Basic Technical Mathematics with Calculus SI Version OTHER PEARSON EDUCATION TITLES OF RELATED INTEREST Basic Technical Mathematics, Tenth Edition, by Allyn J Washington Basic Technical Mathematics with Calculus, Tenth Edition, by Allyn J Washington Introduction to Technical Mathematics, Fifth Edition, by Allyn J Washington, Mario F Triola, and Ellena Reda TENTH EDITION Basic Technical Mathematics with Calculus SI Version Allyn J Washington Dutchess Community College Michelle Boué Toronto Editor-in-Chief: Michelle Sartor Executive Acquisitions Editor: Cathleen Sullivan Marketing Manager: Michelle Bish Program Manager: Patricia Ciardullo Project Manager: Kimberley Blakey Developmental Editor: Mary Wat Media Editor: Charlotte Morrison-Reed Media Producer: Kelly Cadet Production Services: Heidi Allgair, Cenveo ® Publisher Services Permissions Project Manager: Marnie Lamb Photo Permissions Research: Chritina Simpson, Q2A/Bill Smith Text Permissions Research: Electronic Publishing Services, Inc Art Director: Zena Denchik Cover Designer: Alex Li Interior Designer: Cenveo® Publisher Services Cover Image: Gencho Petkov/Shutterstock Credits and acknowledgments for material borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text Original edition published by Pearson Education, Inc., Upper Saddle River, New Jersey, USA Copyright © 2009 Pearson Education, Inc This edition is authorized for sale only in Canada If you purchased this book outside the United States or Canada, you should be aware that it has been imported without the approval of the publisher or the author Copyright © 2015 Pearson Canada 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 Canada Inc., Permissions Department, 26 Prince Andrew Place, Don Mills, Ontario, M3C 2T8, or fax your request to 416-447-3126, or submit a request to Permissions Requests at www.pearsoncanada.ca 10 CKV Library and Archives Canada Cataloguing in Publication Washington, Allyn J., author Basic technical mathematics with calculus : SI version / Allyn J Washington, Michelle Boué Tenth edition Includes indexes ISBN 978-0-13-276283-0 (bound) Mathematics Textbooks. I Boué, Michelle, author II Title QA37.3.W37 2014 510 C2014-900075-8 Copyright © 2010, 2005, 2000, 1995 Pearson Canada Inc., Toronto, Ontario ISBN 978-0-13-276283-0 To Douglas, Julia and Andrea ~Michelle Boué In memory of my loving wife, Millie ~Allyn J Washington This page intentionally left blank Contents Preface xi Basic Algebraic Operations Numbers Fundamental Operations of Algebra Measurement, Calculation, and Approximate Numbers 1.4 Exponents 1.5 Scientific Notation 1.6 Roots and Radicals 1.7 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 32 36 38 41 45 48 Equations, Review Exercises, and Practice Test 51 1.1 1.2 1.3 11 21 26 30 4.3 4.4 4.5 Values of the Trigonometric Functions The Right Triangle Applications of Right Triangles 122 126 131 Equations, Review Exercises, and Practice Test 136 Systems of Linear Equations; Determinants 5.1 5.2 5.3 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 5.4 5.5 5.6 5.7 142 143 146 149 153 160 166 Geometry 55 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 56 60 66 69 74 78 Equations, Review Exercises, and Practice Test 176 Equations, Review Exercises, and Practice Test 81 Functions and Graphs 86 3.1 3.2 3.3 3.4 3.5 3.6 Introduction to Functions More about Functions Rectangular Coordinates The Graph of a Function More about Graphs Graphs of Functions Defined by Tables of Data 87 91 95 97 104 6.3 6.4 6.5 6.6 6.7 6.8 Review Exercises and Practice Test The Trigonometric Functions 4.1 4.2 Angles Defining the Trigonometric Functions 109 112 115 116 119 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 170 181 182 185 190 196 197 202 206 212 Equations, Review Exercises, and Practice Test 216 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 7.4 220 221 225 228 232 Equations, Review Exercises, and Practice Test 236 VII VIII CONTENTS Trigonometric Functions of Any Angle 8.1 8.2 8.3 8.4 Signs of the Trigonometric Functions Trigonometric Functions of Any Angle Radians Applications of Radian Measure 240 241 243 249 253 Equations, Review Exercises, and Practice Test 260 Vectors and Oblique Triangles 9.1 9.2 9.3 9.4 9.5 9.6 Introduction to Vectors Components of Vectors Vector Addition by Components Applications of Vectors Oblique Triangles, the Law of Sines The Law of Cosines 264 265 269 273 277 282 288 Equations, Review Exercises, and Practice Test 292 10 Graphs of The Trigonometric Functions 10.1 10.2 10.3 10.4 10.5 10.6 296 Graphs of y = a sin x and y = a cos x Graphs of y = a sin bx and y = a cos bx Graphs of y = a sin (bx + c) and y = a cos (bx + c) Graphs of y = tan x, y = cot x, y = sec x, y = csc x Applications of the Trigonometric Graphs Composite Trigonometric Curves 297 300 303 307 310 313 Equations, Review Exercises, and Practice Test 317 11 Exponents and Radicals 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 320 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 361 Equations, Review Exercises, and Practice Test 366 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 370 371 373 377 382 385 388 392 Equations, Review Exercises, and Practice Test 396 14 Additional Types of Equations and Systems of Equations 14.1 14.2 14.3 14.4 Graphical Solution of Systems of Equations Algebraic Solution of Systems of Equations Equations in Quadratic Form Equations with Radicals Review Exercises and Practice Test 15 Equations of Higher Degree 15.1 The Remainder and Factor Theorems; Synthetic Division 15.2 The Roots of