 410 411 19 Sequences and the Binomial Theorem 15.1 The Remainder and Factor Theorems; Synthetic Division 15.2 The Roots of an Equation 15.3 Rational and Irrational Roots 413 418 423 Equations Quick Chapter Review Review Exercises Practice Test 429 429 429 430 Matrices: Definitions and Basic Operations Multiplication of Matrices Finding the Inverse of a Matrix Matrices and Linear Equations Gaussian Elimination Higher-order Determinants 17.1 17.2 17.3 17.4 17.5 464 465 469 474 481 412 16.1 16.2 16.3 16.4 16.5 16.6 17 Inequalities Properties of Inequalities Solving Linear Inequalities Solving Nonlinear Inequalities Inequalities Involving Absolute Values Graphical Solution of Inequalities with Two Variables 17.6 Linear Programming 15 Equations of Higher Degree 16 Matrices; Systems of Linear Equations 460 460 461 463 366 396 Quick Chapter Review Review Exercises Practice Test Equations Quick Chapter Review Review Exercises Practice Test vii 431 432 436 441 446 450 454 19.1 19.2 19.3 19.4 Arithmetic Sequences Geometric Sequences Infinite Geometric Series The Binomial Theorem Equations Quick Chapter Review Review Exercises Practice Test 20 Additional Topics in Trigonometry 20.1 20.2 20.3 20.4 20.5 20.6 Fundamental Trigonometric Identities The Sum and Difference Formulas Double-Angle Formulas Half-Angle Formulas Solving Trigonometric Equations The Inverse Trigonometric Functions Equations Quick Chapter Review www.EngineeringBooksPDF.com 509 510 515 519 522 527 527 528 530 531 532 538 543 547 550 554 560 561 viii Contents Review Exercises Practice Test 561 563 21 Plane Analytic Geometry 564 21.1 Basic Definitions 21.2 The Straight Line 21.3 The Circle 21.4 The Parabola 21.5 The Ellipse 21.6 The Hyperbola 21.7 Translation of Axes 21.8 The Second-degree Equation 21.9 Rotation of Axes 21.10 Polar Coordinates 21.11 Curves in Polar Coordinates 565 569 575 580 584 589 595 598 601 605 609 Equations Quick Chapter Review Review Exercises Practice Test 612 613 614 617 22 Introduction to Statistics 22.1 22.2 22.3 22.4 22.5 22.6 22.7 618 Frequency Distributions Measures of Central Tendency Standard Deviation Normal Distributions Statistical Process Control Linear Regression Nonlinear Regression 619 623 627 631 637 642 647 Equations Quick Chapter Review Review Exercises Practice Test 650 651 651 654 23 The Derivative 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 655 Limits The Slope of a Tangent to a Curve The Derivative The Derivative as an Instantaneous Rate of Change Derivatives of Polynomials Derivatives of Products and Quotients of Functions The Derivative of a Power of a Function Differentiation of Implicit Functions Higher Derivatives Equations Quick Chapter Review Review Exercises Practice Test 656 664 667 671 675 680 684 690 693 696 697 697 699 24 Applications of the Derivative 700 24.1 24.2 24.3 24.4 24.5 24.6 24.7 Tangents and Normals Newton’s Method for Solving Equations Curvilinear Motion Related Rates Using Derivatives in Curve Sketching More on Curve Sketching Applied Maximum and Minimum Problems 24.8 Differentials and Linear Approximations 701 703 707 711 715 721 Equations Quick Chapter Review Review Exercises Practice Test 737 737 737 741 25 Integration 25.1 25.2 25.3 25.4 25.5 25.6 726 733 742 Antiderivatives The Indefinite Integral The Area Under a Curve The Definite Integral Numerical Integration: The Trapezoidal Rule Simpson’s Rule 743 745 750 755 758 761 Equations Quick Chapter Review Review Exercises Practice Test 26 Applications of Integration 26.1 26.2 26.3 26.4 26.5 26.6 765 765 765 767 768 Applications of the Indefinite Integral Areas by Integration Volumes by Integration Centroids Moments of Inertia Other Applications 769 773 779 784 790 795 Equations Quick Chapter Review Review Exercises Practice Test 27 Differentiation of Transcendental Functions 800 802 802 804 805 27.1 Derivatives of the Sine and Cosine Functions 27.2 Derivatives of the Other Trigonometric Functions 27.3 Derivatives of the Inverse Trigonometric Functions 27.4 Applications 27.5 Derivative of the Logarithmic Function 27.6 Derivative of the Exponential Function www.EngineeringBooksPDF.com 806 810 813 816 821 825 Contents 27.7 L’Hospital’s Rule 27.8 Applications 