an Equation 15.3 Rational and Irrational Roots 399 400 403 407 410 414 417 418 423 427 Equations, Review Exercises, and Practice Test 433 321 325 329 333 335 Equations, Review Exercises, and Practice Test 339 12 Complex Numbers 12.7 An Application to Alternating-Current (ac) Circuits 341 342 345 348 350 352 355 16 Matrices; Systems of Linear Equations 16.1 16.2 16.3 16.4 16.5 16.6 Matrices: Definitions and Basic Operations Multiplication of Matrices Finding the Inverse of a Matrix Matrices and Linear Equations Gaussian Elimination Higher-Order Determinants 435 436 439 445 449 454 457 Equations, Review Exercises, and Practice Test 463 17 Inequalities 17.1 Properties of Inequalities 17.2 Solving Linear Inequalities 17.3 Solving Nonlinear Inequalities 467 468 472 476 CONTENTS 17.4 Inequalities Involving Absolute Values 17.5 Graphical Solution of Inequalities with Two Variables 17.6 Linear Programming 482 485 488 Equations, Review Exercises, and Practice Test 492 18 Variation 18.1 Ratio and Proportion 18.2 Variation 495 496 500 Equations, Review Exercises, and Practice Test 506 19 Sequences and The Binomial Theorem 19.1 19.2 19.3 19.4 Arithmetic Sequences Geometric Sequences Infinite Geometric Series The Binomial Theorem 510 511 516 520 523 22.2 22.3 22.4 22.5 22.6 22.7 Summarizing Data Normal Distributions Confidence Intervals Statistical Process Control Linear Regression Nonlinear Regression IX 620 628 634 640 646 651 Equations, Review Exercises, and Practice Test 654 23 The Derivative 23.1 23.2 23.3 23.4 23.5 23.6 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 659 660 669 673 677 682 686 690 699 702 Equations, Review Exercises, and Practice Test 528 23.7 23.8 23.9 20 Additional Topics in Trigonometry 531 Equations, Review Exercises, and Practice Test 706 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 532 537 542 545 548 553 Equations, Review Exercises, and Practice Test 558 21 Plane Analytic Geometry 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 562 563 567 573 578 582 587 593 596 599 603 606 24 Applications of the Derivative 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 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 Differentials and Linear Approximations 711 712 714 718 722 727 732 737 743 Equations, Review Exercises, and Practice Test 747 25 Integration 25.1 25.2 25.3 25.4 25.5 Antiderivatives The Indefinite Integral The Area Under a Curve The Definite Integral Numerical Integration: The Trapezoidal Rule 25.6 Simpson’s Rule 752 753 755 760 765 768 771 Equations, Review Exercises, and Practice Test 610 Equations, Review Exercises, and Practice Test 774 22 Introduction to Statistics 26 Applications of Integration 22.1 Tabular and Graphical Representation of Data 615 616 26.1 Applications of the Indefinite Integral 26.2 Areas by Integration 777 778 782 Index Abscissa, 96 Absolute error, 745 Absolute inequality, 468 Absolute value, 3; of complex numbers, 350; in inequalities, 482; order of operations, Acceleration, 279, 704, 719, 778, 842; angular, 256 Accuracy of number, 15, 123; approximate number, 18 Addition: of algebraic expressions, 32; algebraic method of solving three linear equations, 166; of approximate numbers, 19; of complex number, 345, 357; of cubes, 196; of fractions, 206; of matrices, 437; of ordinates, 313; of radicals, 333; of signed numbers, 7; solution of system of equations, 404; of vectors, 266, 273 Adjacent angles, 57 Agnesi, Gaetana, 733 Agnesi, Maria, 849 Algebraic expressions, 24, 32 Algebraic operations, 32 Alternate-exterior angles, 58 Alternate-interior angles, 58 Alternating current, 311, 361, 531 Altitude, 61 Ambiguous case, 286 Ampere, 364 Amplitude of sine curve, 298 Analytic geometry, 562 Angle, 56, 116; acute, 56; bisectors, 61; central, 70; of depression, 132; double-angle formulas, 542; of elevation, 132; halfangle formulas, 545; inscribed, 71; negative, 116, 247; obtuse, 56; phase, 303; quadrantal, 117, 247; radian measure, 71; reference, 243; right, 56; of rotation, 600; standard position, 117; straight, 56; sum and difference of two, 537 Angular acceleration, 256 Angular velocity, 255 Antiderivative, 753 Antilogarithm, 382 Apollo 11, 503 Applied maximum and minimum problems, 737 Approximate numbers, 15, 17, 121; operations, 18 Arc, 70 Arc length, 253 Area: of circle, 70; of circular sector, 254; under a curve, 760; element of, 782; of geometric figures, 61; by integration, 782; lateral, 78; of quadrilateral, 67; of triangle, 62; between two curves, 783 Argument of complex number, 350 Aristotle, 86 Arithmetic mean, 620 D.12 www.downloadslide.net 1.1 Numbers Arithmetic sequence, 511 Array, 616 Associative law, 6, 438 Asymptote, 100, 588, 733 Attribute, 616; control chart for, 641 Auxiliary equation of differential equation, 970; complex roots, 974; repeated roots, 973 Average value, 807 Average velocity, 659 Axes: rotation of, 599; translation of, 593 Axes, coordinate, 96 Axis: of ellipse, 583; of hyperbola, 588; of parabola, 578; polar, 603 Babbage, Charles, 417 Bar, 34 Barrow, Isaac, 752 Base: of exponents, 21, 371; of logarithms, 382, 385; of solid, 78; of trapezoid, 66; units, 13 Basic identities, 532 Becquerel, Henri, 964 bel, 383 Bell, Alexander Graham, 383 Benford’s law, 398 Benz, Karl, 737 Bernoulli, Johann, 837 Binomial, 33 Binomial formula, 523, 525 Binomial series, 526 Binomial theorem, 523, 682 Braces, 34 Brackets, 8, 25, 34 Briggs, Henry, 370 Calculator, 11, 104, 150, 245, 370, 448; scientific notation, 28; solving systems of equations, 401 Calculus, 659, 667, 711 Cancellation, 199 Capacitance, 