828 832 Equations Quick Chapter Review Review Exercises Practice Test 835 836 836 839 28 Methods of Integration 840 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8 28.9 The General Power Formula The Basic Logarithmic Form The Exponential Form Basic Trigonometric Forms Other Trigonometric Forms Inverse Trigonometric Forms Integration by Parts Integration by Trigonometric Substitution Integration by Partial Fractions: Nonrepeated Linear Factors 28.10 Integration by Partial Fractions: Other Cases 28.11 Integration by Use of Tables 841 843 847 850 854 858 862 866 Equations Quick Chapter Review Review Exercises Practice Test 880 881 881 883 29 Partial Derivatives and Double Integrals 29.1 29.2 29.3 29.4 Functions of Two Variables Curves and Surfaces in Three Dimensions Partial Derivatives Double Integrals 885 888 894 898 902 902 902 903 30 Expansion of Functions in Series 904 Infinite Series Maclaurin Series Operations with Series Computations by Use of Series Expansions Taylor Series Introduction to Fourier Series More About Fourier Series Equations Quick Chapter Review Review Exercises Practice Test 872 877 884 Equations Quick Chapter Review Review Exercises Practice Test 30.1 30.2 30.3 30.4 30.5 30.6 30.7 869 905 909 913 917 920 923 930 935 936 937 938 31 Differential Equations 31.1 31.2 31.3 31.4 Solutions of Differential Equations Separation of Variables Integrating Combinations The Linear Differential Equation of the First Order 31.5 Numerical Solutions of First-order Equations 31.6 Elementary Applications 31.7 Higher-order Homogeneous Equations 31.8 Auxiliary Equation with Repeated or Complex Roots 31.9 Solutions of Nonhomogeneous Equations 31.10 Applications of Higher-order Equations 31.11 Laplace Transforms 31.12 Solving Differential Equations by Laplace Transforms ix 939 940 942 945 947 950 953 959 963 966 971 978 983 Equations Quick Chapter Review Review Exercises Practice Test 987 988 988 990 APPENDIX A Solving Word Problems A.1 APPENDIX B Units of Measurement; The Metric System A.2 B.1 B.2 Introduction Reductions and Conversions A.2 A.5 APPENDIX C The Graphing Calculator A.8 C.1 C.2 C.3 C.4 A.8 A.8 A.12 A.16 Introduction The Graphing Calculator Graphing Calculator Programs The Advanced Graphing Calculator APPENDIX D Newton’s Method A.24 APPENDIX E A Table of Integrals A.25 Answers to Odd-Numbered Exercises and Quick Chapter Reviews B.1 Solutions to Practice Test Problems C.1 Index of Applications D.1 Index of Writing Exercises D.11 Index D.13 www.EngineeringBooksPDF.com D.10 Index of Applications Ocean temperature, 616(94) Temperature and altitude, 507(54) Temperature change in wall, 573(57) Temperature in blast furnace, 749(57) Temperature readings, 6(43), 11(61), 32(62), 40(4), 41(43), 101(64), 104(1), 142(4), 473(51), 474(56), 518(51), 573(53), 615(86), 773(32), 800(34), 828(52), 945(43), 952(18), 989(75) Thermal expansion, 40(3), 47(96), 48(21), 169(41), 179(71), 212(101), 430(8), 508(3), 650(6), 679(45), 706(32), 847(42), 897(38), 898(41) Thermal resistance, 47(43) Thermodynamics Absolute zero, 647(14) Adiabatic expansion of gas, 391(32), 502(10), 574(71), 689(51), 989(70) Boyle’s law, 498(27), 714(29), 847(44) Energy in metal at lower temperature, 758(53) Entropy, 882(73) Gas-liquid equilibrium, 418(59) Gas molecules, 835(43), 843(39) General gas law, 504(54), 650(3) Isothermal change, 373(71), 895(46, 47) Otto cycle, 320(76) Radiant energy emission, 32(64), 179(68), 184(78), 507(60) www.EngineeringBooksPDF.com Speed of oxygen molecules, 504(51) Thermodynamics, 47(86), 189(62), 191(30, 33), 205(60), 210(43) Thermodynamic temperature, 394(100), 527(54), 847(46), 887(33), 894(41) Vapor pressure, 834(29) Volume expansivity, 903(38) Wastewater Technology Chemical waste holding tank, 804(53) Draining holding tank, 226(6), 503(41), 521(31) Wastewater processing, A.7(45) Wastewater pump, 487(55) INDEX OF WRITING EXERCISES The final exercise in each chapter is a writing exercise Each of these 22 exercises will require at least a paragraph to provide a good explanation of the problem presented Also there are over 410 other exercises throughout the text marked W before the exercise number or instructions that require at least