361, 781 Capacitor, voltage across, 779 Carbon dating, 964 Cardioid, 607 Catenary, 392 Cauchy, Augustin-Louis, 399, 667 Cayley, Arthur, 435 Center of mass, 793 Central angle, 70 Central line, 642 Central tendency, 620 Centre of circle, 69 Centre of mass, 793 Centroid, 61, 793; of solid of revolution, 797; of thin, flat plate, 795 Chain rule, 691 Change, time rate of, 827 Character of roots, 230 D.12 Charles’ law, 500 Chebychev’s theorem, 626 Chord, 69 Circle, 69, 400, 573 Circular paraboloid, 901 Circumference, 70 Class, 616 Class mark, 617 Coefficient, 33, 155 Cofunction, 128 Colossus computer, 214 Column of determinant, 160 Common difference, 511 Common logarithms, 382 Common ratio, 516 Commutative law, 6, 440 Complementary angles, 57, 127 Complementary solution of differential equation, 977 Completing the square, 225, 575, 595 Complex fraction, 209 Complex number, 2, 341, 343, 345; division, 356; exponential form, 352; multiplication, 355; polar form, 350 Complex plane, 348 Complex roots, 424, 430 Components of vector, 269, 273 Composite function, 690 Composite trigonometric curves, 313 Computer, 903 Concavity, 728 Conditional equation, 41 Conditional inequality, 468 Cone, 78 Confidence intervals, 634 Congruent triangles, 64 Conic sections, 400, 598 Conjugate: axis of hyperbola, 588; of complex number, 344 Constant, 4, 963; combining, 877; derivative of, 682; of integration, 755, 758; of proportionality, 500, 501 Constraint, 488 Continuity, 660 Contradiction, 42 Control charts, 641 Convergent, 916 Conversion of angle measurements, 249 Conversion of angles, 116 Conversion of units, 116 Coordinates: cylindrical, 903; polar, 603; rectangular, 95 Copernicus, Nicolaus, 586 Correlation, 651 Corresponding angles, 58, 119 Corresponding segments, 58 Corresponding sides, 63, 119 Cosecant: of angle, 120; graph of, 307; integration of, 859 www.downloadslide.net Index Cosine: of angle, 120; derivative of, 823; of double angle, 542; graph of, 297; half angle, 545; half-range Fourier series, 944; integration of, 863; of sum of two angles, 537 Cosines, law of, 282, 288 Cotangent: of angle, 120; graph of, 307; integration of, 859 Coterminal angles, 116, 243 coulomb, 364 Cramer, Gabriel, 142, 162 Cramer’s rule, 142, 162, 172, 460 Critical value, 477 Cube, 78; difference, 196; special products, 183–184; sum, 196 Cube roots, 358 Cumulative frequency, 618 Current, 361, 780 Curve: area between two, 783; area under, 760; finding, given slope, 963; graphing smooth, 109; identifying, 601; in polar coordinates, 606; sketching, 727, 732, 825, 841; slope of tangent to, 669; in three dimensions, 899 Curve in space, 900 Curvilinear motion, 718, 828 Cycle, logarithmic scale, 393 Cyclotron, 604 Cylinder, 78 Cylindrical coordinates, 903 Cylindrical shell, 789 Cylindrical surface, 903 d’Alembert, Jean, 264 Damped simple harmonic motion, 983 Dantzig, George, 467 Decibel, 383 Decimal, repeating, 2, 521 Decision variables, 488 Definite integral, 765 Deflection of beams, 986 Degree: of differential equation, 950; as measure of angle, 56, 116; of polynomial, 33; of term, 33 Delta method, 670 DeMoivre, Abraham, 357 DeMoivre’s theorem, 357 Denominate number, 4, 323 Denominator: rationalizing of, 331 Dependent system of equations, 152, 163 Dependent variable, 87 Derivative, 659, 673; of a constant, 682; of a constant times a function, 683; of cosine function, 815; curve sketching, 727; of exponential function, 834; higher, 702; of implicit function, 700; of inverse trigonometric functions, 822; of logarithmic function, 830; partial, 895; of a polynomial, 682; of a power of a function, 690; of a product, 686; of a quotient, 686; as a rate of change, 677; second, 702; of sine function, 815; of a sum, 684; of trigonometric functions, 819 Derived units, 13 Descartes, René, 320, 343, 400, 427, 562 Descartes’ rule of signs, 428 Determinant, 142; higher-order, 457; properties of, 459; second-order, 160; third-order, 170, 459 Deviation, 647; standard, 623 Diagonal, 66 Diagonal of determinant, 161 Diameter, 69 Difference: of matrices, 438; of squares, 665 Difference, common, 511 Difference engine, 417 Differential, 743, 827 Differential calculus, 659 Differential equation, 949; Laplace transforms, 994; linear, 969; numerical solutions, 960 Differentiation, 674; of implicit functions, 699; series formed by, 925; of transcendental functions, 814 Diocles, 220 Direct current, 531 Direction, 720 Directrix of parabola, 578 Direct variation, 500 Disc, 788 Displacement, 268, 778; of sine curve, 304 Distance, related rates and, 723 Distance formula, 563, 739 Distances, related rates and, 724 Distributive law, Divergent, 916 Dividend, 39 Division: of algebraic expressions, 38; of complex numbers, 345, 356; of fractions, 202; of radicals, 336; with remainder, 418; of signed numbers, 7; synthetic, 419; by zero, 9, 99 Divisor, 39 Domain, 91, 675, 733 Double-angle formulas, 542 Double integral, 895, 909 e (irrational number), 352, 385, 834 Edison, Thomas, 296, 362, 531 Einstein, Albert, 45 Electric circuits, 965, 985, 995 Element: of area, 782; of determinant, 160; of matrix, 436; of volume, 788, 910 Elements (Euclid), 510 Elimination by addition or subtraction, 154, 404 Ellipse, 400, 582, 594, 601 Elliptic hyperboloid, 901 Engineering notation, 28 ENIAC, 214 Equal sign (=), 19 Equations, 41; conditional, 41; curves sketched from, 606; differential, 950; exponential, 388; graphical solution, 104, 146; higher-degree, 417; involving D.13 fractions, 212; linear, 142; literal, 45; logarithmic, 389; polynomial, 424; quadratic form, 407; with