one complete sentence (up to a paragraph) to provide the explanation needed Following is a listing of these writing exercises The: first number shown is the page number, and the numbers in parentheses are the exercise numbers The * denotes the final review exercise of the chapter Chapter 1: Basic Algebraic Operations 5(21, 22, 33) 6(34, 35, 36, 43) 10(52, 53, 54, 55, 56) 11(60) 15(47, 48) 16(52, 58) 21(58, 60) 25(2) 26(53) 30(55, 56, 59, 60) 32(55, 56) 38(44) 47(109, 110) 48(135*) Chapter 2: Geometry 53(47,48) 58(30, 32, 33, 34) 62(25, 26, 33) 63(45) 66(33, 36, 37, 38) 67(55) 70(1, 2, 3, 4) 71(17, 18, 19, 20) 77(43, 44) 79(77*) Chapter 3: Functions and Graphs 84(29, 30, 31, 32, 33, 34, 35, 36, 49) 87(11, 12) 88(43, 52) 90(32) 95(57) 96(61, 62, 65, 66) 101(60, 62, 65, 66) 105(45, 55, 56) 106(57, 60) 107(91*) 119(47, 48, 49, 50) 123(44) 130(54, 56) 133(92, 99*) 327(70) 330(48, 49) 333(62) 336(103*) Chapter 5: Systems of Linear Equations; Determinants Chapter 12: Complex Numbers 153(61, 62) 163(32) 171(41, 42, 43, 44) 172(82) 174(99*) Chapter 6: Factoring and Fractions 179(84) 183(61, 62) 189(76) 191(38, 39, 40) 195(65, 66, 67, 68) 205(58) 213(133*) 341(71,72,74) 343(62, 63, 64) 345(35, 36, 37, 38) 356(52) 362(24) 365(101*) Chapter 13: Exponential and Logarithmic Functions 373(62, 64, 76) 377(57, 58, 59, 65) 383(38) 394(107*) Chapter 7: Quadratic Equations Chapter 14: Additional Types of Equations and Systems of Equations 219(5, 6) 225(44) 230(25, 26, 27, 28, 29, 30) 232(80) 233(85*) 399(36) 405(10) 409(19, 22, 41) 410(43, 44, 56) 411(75*) Chapter 8: Trigonometric Functions of Any Angle Chapter 15: Equations of Higher Degree 242(49, 50, 51, 52) 247(72) 251(18) 253(59) 255(83) 256(95*) Chapter 9: Vectors and Oblique Triangles 261(5, 6, 7, 8) 275(26) 286(22, 24, 27) 287(42) 288(38) 290(73*) Chapter 10: Graphs of the Trigonometric Functions 298(51, 52) 301(35, 36) 302(45, 46, 47, 48) 307(19) 312(39, 40) 314(76, 87*) Chapter 4: The Trigonometric Functions Chapter 11: Exponents and Radicals 115(40) 118(45, 46) 319(58, 60) 323(64, 65) 418(47, 48, 53, 54) 422(27, 28) 428(48) 430(47, 48, 67*) Chapter 16: Matrices; Systems of Linear Equations 435(35) 436(44) 440(17, 18, 38) 445(36, 38) 450(31, 32) 462(68) 463(91*) Chapter 17: Inequalities 468(1) 469(25, 26, 27, 28, 48, 51) 480(45, 46) 483(33, 41) 491(20) 492(57, 58) 493(85*) Cbapter 18: Variation 503(13, 14, 15, 16, 42, 43) 504(44) 508(79*) D.11 www.EngineeringBooksPDF.com D.12 Index of Writing Exercises Chapter 19: Sequences and the Binomial Theorem 514(35, 36, 62) 518(29, 58, 59, 60) 521(30) 522(38) 526(39, 47) 530(97*) Chapter 20: Additional Topics in Trigonometry 538(67, 68) 542(28, 45) 546(53) 559(1, 71, 72) 560(76) 563(125*) Chapter 21: Plane Analytic Geometry 573(39, 40, 49, 50, 57) 574(60, 63) 579(49, 50, 53) 583(44) 588(37, 38) 589(50) 594(46, 48) 600(39, 40, 41, 42) 608(64) 612(55, 56) 617(113*) Chapter 22: Introduction to Statistics 623(31, 32) 627(43, 44, 45, 46) 636(1, 8, 21, 22, 34) 641(3, 4) 650(12) 654(61*) Chapter 23: The Derivative 662(21, 22, 23, 24) 663(68, 69, 70) 671(39) 675(32) 679(40, 42) 682(36) 689(44) 698(49, 50, 72) 699(88, 91*) Chapter 24: Applications of the Derivative 703(19) 706(19, 20, 22, 23) 710(30) 720(33, 34, 35, 36, 37, 38, 41) 721(54) 725(19, 20, 25) 736(38, 42) 739(58) 741(87*) Chapter 25: Integration 744(71) 749(41, 42, 43, 44, 45, 46) 754(24, 25, 26, 27, 28) 757(35, 39) 758(50) 761(17) 764(2) 766(27, 28, 35, 39) 767(63*) Chapter 26 Applications of Integration 773(24) 778(31, 36) 784(27, 28, 32) 794(28) 800(31, 36) 804(59*) Chapter 27: Differentiation of Transcendental Functions 809(39, 40) 810(42) www.EngineeringBooksPDF.com 813(39, 40) 816(35, 36, 40) 824(37, 38, 49, 50, 51) 827(33, 34) 831(45) 834(15, 16) 839(105*) Chapter 28: Methods of Integration 846(32) 850(30) 853(31) 857(41) 861(31, 32, 33, 34, 35) 865(1, 2) 868(1, 2) 882(49, 50, 53) 883(87*) Chapter 29: Partial derivatives and Double Integrals 887(27) 893(33) 897(40) 898(44) 902(28) 903(35, 49, 51*) Cbapter 30: Expansion of Functions in Series 908(40) 912(31, 32) 920(33, 37) 923(39, 40, 41, 42) 938(77*) Chapter 31: Differential Equations 945(15) 947(15) 950(29, 30) 952(1, 5) 966(38, 39) 982(28) 989(79) 