radicals, 410; roots of, 419, 423; second-degree, 596; solving graphically, 104–106; systems of linear, 144, 160, 166, 401, 460; systems of quadratic, 400; trigonometric, 548 Equilibrium, 279 Eratosthenes, 115 Estimating, 19 Euclid, 55, 510 Euler, Leonhard, 240, 352, 711, 814, 895, 949, 960 Euler’s formula, 925 Euler’s method, 960 Even function, 940 Exact number, 15; and approximate numbers, 19 Explicit functions, 699 Exponential equations, 388 Exponential form, 856; of complex number, 352 Exponential function, 370; derivative of, 834; graph of, 392; integration of, 854 Exponential value, 928 Exponents, 21, 320, 322 Extraneous roots, 407 Extraneous solution, 214 Extrapolation, 110 Extreme point, 233 Faces, 78 Factor, 32, 197, 419; canceling, 199 Factorial notation, 524 Factoring, 181, 185; common factor and difference of squares, 185; complete, 188; difference of two squares, 187; by grouping, 188, 194; quadratic equations, 222; trinomials, 190 Factor theorem, 419 Family of curves, 963 Farad, 364 Feasible point, 488 Fermat, Pierre de, 562, 752 First-derivative test, 727 First-quadrant angle, 116 Focus: of ellipse, 582; of hyperbola, 587; of parabola, 578 Folium, 701 Force: liquid pressure, 806; related rates, 724; unit of, 277 Forced vibrations, 985 Formula, 45; binomial, 523; distance, 563; quadratic, 228 Fourier, Jean, 915, 934 Fourier series, 934, 940, 980 Fraction, 2–3, 181; addition, 30, 206; complex, 209; division, 202; equations, 212; equivalent, 197; as exponent, 325; multiplication, 202; partial, 879, 883; simplest form, 198 www.downloadslide.net D.14 Index Frequency, 312; arithmetic mean, 621; distribution, 616; polygon, 617 Frustum, 79 Function, 87, 554; average value, 807; composite, 690; exponential, 371; finding zeros of, 104; functional notation, 88; implicit, 699; interval notation, 91; inverse, 375, 553; logarithmic, 371; quadratic, 220; series formed by functional notation, 924; trigonometric, 120, 240; of two variables, 896; from verbal statements, 92–94 Fundamental laws of algebra, Fundamental principle of fractions, 197 Fundamental theorem of algebra, 417, 423 Gabor, Dennis, 413 Galileo, 86, 220 Galton, Francis, 615 Gauss, Karl, 343, 417, 423, 454, 513, 628 Gaussian elimination, 435, 454 Gauss-Jordan method, 446 General equation: of circle, 575; quadratic, 220; of straight line, 568 General power formula, 850 General solution of differential equation, 950 Geometric sequence, 510, 516 Geometry, 55 Googol, 29 Graph: derivatives to find features of, 729; of exponential function, 371, 392; of function, 97; functions, 733; on graphing calculator, 104–107; inequalities, 468; of linear function, 146; of logarithmic function, 375; on logarithmic paper, 393; in polar coordinates, 606; of quadratic function, 232; sketch a graph, 841; of specific values, 619; of trigonometric equations, 549; of trigonometric functions, 296; using intercepts and traces to sketch, 900 Graphical representation of complex numbers, 348 Graphical solution of equations, 104, 149, 232, 400 Graphical solution of inequalities, 481, 485 Graunt, John, 615 Gravitation, universal law of, 495, 501 Gravity Recovery and Interior Laboratory (GRAIL), 495 Great circle, 254 Greater than, 4, 468 Grouping: factoring trinomial by, 194–195 Half-angle formulas, 545 Half-line, 56 Half-range expansion, 943 Half-wave rectifier, 938 Halley, Edmond, 615 Harmonic sequence, 515 Heaviside, Oliver, 949 Height: of solid geometric figures, 78; of triangle, 61 henry, 364 Hero’s formula, 62 Hertz, 312 Hertz, Heinrich, 296, 311, 341 Hexagon, 60 Higher-order determinants, 457 Hipparchus, 115 Histogram, 617 Holography, 413 Homogeneous differential equation, 969 Hooke, Robert, 804 Hooke’s law, 804 Hyperbola, 101, 401, 587, 594, 601 Hyberbolic cosine, 837 Hypotenuse, 60, 120 Identity, 7, 34, 39, 42; matrix, 441; trigonometric, 532 Imaginary axis, 348 Imaginary number, 2, 31, 342; graphing, 100 Imaginary roots, 31 Impedance, 362 Implicit function, 699; derivative of, 700; differentiation of, 699 Improper integral, 989 Inclination, 564 Inconsistent system of equations, 151, 163 Increment, 744 Indefinite integral, 755, 778 Independent variable, 87 Indeterminate, Indeterminate form, 837 Index: of radical, 329 Inductance, 361 Inequalities, 4, 467; algebraic solution of, 468; graphical solution of, 485; involving absolute values, 485; properties of, 469; with two variables, 485 Infinite series, 520, 916 Infinity, 521, 733; limit as x approaches, 664 Initial point of vector, 266 Initial side of angle, 116 Instantaneous acceleration, 704 Instantaneous rate of change, 659, 672, 677 Instantaneous velocity, 659, 678 Integer, Integral: approximating with Simpson’s rule, 772; definite, 765; double, 895, 909; indefinite, 755 Integrating combinations, 955 Integration, 752; areas by, 782; of exponential forms, 854; of inverse trigonometric forms, 867; limits of, 765; of logarithmic forms, 850; methods of, 849; by partial fractions, 879, 883; by parts, 871; of powers, 850; as summation, 763; by tables, 888; of trigonometric forms, 854, 859; by trigonometric substitution, 876; by use of series, 926; volumes, 788 Intercepts, 104, 148, 733 Intersect, 56 Interval of convergence, 920 Inverse functions, 375, 553; trigometric functions, 123, 554 Inverse Laplace transform, 992 Inverse logarithm, 383 Inverse matrix, 441, 445 Inverse trigonometric functions, 123, 246, 553; derivatives of, 822; integral forms, 867 Inverse variation, 500 Irrational numbers, Irrational roots, 430 Iterative method, 715 Jacobi, Carl, 905 Joint variation, 501 j-operator, 