990(95*) INDEX Abscissa, 89 Absolute error, 738 Absolute inequality, 465 Absolute value, 3; of complex numbers, 346; in inequalities, 481 Acceleration, 695, 708, 767 Accuracy of number, 12, 117 Addition: of algebraic expressions, 27; of complex numbers, 341; of fractions, 200; of matrices, 433; of ordinates, 308; of radicals, 328; of signed numbers, 7; of vectors, 259, 266 Adjacent angles, 51 Agnesi, Maria, 722, 840 Algebraic expressions, 26 Alternate-exterior angles, 51 Alternate-interior, angles, 51 Alternating current, 306, 357 Alternating series, 911 Altitude, 54 Ambiguous case, 280 Ampere, 360 Amplitude of sine curve, 293 Analytic geometry, 564 Angle, 50, 109; acute, 50; central, 64; of depression, 124; of elevation, 124; inscribed, 64; negative, 109, 242; obtuse, 50; phase, 293; quadrantal, 110, 241; reference, 237; right, 50; of rotation, 602; standard position, 110; straight, 50 Angular velocity, 249 Antiderivative, 743 Antilogarithm, 378 Applied maxima and minima, 726 Approximate numbers, 12, 114 Arc, 64 Arc length, 248 Area: of circular sector, 249; of geometric figures, 55; by integration, 773; lateral, 72; under a curve, 750 Argument of complex number, 346 Aristotle, 80 Arithmetic mean, 624 Arithmetic sequence, 510 Array, 619 Associative law, 6, 434 Asymptote, 93, 303, 590, 721 Attribute, 646 Auxiliary equation of differential equation, 960; complex roots, 964; repeated roots, 963 Average value, 798 Average velocity, 672 Axes, coordinate, 89, 888 Axis: of ellipse, 585; of hyperbola, 590; of parabola, 580; polar, 605 Babbage, Charles, 412 Bar, 8, 28 Barrow, Isaac, 742 Base: of exponents, 16; of logarithms, 369, 378, 381; of solid, 71; of trapezoid, 60; units, A.2 Binomial, 27 Binomial formula, 524 Binomial series, 525, 913 Binomial theorem, 522, 676 Bit, Bode diagram, 391 Braces, 28 Brackets, 28 Briggs, Henry, 366 Byte, Calculator, 11, 96, 116, 239, 367, 443, 472 Calculator operations (complete listing of operations), A.10 Cancellation, 192 Capacitance, 357, 771 Cardioid, 609 Catenary, 387 Cauchy, Augustin-Louis, 395, 662 Cayley, Arthur, 431 Center of mass, 784 Central angle, 64 Central line, 638 Central tendency, 623 Centroid, 54, 784 Chain rule, 684 Character of roots, 224 Checking answers, 199 Chord, 63 Circle, 63, 396, 575 Circumference, 63 Class, 619 Class mark, 620 Coefficient, 27 Cofunction, 120 Column of determinant, 154 Common logarithms, 378 Commutative law, 6, 434 Complementary angles, 50, 119 Complementary solution of differential equation, 966 Completing the square, 220, 228, 577, 596, 981 Complex fraction, 204 Complex number, 2, 339, 915 Complex plane, 344 Complex roots, 419, 426, 964 Components of vector, 262 Composite trigonometric curves, 307 Computer, 464, 892 Concavity, 717 Conditional equation, 35 Conditional inequality, 465 Cone, 71 Congruent triangles, 57 Conic sections, 396, 600 Conjugate: axis of hyperbola, 590; of complex number, 340 Constant, 4: derivative, of, 677; of integration, 745; 748, 943; of proportionality, 499, 500 Constraint, 487 Continuity, 656 Contradiction, 35 Control charts, 637 Convergence, 905, 909 Conversion of units, 20, 109, 244, A.5 Coordinates: cylindrical, 892; polar, 605; rectangular, 89, 888 Correlation, 647 Corresponding angles, 51, 56, 112 Corresponding segments, 51 Cosecant: of angle, 113; derivative of, 811; graph of, 302; integration of, 852 Cosine: of angle, 113; derivative of, 808; of double angle, 543; graph of, 292; half angle, 547; integration of, 851; series, 934; series expansion of, 913; of sum of two angles, 538 Cosines, law of, 283 Cotangent: of angle, 113; derivative of, 811; graph of, 302; integration of, 852 Coterminal angles, 109 Coulomb, 360 Cramer, Gabriel, 135 Cramer’s rule, 155, 165, 457 Critical value, 475, 715 Cube, 71 Cumulative frequency, 621 Current, 357, 770 Curve: fitting, 662; least squares, 644; sketching, 715, 721, 817, 832 Curve in space, 890 Curvilinear motion, 707 Cycle of sine curve, 300 Cylinder, 71 Cylindrical coordinates, 892 d’Alembert, Jean, 257 Damped harmonic motion, 972 Dantzig, George, 464 Decibel, 385 D.13 www.EngineeringBooksPDF.com D.14 Index Decimal, repeating, 2, 520 Decreasing function, 715 Definite integral, 755 Deflection of beams, 975 Degree: of differential equation, 940; as measure of angle, 50, 109; of polynomial, 27 Delta notation, 733 DeMoivre’s theorem, 354 Denominate number, 4, 317, A.2 Denominator: common, 150; rationalizing of, 326, 332 Dependent system of equations, 146, 151, 158 Dependent variable, 81 Derivative: 667, 673; of a constant, 676; of exponential function, 825; of implicit function, 690; of inverse trigonometric functions, 813; of logarithmic function, 821; partial, 894; of a polynomial, 675; of a power, 676, 684; of a product, 680; of a quotient, 681; as a rate of change, 673; with respect to time, 707; of trigonometric functions, 806 Descartes, René, 89, 315, 339, 395, 424, 564 Descartes’ rule of signs, 424 Determinant, 154; higher-order, 454; properties of, 455; second-order, 154; third-order, 164, 454 Deviation, 644; standard, 627 Diagonal, 60 Diagonal of determinant, 154 Diameter, 63 Difference, common, 510 Differential, 733 Differential equation, 940; for an electric current, 955, 974; homogeneous, 959; linear, 947, 959; nonhomogeneous, 959, 966; numerical solutions, 950 Differentiation, 667 Diocles, 214 Direct variation, 499 Directrix of parabola, 580 Displacement, 261, 769; of sine curve, 299 Distance formula, 565 Distributive law, Divergence, 905 Dividend, 33 Division: of algebraic expressions, 32; of complex numbers, 341, 352; of fractions, 196; of radicals, 331; of signed numbers, 8; synthetic, 414; by zero, Divisor, 33 Domain, 84, 293, 668, 721 Double-angle formulas, 543 Double integral, 898 e (irrational number), 348, 381, 821 Eccentricity, 588, 594 Edison, Thomas, 291, 357, 531 Einstein, Albert, 38 Electric circuits, 357, 770, 955, 974 Element: of area, 774; of determinant, 154; of matrix, 432; of volume, 779, 899 Elimination by addition or subtraction, 149, 400 Ellipse, 396, 584, 595, 603 ENIAC, 208 Equations, 35; auxiliary, 960; conditional, 35; differential, 940; exponential, 384; graphical solution, 97, 143; higherdegree, 412; involving fractions, 206; linear, 136; literal, 38; logarithmic, 384; parametric, 309, 611, 707; polar, 606; polynomial, 419; quadratic, 215, 403; quadratic form, 403; with radicals, 406; roots of, 414, 418, 703; second-degree, 598; systems of linear, 137, 143, 154, 159, 451, 458; systems of quadratic, 397, 399; trigonometric, 550 Equilibrium, 272 Estimating, 15 Euclid, 49, 509 Euler, Leonhard, 234, 348, 700, 805, 840, 939, 950 Euler’s method, 950 Even function, 930 Exact number, 12 Exponential complex number form, 348, 915 Exponential equations, 384 Exponential function, 367; derivative of, 825; integration of, 847; series expansion, of, 913 Exponents, 16, 316 Extraneous roots, 208, 404 Extrapolation, 104 Extreme point, 227 Factor, 26, 179, 192, 414; prime, 117 Factor theorem, 414 Factorial notation, 523 Factoring, 179; by grouping, 182, 188 Family of curves, 944 Farad, 360 Feasible point, 487 Fermat, Pierre de, 80, 564, 655, 742 First-derivative test, 716 Focus of ellipse, 584; of hyperbola, 589; of parabola, 580 Formula, 38; binomial, 524; distance, 565; quadratic, 222 Fourier, Jean, 904, 924 Fourier series, 923, 930 Fraction, 3, 191; complex, 204; equivalent, 191; as exponent, 320; partial, 865, 982; simplest form, 192 Frequency, 306; distribution, 619; polygon, 620 Frustum, 72 Function, 81, 555; average value, 798; composite, 684; derivative of, 667; exponential, 367; finding zeros of, 97; implicit, 694; integration of, 745; inverse, 372, 554; limit of, 658; linear, 35; www.EngineeringBooksPDF.com logarithmic, 367; quadratic, 215; trigonometric, 113, 120, 235; of two variables, 885 Fundamental principle of fractions, 191 Fundamental theorem of algebra, 419 Galileo, 80, 214 Galton, Francis, 618 Gauss, Karl, 339, 412, 419, 431, 442, 450, 512 Gaussian elimination, 444 Gauss-Jordan method, 442 General equation: of circle, 577; quadratic, 215; of straight line, 571 General solution of differential equation, 940 Geometric sequence, 515, 907 Geometry, 49 Googol, 23 Graph: of exponential function, 367; of function, 91; of inverse trigonometric functions, 556; of linear equation, 137; of logarithmic function, 371; on logarithmic paper, 388; in polar coordinates, 609; of quadratic function, 226; of trigonometric