364 Jordan, Wilhelm, 446 joule, 322 Joule, James Prescott, 322 kelvin, 326 Kirchhoff, Gustav, 142 Kirchhoff’s current law, 142 Kirchhoff’s voltage law, 142 Kutta, Martin, 961 Lagrange, Joseph Louis, 264, 915 Laplace, Pierre, 370, 628, 949, 989 Laplace transforms, 989, 994 Latitude, 254 Law: of cosines, 282, 288; of sines, 282 Least squares, method of, 647 Least-squares line, 648 Leibniz, Gottfried Wilhelm, 160, 765, 814, 849, 895 Lemniscate, 608 Length, 66 Less than, 4, 468 L’Hospital, Marquis de, 837 L’Hospital’s rule, 837 Libby, Willard, 964 Like terms, 33 Limaỗon, 607 Limit, 520, 660; e as a, 834; of a function, 662; numerical verification of, 839; of sin u>u, 815 Limits of integration, 765 Line, 56; slope of, 563 Linear approximation, 745 Linear differential equation, 957, 969 Linear equation, 143; graph of, 146; matrices, 449; solving by determinants, 460 Linear extrapolation, 110 Linear factors, repeated, 883 Linear inequalities, 472 Linear interpolation, 110 Linearity property, 991 Linearization, 745 Linear programming, 467, 488 www.downloadslide.net Index Linear regression, 646 Linear simultaneous systems, 144 Lissajous, Jules, 315 Lissajous figures, 314 Literal number, Locus, 573 Logarithmic equations, 389 Logarithmic function, 373; derivative of, 830; integral form, 850 Logarithmic paper, 393 Logarithmic scale, 392 Logarithmic value, 929 Logarithms, 370, 385; to base 10, 382; basic form, 852; computations by, 383; natural, 385; of a product, 378; properties of, 377 Longitude, 254 Lord Kelvin, 326 Lower control limit (LCL), 642 Lowest common denominator (LCD), 206 Maclaurin, Colin, 142, 921 Maclaurin series, 919 Magnitude, 720 Major axis of ellipse, 583 Matrix, 171, 435; addition, 437; elements of, 436; identity, 441; inverse, 441, 445; linear equations, 449; multiplication, 439; square, 436; subtraction, 438; zero, 436 Maximum and minimum problems, 727 Maximum points, 233, 727 Maxwell, James, 341 Mean: arithmetic, 620; large sample confidence intervals, 635 Measurement, 12; control chart for, 641; estimating errors in, 745 Median, 61, 620 Members of inequality, 468 Method of least squares, 647 Method of partial fractions, 879 Method of undetermined coefficients, 977 Middle term, 190 Minimum points, 233, 727 Minor, 458 Minor axis of ellipse, 583 Minute (measure of angle), 116 Mode, 116, 622 Modulus of complex number, 350 Mohr’s circle, 577 Moivre, Abraham de, 628 Moment of inertia, 799; of a mass, 793; of a solid, 801 Monomial, 33; common factors, 186; dividing, 38; multiplying, 36 Motion, in resisting medium, 966 Multinomial, 33 Multiplication: of algebraic expressions, 36; of complex numbers, 345, 355; of fractions, 202; of matrices, 439; order of operation, 322; of radicals, 335; scalar, 438; series formed by, 924; of signed numbers, Multiplicity of a root, 222 Napier, John, 370 Natural logarithm, 385 Negative angle, 116, 247 Negative direction, Negative exponents, 23, 124; solving an equation containing, 408 Negative numbers, 343; plotting, 98 Nested parentheses, 35 Newton (N), 277 Newton, Sir Isaac, 277, 279, 320, 435, 495, 501, 659, 715, 814, 849, 983 Newton’s method, 714, 826, 842 Niépce, Joseph, 409 Nightingale, Florence, 615 Nonhomogeneous differential equations, 969, 977 Nonlinear inequalities, 476 Nonlinear regression, 651 Nonrepeated linear factors, 879 Nonrepeated quadratic factors, 885 Normal distribution, 628 Normal line, 712 Number: approximate, 15, 18; changing to scientific notation, 27; complex, 2, 341; denominate, 4; exact, 15; imaginary, 2, 31, 343; irrational, 2; natural, 2; negative, 2; rational, 2; real, 2; roots, 30 Numerical coefficient, 33 Numerical integration, 768 Objective function, 488 Oblique triangle, 264, 282, 290 Octant, 899 Odd function, 940 Ogive, 618 Ohm’s law, 361 Operations with zero, Operator, 969 Optimal solution, 489 Order: of differential equation, 950; of operations, 8, 24; of radical, 329; of trigonometric functions, 125 Ordinate, 96 Ordinates, addition of, 313 Origin, 3, 95 Orthogonal trajectories, 963 Parabola, 220, 232, 400, 578, 594, 601 Parallel, 56, 565 Parallelogram, 66; method of adding vectors, 267 Parameter, 616, 718 Parametric equations, 314 Parametric form, 718 Parentheses, 8, 34; nested, 35 Partial derivative, 895, 905 Partial fractions, 879 Partial sum, 916 Particular solution of differential equation, 950, 977 pascal, 322 D.15 Pascal, Blaise, 525, 777 Pascal’s triangle, 525 Pentagon, 60 Perfect square, 30, 330 Perimeter, 61; of quadrilateral, 67; of triangle, 61 Period of sine curve, 298 Perpendicular, 56, 565 Phase angle, 303–306, 363 Phase shift, 304 Phasor, 364 Pi, 70 Pixel, 104 Plane, 56, 168, 900 Point, 56; continuous at a, 660; of inflection, 729; locating, 96, 348; polar coordinates of, 603 Point estimate, 634 Point-slope form of straight line, 567 Polar axis, 603 Polar coordinates, 603; curves, 606 Polar form of complex number, 350; addition, 357; division, 356; multiplication, 355 Pole, 603 Polygon, 60; method of adding vectors, 266 Polynomial, 33; antiderivative, 753; derivative of, 682; dividing, 39–40; indefinite integral, 755; multiplying, 36; rational roots, 427 Polynomial function, 418 Polyphase generator, 531 Population, 616 Power: antiderivative of, 753; of complex number, 357; derivative of, 682, 686; general power formula, 850; integration of, 850; of number, 25; series, 920 Power rule, 756 Precision, 15, 18 Pressure, 777; liquid, 806 Prime factor, 185 Principal root, 30 Prism, 78 Product: of complex numbers, 345, 355; derivative of, 687; logarithm of, 378; of matrices, 438; special, 182 Progression: arithmetic, 511; geometric, 516 Projection, 310 Proportion, 43, 496; large sample confidence intervals, 638 Pyramid, 78 Pythagoras, 62 Pythagorean theorem, 120 Quadrant, 95 Quadrantal angle, 117, 247 Quadratic equation, 220, 407; solving graphically, 235 Quadratic equation in form, 407 Quadratic factors, nonrepeated, 885 Quadratic formula, 228 Quadrilateral, 60, 66 www.downloadslide.net D.16 Index Quantitative variable, 616 Quotient, 40; of complex numbers, 345, 356; derivative of, 688; of polynomials, 881 Radian, 71, 117, 249, 297 Radicals, 320, 329; addition of, 333; division of, 336; equations with, 410; multiplication of, 335; simplest form, 329, 331; subtraction of, 333 Radical sign, 30 Radicand, 329 Radioactivity, 964 Radius, 69, 573; of gyration, 799; related rates, 723 Range, 91, 106, 554, 623, 733 Rate of change, 672 Ratio, 43, 496; equality, 119 Rationalizing: denominator, 331; numerator, 331, 665 Rational number, Rational roots, 427 Raw data, 616 Ray, 56 Reactance, 361 Real axis, 348 Real numbers, 2, 91 Reciprocal, 4, 124 Rectangle, 66, 409; area under a curve, 760 Rectangular coordinate system, 95, 604, 899 Rectangular form of complex number, 343 Rectangular solid, 78 Rectifier: half-wave, 938 Reference angle, 244 Regression, 646 Related rates, 722 Relation, 93 Relative error, 745 Relative frequency, 617 Relative maximum and minimum points, 727 Remainder, 418 Remainder theorem, 418 Repeated quadratic factors, 887 Repeating decimal, 521 Resistance, 361 Resolving vector, 269 Resonance, 365 Resultant of vectors, 266 Rhombus, 66 Rice, Kellogg, 489 Right triangle, 62, 126 Roemer, Olaf, 591 Root-mean-square value, 866 Roots, 30, 427, 430; complex, 974; of complex numbers, 358; double, 222; of equation, 419, 423; extraneous, 407; of linear equations, 143; of polynomial equation, 429; of quadratic equation, 222; rational, 427; repeated, 973; repeated complex, 975 Rose, 608 Rotation of angles, 116 Rotation of axes, 599 Rounding off, 17 Row of determinant, 160 Row operations, 446 Runge, Carl, 961 Runge-Kutta method, 961 Salt solution, 965 Sample, 616 Sampling distributions, 632 Scalar, 265; multiple of vector, 267; multiplication, 437 Scale drawing, 63 Scientific notation, 26 Scott, David, 86 Secant, 69; of angle, 120; graph of, 307; integration of, 863; line, 69, 669 Second (measure of angle), 116 Second-degree equation, 596 Second derivative, 702 Second-derivative test, 729 Section, 901 Sector, 70, 254 Segment, 56, 71 Semilogarithmic (semilog) paper, 392 Semimajor axis of ellipse, 583 Sense of inequality, 468 Separation of variables, 952 Sequence, 510, 511; arithmetic, 511; finite, 511; geometric, 516; infinite, 916 Series, 520, 923; binomial, 523; computations with, 928; Fourier, 915, 934; Maclaurin, 921 Shell (element of volume), 789 Shewhart, Walter, 615 Shifing a graph, 106 Signed numbers, Significant digits, 15, 27, 123 Signs: factors differing only in, 200; laws of, 7; of trigonometric functions, 241 Signs of inequality, Sikorsky, Igor, 284 Similar terms, 33 Simple harmonic motion, 310, 982, 995 Simpson, Thomas, 75, 716 Simpson’s rule, 75, 771 Sine: of angle, 120; derivative of, 823; graph of, 297; of half-angle, 545; half-range Fourier series, 944; integration of, 863; inverse, 553; of sum of two angles, 537; Taylor series, 932 Sines, law of, 282 SI prefixes, 14; engineering notation, 28; scientific notation, 28 Sixth roots: by DeMoivre’s theorem, 359 Slant height, 78 Slope, 146, 563; curve, finding, 963; of linear function, 146; of tangent line, 669, 818 Slope-intercept form of straight line, 147, 567 Slug, 966 Solution: of differential equation, 950; of equation, 144; of inequality, 468, 485; of linear equation, 143; of quadratic equation, 222; of system of linear equations, 144, 149, 160, 166, 450, 460; of triangle, 126, 283; of trigonometric equations, 548; of two sides and angle opposite one of them, 285 Special products, 182 Sphere, 78 Spread, measures of, 623 Square, 66 Square matrix, 436 Square root, 30, 181; antiderivative, 753; by DeMoivre’s theorem, 359; evaluating using Taylor series, 932; solving an equation, 408 Squares, difference of, 665 Square wave, 936 Standard deviation, 623 Standard equation: of circle, 573; of ellipse, 583; of hyperbola, 588; of parabola, 578 Standard error of the mean, 632 Standard errors, 632 Standard normal distribution, 630; table of areas, 631 Standard position of angle, 117 Statistical process control, 640 Statistics, 615, 616 Steady-state solution, 986 Straight line, 109, 146, 567 Subscripts, 45, 46 Substitution, 41; elimination by, 150, 404; solution of a system of equations by, 153, 403; trigonometric, 876 Subtraction: of algebraic expressions, 32; algebraic method of solving three linear equations, 166; of complex numbers, 345; of cubes, 196; of fractions, 208; of matrices, 438; of radicals, 333; of signed numbers, 7; solution of system of equations, 404; of vector, 267 Summation symbol, 622 Sum of n terms, 513, 517 Supplementary angles, 57 Supplementary units, 13 Surface, 899 Symbols of grouping, 8, 34 Symmetry, 574, 733 Synthetic division, 419 System: of linear equations, 144, 149, 160, 166, 403, 450, 460; of quadratic equations, 400 Tables, integration by use of, 888 Tangent, 69; of angle, 120; to curve, 669; derivative of, 824; of double angle, 542; to earth’s surface, 930; graph of, 307; integration of, 863; line, 69, 669, 712 Taylor, Brook, 931 Taylor series, 931 www.downloadslide.net Index Temperature, 722 Terminal point of vector, 266 Terminal side of angle, 116 Terms, 32, 190; common factor same