functions, 292 Graphical representation of complex numbers, 344 Graphical solution of equations, 97, 143, 229, 397 Graphical solution of inequalities, 479, 484 Graunt, John, 618 Great circle, 248 Greater than, 4, 465 Grouping symbols, 8, 28 Half-angle formulas, 547 Half-range expansion, 933 Halley, Edmond, 618 Harmonic sequence, 514 Heaviside, Oliver, 939 Henry, 360 Hero’s formula, 55 Hertz, 311 Hertz, Heinrich, 291, 306, 337 Hexagon, 53 Higher-order derivatives, 693 Hipparchus, 108 Histogram, 620 Homogeneous differential equation, 959 Hooke’s law, 795 Hyperbola, 93, 397, 589, 595, 603 Hypotenuse, 53, 113 Identity, 7, 28, 32, 35; matrix, 437; trigonometric, 532 Imaginary axis, 344 Imaginary number, 2, 25, 338 Impedance, 358 Implicit function, 690; derivative of, 690 Improper integral, 978 Inclination, 566 Index Inconsistent system of equations, 145, 151, 158, 162 Increasing function, 715 Increment, 733 Indefinite integral, 745 Independent variable, 81, 885 Indeterminate, Indeterminate form, 828 Inductance, 357 Inequalities, 4, 465, 715; algebraic solution of, 469; graphical solution of, 479; involving absolute values, 481; properties of, 466; with two variables, 484 Infinite series, 519, 905 Infinity, 519, 660, 721 Inflection, point of, 717 Initial point of vector, 259 Initial side of angle, 109 Instantaneous rate of change, 666, 671 Instantaneous velocity, 672 Integer, Integrating combinations, 945 Integral: definite, 755; double, 898; improper, 978; indefinite, 745 Integration, 745; constant of, 745; of exponential forms, 847; of inverse trigonometric forms, 858; of logarithmic forms, 843; methods of, 840; numerical, 758; by partial fractions, 869; by parts, 862; of powers, 746, 841; by use of series, 914; as a summation process, 753; by tables, 877; of trigonometric forms, 850, 854; by trigonometric substitution, 866 Intercepts, 97, 141, 721 Interpolation, 103 Inverse differentiation, 743 Inverse functions, 372, 554 Inverse Laplace transform, 981 Inverse matrix, 438, 441 Inverse trigonometric functions, 116, 240, 554; derivatives of, 813; integral forms, 858 Inverse variation, 499 Irrational numbers, 2, 520 Irrational roots, 223, 426 Iterative method, 704 Joint variation, 500 j-operator, 360 Jordan, Wilhelm, 442 Joule, 317 Kelvin, 321 Kirchhoff, Gustav, 135 Lagrange, Joseph Louis, 257, 904 Laplace, Pierre, 366, 939, 978 Laplace transforms, 978 Latitude, 248 Law: of cosines, 283; of sines, 277 Least-squares line, 644 Leibniz, Gottfried Wilhelm, 154, 655, 742, 755, 884 Lemniscate, 610 Less than, 4, 465 L’Hospital, Marquis de, 828 Like terms, 27 Limaỗon, 609 Limit, 519, 658; e as a, 821; of sin u>u, 805 Limits of integration, 755 Line, 50 Linear approximation, 733 Linear differential equation, 947, 959 Linear equation, 136; graph of, 137 Linear extrapolation, 104 Linear interpolation, 103 Linear programming, 464, 487 Linear regression, 636 Linear simultaneous systems, 137, 143, 154, 159, 451, 458 Linearization, 735 Lissajous figures, 309 Literal number, Local maximum and minimum points, 716 Locus, 575 Logarithmic equations, 385 Logarithmic function, 369; derivative of, 821; integral form, 843; series expansion of, 913 Logarithmic paper, 388, 574 Logarithms, 373, 381; to base 10, 378; to bases other than 10, 381; computations by, 379; natural, 381; properties of, 373 Longitude, 248 Lower control limit, 638 Lowest common denominator, 150 Maclaurin, Colin, 135, 910 Maclaurin series, 910 Major axis of ellipse, 585 Matrix, 165, 432; elements of, 432; identity, 437; inverse, 438, 441; square, 432, zero, 432 Maximum and minimum problems, 726 Maximum points, 227, 716 Maxwell, James, 337 Mean, arithmetic, 624 Median, 54, 623 Members of inequality, 465 Metric system, 22, A.2 Midrange, 627 Minimum points, 227, 716 Minor, 455 Minor axis of ellipse, 585 Minute (measure of angle), 109 Mode, 109, 310, 611, 625 Modulus of complex number, 346 Mohr’s circle, 579 Moment of inertia, 790; of a mass, 784 Monomial, 27 Multiplication: of algebraic expressions, 30; of complex numbers, 341, 351; of fractions, 196; of matrices, 436; www.EngineeringBooksPDF.com D.15 of radicals, 330; scalar, 434; of signed numbers, Napier, John, 366 Natural logarithm, 381 Negative angle, 109, 242 Negative exponents, 18, 316 Negative numbers, Nested parentheses, 29 Newton, Sir Isaac, 271, 272, 315, 494, 502, 755, 704, 742, 972 Newton, 271 Newton’s method, 703, 817, 837, A.16 Nightingale, Florence, 618 Nonhomogeneous differential equations, 959, 966 Nonlinear inequalities, 474 Nonlinear regression, 647 Normal distribution, 631 Normal line, 701 Number: complex, 2, 339; denominate, 4, A.2; imaginary, 2, 25, 339; irrational, 2, 520; natural, 2; negative, 2; rational 2, 520; real, Numerical integration, 758 Objective function, 487 Oblique triangle, 276, 285 Octant, 888 Odd function, 930 Ogive, 621 Ohm, 357 Operations with zero, Operator, 959 Order: of differential equation, 940; of operations, 8, 19; of radical, 324 Ordinate, 89 Ordinates, addition of, 308 Origin, 3, 89, 888 Orthogonal trajectories, 953 Parabola, 93, 227, 396, 580, 595, 603 Parallel, 50, 567 Parallelogram, 60 Parametric equations, 309, 611, 707 Parentheses, 8, 28; nested, 29 Partial derivative, 894 Partial fractions, 869, 873, 982 Partial sum, 905 Particular solution of differential equation, 940, 944, 966, 978, 983 Pascal, Blaise, 768 Pascal, 317 Pascal’s triangle, 524 Pentagon, 53 Perfect square, 24, 324 Perimeter, 54 Period of sine curve, 295 Perpendicular, 50, 567 Phase of sine curve, 298 Phase shift, 299 Phasor, 360 D.16 Index Pixel, 97 Plane, 50, 162, 889 Point, 50; of inflection, 717 Point-slope form of straight line, 569 Polar axis, 605 Polar coordinates, 605 Polar form of complex number, 346 Pole, 605 Polygon, 53 Polynomial, 27, 423; derivative of, 675 Population, 627 Power, 16; of complex number, 353; derivative of, 676, 684; integration of, 745, 841; series, 909 Precision, 12 Pressure, 797 Prime factor, 179 Principal root, 24 Prism, 71 Product, 8, 26; of complex numbers, 341, 351; derivative of, 680; of matrices, 436; special, 176 Programs, calculator, A.12 Progression, arithmetic, 510; geometric, 515 Projection, 305 Proportion, 37, 495 Pyramid, 71 Pythagoras, 2, 56 Pythagorean theorem, 56, 113, 266 Quadrant, 89 Quadrantal angle, 110, 241 Quadratic equation, 215, 403 Quadratic equation in form, 403 Quadratic formula, 222 Quadrilateral, 53, 60 Quotient, 8, 33; of complex numbers, 341, 352; derivative of, 681 Radian, 65, 109, 243, 292 Radicals, 24, 324; addition of, 328; division of, 331; equations with, 406; multiplication of, 330; simplest form, 324; subtraction of, 328 Radicand, 324 Radioactive decay, 954 Radius, 63, 575; of gyration, 790; vector, 113, 605 Range, 84, 98, 293, 555, 638, 721 Rate of change, 666, 671 Ratio, 37, 495; common, 577 Rational number, 2, 520 Rational roots, 223, 423 Rationalizing: denominator, 326, 332; numerator, 332, 670 Raw data, 619 Ray, 50 Reactance, 357 Real axis, 344 Real numbers, Reciprocal, 4, 117 Rectangle, 60 Rectangular coordinate system, 89, 888 Rectangular form of complex number, 339 Rectangular solid, 71 Rectifier: half-wave, 928; full-wave, 929 Reduction, A.5 Reference angle, 237 Regression, 636 Related rates, 711 Relation, 87, 555 Relative error, 734 Relative frequency, 620 Remainder, 33 Remainder theorem, 413 Repeating decimal, 520 Resistance, 357 Resolution of vector, 262 Resonance, 361 Resultant of vectors, 259 Rhombus, 60 Right triangle, 53, 119 Root-mean-square value, 856 Roots, 24; of complex numbers, 353; double, 216; of equation, 136, 404, 418; extraneous, 208, 404; irrational, 426; of linear equations, 136; of quadratic equations, 216; rational, 423 Rose, 610 Rotation of axes, 601 Rounding off, 12 Row of determinant, 154 Runge-Kutta method, 951 Sample, 627 Scalar, 258; multiplication, 434 Scale drawing, 57 Scientific notation, 21 Secant: of angle, 113; derivative of, 811; graph of, 302; integration of, 852; line, 63, 664 Second (measure of angle), 109 Second-degree equation, 598 Second derivative, 697 Second-derivative test, 717 Section, 890 Sector, 64, 249 Segment, 64 Semilogarithmic paper, 388, 574 Sense of inequality, 465 Separation of variables, 942 Sequence, arithmetic, 510; finite, 510; geometric, 515; infinite, 510, 905 Series, 519, 905; alternating, 911; binomial, 525; computations with, 917; Fourier, 923, 930; Maclaurin, 909; power, 909; Taylor, 920 Shell (element of volume), 780 Shewhart, Walter, 618 Shifing a graph, 99 Signed numbers, www.EngineeringBooksPDF.com Significant digits, 12, 117, 346 Signs: laws of, 7; of trigonometric functions, 235 Similar terms, 27 Simple harmonic motion, 304, 971 Simpson, Thomas, 69, 762, A.16 Simpson’s rule, 69, 761 Sine: of angle, 113; derivative of, 807; of double angle, 563; graph of, 292; of half-angle, 547; integration of, 850; inverse, 554; series, 934; series expansion 913; of sum of two angles, 538 Sines, law of, 277 Slant height, 71 Slope, 139, 566; of tangent line, 664 Slope-intercept form of straight line, 140, 571 Solution: of differential equation, 940; of equation, 36, 47, 136, 703; of inequality, 465, 481; of linear equation, 136; of quadratic equation, 216; of system of linear equations, 137, 143, 154, 159, 446, 451, 457; of triangle, 119, 274; of trigonometric equations, 550 Special products, 176 Sphere, 71 Square, 60 Square root, 24 Square wave, 926 Squares, least, 644 Standard deviation, 628 Standard equation: of circle, 575; of ellipse, 585; of hyperbola, 590; of parabola, 580 Standard errors, 635 Standard normal distribution, 632 Standard position of angle, 110 Statistical process control, 637 Statistics, 618 Steady-state solution, 972 Straight line, 93, 139, 569 Subscripts, 29, 39 Substitution, 19; elimination by, 147, 399; trigonometric, 866 Subtraction, 7; of algebraic expressions, 27; of complex numbers, 341; of fractions, 200; of matrices, 434; of radicals, 328; of signed numbers, 7; of vector, 260 Summation sign, 625, 905 Supplementary angles, 50 Surface, 889 Symbols of grouping, 8, 28 Symmetry, 576, 721 Synthetic division, 414 System: of linear equations, 137, 143, 154, 159, 446, 451, 458; of quadratic equations, 397, 399 Tangent: of angle, 113; derivative of, 810; of double angle, 543; graph of, 302; integration of, 852; line, 63, 664, 701; of sum of two angles, 539 Index Taylor, Brook, 921 Taylor series, 920 Terms, 26, 185; of sequences, 510, 515; similar, 26 Terminal point of vector, 259 Terminal side of angle, 109 Tesla, Nikola, 531 Trace, 98, 890 Transcendental functions, 805; derivatives of, 805 Transient term, 974 Translation of axes, 595 Transversal, 51 Transverse axis of hyperbola, 590 Trapezoid, 68 Trapezoidal rule, 67, 757 Triangle, 53; equilateral, 53; isosceles, 53; oblique, 276, 285; Pascal’s, 524; right, 53, 119; scalene, 53; similar, 56, 112; solution of, 119, 276 Trigonometric equations, 550 Trigonometric form of complex numbers, 346 Trigonometric functions, 113, 120, 235; of angles of right triangle, 120; of any angle, 237; derivatives of, 806; graphs of, 292; integration of, 850, 854; inverse, 116, 240, 554; of negative angles, 242; of small angles, 253 Trinomial, 27, 184 Undetermined coefficients, 967 Units, 20, 358, A.2 Unknown, 35 Upper control limit, 638 U.S Customary system, A.2 Variable, 4, 640; dependent, 81; independent, 81, 885 Variables, separation of, 942 Variation, 499 Vectors, 258, 707 Velocity, 249, 258, 672, 695, 707, 769; angular, 249; average, 672; instantaneous, 672; linear, 249 www.EngineeringBooksPDF.com D.17 Vertex: of angle, 50, 109; of ellipse, 585; of hyperbola, 590; of parabola, 227, 580 Vertical angles, 51 Vertical line test, 94 Viete, Franỗois, 175 Voltage, 357, 770 Volumes: of geometric figures, 71; by integration, 779; under a surface, 899 Wallis, John, 315 Watt, James, 618 Watt, 93 Westinghouse, George, 357, 531 Word problems, 41, 86, 124, 144, 271 Work, 795 x-intercept, 97, 141 y-intercept, 140, 571 Zero, 2; as exponent, 18, 316; of a function, 97, 414; matrix, 432; operations with, Zoom, 98 NOTES www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com .. .Basic Technical Mathematics with Calculus www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com TENTH EDITION Basic Technical Mathematics with Calculus. .. in any subsequent work in mathematics New to This Edition The tenth edition of Basic Technical Mathematics with Calculus includes all the basic features of the earlier editions Specifically, among... D.13 www.EngineeringBooksPDF.com This page intentionally left blank www.EngineeringBooksPDF.com Preface Scope of the Book Basic Technical Mathematics with Calculus, Tenth Edition, is intended primarily