as, 186; of fractions, 199; of sequences, 511, 916; similar, 32 Tesla, Nikola, 531 3D, compared to 2D, 902 Time rate of change, 827, 842 Trace, 105, 900 Transcendental functions, 814; derivatives of, 815; differentiation of, 814 Transient term, 986 Translation of axes, 593 Transversal, 58 Transverse axis of hyperbola, 588 Trapezoidal rule, 74, 768 Triangle, 60; congruent, 64; equilateral, 60; isosceles, 60; oblique, 264, 288; Pascal’s, 525; right, 60, 126; scalene, 60; similar, 63, 119; solution of, 126, 283 Trigonometric equations, 548 Trigonometric form of complex numbers, 350 Trigonometric functions, 115, 120; of angles measured in degrees, 122; of angles of right triangle, 120; of any angle, 240; derivatives of, 819; graphs of, 296; integration of, 859, 876; inverse, 553; of negative angles, 246; signs, 241 Trigonometric identities, 532 Trigonometric value, 928 Trinomial, 33, 190; factoring, 190 2D, compared to 3D, 902 Unbiased rounding, 17 Uncertainty, of measured value, 12 Units: conversion, 14; writing, 13 Units of measurement, 12, 322 Universal law of gravitation, 495, 501 Unknown, 41 Upper control limit (UCL), 642 Variable, 4–5, 616; dependent, 87; independent, 87 Variables, functions of two, 896 Variables, separation of, 952 Variation, 495, 500, 739 Vectors, 264 Velocity, 720, 778; angular, 255; approximation, 929; average value, 807; linear, 255; parametric form, 718 Vertex: of angle, 56, 116; of ellipse, 583; of hyperbola, 588; of parabola, 233, 578 Vertical angles, 57 Vertical asymptotes, 308 D.17 Vertical line test, 102 Viốte, Franỗois, 181 volt, 364 Voltage, 361, 363, 722; across capacitor, 779; variations caused by, 641 Volumes: element of, 910; of geometric figures, 78; by integration, 788; under a plane, 911; related rates, 723; under a surface, 910, 911 Wallis, John, 320, 520 watt, 101 Watt, James, 615 Weighted mean, 621 Westinghouse, George, 362, 531 Width, 66 Witch of Agnesi, 733 Word problems, 48, 106, 151 Work, 804 x-axis, 95; areas below, 785 x-intercept, 104, 146 y-axis, 95 y-intercept, 147, 568 Zero, 2; as exponent, 23; matrix, 436; operations with, 9; trailing, 16 Zoom, 105 Notes www.downloadslide.net www.downloadslide.net Notes Notes www.downloadslide.net www.downloadslide.net Notes Notes www.downloadslide.net www.downloadslide.net M AT H N O T E S A Study Chart for Technical Mathematics JOHN JENNESS ALGEBRA ORDER OF OPERATIONS (BEDMAS) Simplify contents of Brackets [ ], parentheses ( ), and braces { } working from the innermost outward, and working separately above and below the fraction lines Simplify Exponents and Roots working from left to right Do Multiplication and Division in the order that they appear from left to right Do Addition and Subtraction in the order that they appear from left to right PROPERTIES OF NUMBERS Commutative Law: aϩb ϭ bϩa and ab ϭ ba Associative Law: aϩ(bϩc) ϭ (aϩb)ϩc and a(bc) ϭ (ab)c Note: Commutative Law does not apply to Subtraction or Division Distributive Law: a(bϩc) ϭ abϩac Signs: aϩ(Ϫb) ϭ aϪb and aϪ(Ϫb) ϭ aϩb and aϪb ϭ Ϫ(bϪa) NUMBER SET DEFINITIONS Natural (or Counting) Numbers: {1, 2, 3, 4, } Whole Numbers: { 0, 1, 2, 3, 4, } Integers: { , –3, –2, –1, 0, 1, 2, 3, } Rational Numbers: {x/y such that x, y are integers but y ϶ 0} Irrational Numbers: x is a real number but not a Rational number {e.g , e} Real Numbers include both Rational and Irrational Numbers Imaginary Numbers: of the form xj (or xi) where x is a real number and j is a number such that j2 ϭ Ϫ1 Complex Numbers: of the form: x ϩ yj where x is a real number and yj is an imaginary number EXPONENTS, ROOTS AND RADICALS amϫan ϭ amϩn m a am ,a ϭ amϪn, a or n ϭ a anϪm an m n mn (a ) ϭ a (ab)n ϭ anbn a n an ϭ n (b 0) b b a0 ϭ (a 0) aϪn ϭ n (a 0) a a1/n ϭ nΊහ a a )m am/n ϭ nΊaහm ϭ (nΊහ n හ Ί an ϭ a n හ b ϭ nΊaහb Ί a nΊහ BRITISH COLUMBIA INSTITUTE OF TECHNOLOGY mΊහ nΊහ හ හ a ϭ mnΊහ a a ϭ n හ (b 0) nΊහ b b nΊහ a Ί FACTORING AND SPECIAL PRODUCTS F.O.I.L rule: (aϩb)(cϩd) ϭ acϩadϩbcϩbd a(xϩy) ϭ axϩay (xϩy)(xϪy) ϭ x2Ϫy2 (xϩy)2 ϭ x2ϩ2xyϩy2 (xϪy)2 ϭ x2Ϫ2xyϩy2 (xϩa)(xϩb) ϭ x2ϩ(aϩb)xϩab (axϩb)(cxϩd) ϭ acx2ϩ(adϩbc)xϩbd (xϩy)3 ϭ x3ϩ3x2yϩ3xy2ϩy3 (xϪy)3 ϭ x3Ϫ3x2yϩ3xy2Ϫy3 x3ϩy3 ϭ (xϩy)(x2Ϫxyϩy2) x3Ϫy3 ϭ (xϪy)(x2ϩxyϩy2) QUADRATIC EQUATION හහහ ϪbϮΊහ b2Ϫ4ac ax2ϩbxϩc ϭ has solution form x ϭ _ 2a Four possible solutions based on discriminant D ϭ b2Ϫ4ac – If D Ͼ and a perfect square, then roots are real, rational, and unequal – If D Ͼ and not a perfect square, then roots are real, irrational, and unequal – If D ϭ 0, then roots are real, rational, and equal – If D Ͻ 0, then roots contain imaginary numbers, and are unequal COMPLEX NUMBERS – The symbol j represents the imaginary number හ1 such that j ϭ Ϫ1 ΊϪ හa ϭ jΊහ a where a Ͼ0 – Note: ΊϪ Operations with complex numbers: Addition: (aϩbj)ϩ(cϩdj) ϭ (aϩc)ϩ(bϩd)j Subtraction: (aϩbj)Ϫ(cϩdj) ϭ (aϪc)ϩ(bϪd)j Multiplication: (aϩbj)ϫ(cϩdj) ϭ (acϪbd)ϩ(adϩbc)j Division: (aϩbj)(cϪdj) (acϩbd)ϩ(bcϪad)j aϩbj _ ϭ ϭ cϩdj (cϩdj)(cϪdj) c2ϩd2 Rectangular form: xϩyj Polar form: r(cosϩjsin) ϭ rЄ Product in polar form: r1(cos1ϩjsin1)ϫr2(cos2ϩjsin2) ϭ (r1Є1)(r2Є2) ϭ r1r2Є(1ϩ2) Quotient in polar form: r r1(cos1ϩjsin1) _ r Є _ ϭ 1 ϭ Є(1Ϫ2) r2(cos2ϩjsin2) r2Є2 r2 EXPONENTS AND LOGARITHMS Exponential function: y ϭ bx Logarithmic function: y ϭ logbx Properties of logarithms: logbxy ϭ logbxϩlogby logb _x ϭ logbxϪlogby y logb (xn) ϭ nlogbx logb1 ϭ logbb ϭ logb(bn ) ϭ n Changing bases of logarithms: logax logbx ϭ _ logab logx 1nx ϭ loge lnx logx ϭ 1n10 VARIATION, RATIO AND PROPORTIONS a c ratio or proportion: _ ϭ _ b d direct variation: y ϭ kx inverse variation: y ϭ k/x joint variation: y ϭ kxz where: x, y, z are variables, k is constant of proportionality, and k GEOMETRY PLANE SHAPES Triangles: sides, all angles add to 180° Scalene: no two sides are equal in length Isosceles: two sides are equal in length Equilateral: all sides are equal in length Right: one angle is 90° Area: A ϭ 1/2bh Hero’s formula: A ϭ Ίහහහහහහහහ s(sϪa)(sϪb)(sϪc) where s ϭ _ (aϩbϩc) Quadrilaterals: sides Square: all sides are equal in length, all angles 90° Area: A ϭ s2 s Rhombus: all sides are equal in length, angles not 90° Area: A ϭ bh h b Rectangle: opposite sides are equal in length, all angles 90° Area: A ϭ lw l w www.downloadslide.net Parallelogram: opposite sides are equal in length, angles not 90° Area: A ϭ bh h Surface area: A ϭ r 2ϩrs Volume: V ϭ _ r 2h Lateral surface area (excluding base): S ϭ rs Regular pyramid: s Circle: 360° ϭ 2radians ϭ 1revolution 180° and 1radian ϭ _ Ϸ57.30° SOLIDS (x, y) h r w e Surface area: A ϭ 6e2 Volume: V ϭ e3 Right circular cylinder: h r O h Lateral surface area (excluding ends): S ϭ (base perimeter)ϫh Volume: V ϭ (base area)ϫh Right circular cone: s h r x Basic relationships: y side opposite sin ϭ r ϭ hypotenuse x side adjacent cos ϭ r ϭ hypotenuse y side opposite tan ϭ x ϭ side adjacent r hypotenuse sec ϭ x ϭ side adjacent r hypotenuse csc ϭ y ϭ side opposite x side adjacent cot ϭ y ϭ side opposite csc ϭ Surface area: A ϭ 2r2ϩ2rh Volume: V ϭ r2h Lateral surface area (excluding base): S ϭ 2rh Right prism: y l sec ϭ cot ϭ tan ϭ cot ϭ sin cos tan sin cos cos sin sin2ϩ cos2ϭ1 1ϩtan2 ϭ sec2 1ϩcot2 ϭ csc2 Sum and difference identities: sin(␣Ϯ) ϭ sin␣ cosϮcos␣ sin cos(␣Ϯ) ϭ cos␣ cosϯ sin␣ sin tan␣Ϯtan tan(␣Ϯ) ϭ 1ϯtan␣ tan Double-angle formulae: sin2␣ ϭ 2sin␣ cos␣ cos2␣ ϭ cos2␣Ϫsin2␣ ϭ 2cos2␣Ϫ1 ϭ 1Ϫ2sin2␣ 2 0ՅsecϪ1xՅ secϪ1x Յ csc−1x Յ csc−1 x 2 0 – Unless specified otherwise all angles are measured counter-clockwise from the positive x axis – Note: When manipulating angles using a calculator, verify the quadrant of the answer Positive functions: First quadrant: all Second quadrant: sin and csc Third quadrant: tan and cot Fourth quadrant: cos and sec a b c Law of Sines: sin A ϭ sin B ϭ sin C Surface area: 4r2 Volume: r3 y Surface area: A ϭ 2lwϩ2lhϩ2wh Volume: V ϭ lwh Cube: 0ϽcotϪ1xϽ TRIG FUNCTIONS OF ANY ANGLE r TRIGONOMETRIC FUNCTIONS Rectangular solid: Ϫ ϽtanϪ1xϽ − Sphere: Perimeter (or circumference): c ϭ 2r Area: A ϭ r2 Arc length: s ϭ r where r is radius, is angle in radians Ί 0ՅcosϪ1xՅ Lateral surface area (excluding base): S ϭ (base perimeter)ϫs Volume: (base area)ϫh b2 + cosα α cos ϭϮ Ϫ ՅsinϪ1xՅ h b1 h Ί Inverse trigonometric functions: y ϭ sinϪ1x Ϫ ՅyՅ b Trapezoid: sides of unequal length, angles not 90° Area: A ϭ 1/2h(b1ϩb2) Half-angle formulae: හහහහ sin ␣ ϭϮ 1Ϫcos␣ 2 x Law of Cosines: a2 ϭ b2ϩc2Ϫ2bc cos A b2 ϭ a2ϩc2Ϫ2ac cos B c2 ϭ a2ϩb2Ϫ2ab cos C VECTORS ADDITION Step 1: break original vectors into → x and y components: A ϭ AxϩAy Step 2: add x pieces to x pieces to get Rx, add y pieces to y pieces to get Ry Step 3: apply Pythagorean formula to get magnitude resultant R, R Step 4: use ϭ tanϪ1 y to get angle Rx GRAPHS OF TRIG FUNCTIONS y 2 b a c Ϫb 2 b Ϫa x c Ϫb y ϭ a sin (bx ϩ c), c Ͼ For each a Ͼ 0, b Ͼ (a) y Since c Ͻ 0, Ϫc/b is positive 2 b a Ϫa 2 b c Ϫb c Ϫb x y ϭ a sin (bx ϩ c), c Ͻ (b) amplitude ϭ ΈaΈ period ϭ 2 b c displacement ϭ Ϫ b PLANE ANALYTIC GEOMETRY හහහහසසසසසසස 2 distance formula: d ϭ Ί(x 2Ϫx1) ϩ(y2Ϫy1) y2Ϫy1 slope: m ϭ and m ϭ tan (0°ՅϽ180°) x2Ϫx1 www.downloadslide.net parallel lines: m1 ϭ m2 perpendicular lines: m2 ϭ Ϫ m1 or m1m2 ϭ Ϫ1 straight line: point slope form: yϪy1ϭ m(xϪx1) or slope-intercept form: y ϭ mxϩb or general form: AxϩByϩC ϭ circles: centred on origin: x2ϩy2 ϭ r2 centred on: (h, k): (xϪh)2ϩ(yϪk)2 ϭ r2 parabolas: centred on origin parallel to x axis: y2 ϭ 4px centred on origin parallel to y axis: x2 ϭ 4py ellipses: centred on origin major axes parallel to x2 y2 x axis: ϩ ϭ 1, a Ͼ b a b centred on origin major axes parallel to y2 x2 y axis: ϩ ϭ 1, a Ͼ b a b hyperbolae: centred on origin foci on x axis: x2 y2 Ϫ ϭ 1, a Ͼ b a2 b2 centred on origin foci on y axis: y2 x2 Ϫ ϭ 1, a Ͼ b a2 b2 translation of axes: x ϭ x'ϩh and y ϭ y'ϩk FUNCTIONS BASIC DEFINITIONS A function is defined as a relationship between two variables such that for every value of the first (independent) variable, there is only one corresponding value of the second (dependent) variable The complete set of possible values of the independent variable is called the domain of the function The corresponding complete set of dependent variable values is called the range of the function Linear equation in one unknown: axϩb ϭ Linear equation in two unknowns: axϩby ϭ c LINEAR EQUATIONS AND DETERMINANTS Given two linear equations of the form: a1xϩb1y ϭ c1 a2xϩb2y ϭ c2 The determinant of the second order is defined as: Έa b1Έ Έa b2Έ ϭ a1b2Ϫa2b1 Cramer’s Rule gives solution forms: Έa1 c1Έ Έc1 b1Έ Έc b Έ Έa c Έ x ϭ 2 and y ϭ 2 Έa1 b1Έ Έa1 b1Έ Έa2 b2Έ Έa2 b2Έ EQUATIONS OF HIGHER DEGREE a0xnϩa1xnϪ1ϩ…ϩan Polynomial function: f(x) ϭ Remainder theorem: f(x) ϭ (xϪr)q(x)ϩR where f(r) ϭ R factor of an Rational roots: rr ϭ factor of a0 MATRICES A matrix is any rectangular array of numbers If the number of rows and columns is equal, then it is a square matrix A determinant is a specific value associated with a square matrix Basic laws for matrices: Commutative law: AϩB ϭ BϩA Associative law: Aϩ(BϩC) ϭ (AϩB)ϩC k(AϩB) ϭ kAϩkB Aϩ0 ϭ A AAϪ1 ϭ AϪ1A ϭ I A system of linear equations: a1xϩb1y ϭ c1 a2xϩb2y ϭ c2 can be represented in matrix form as: AX ϭ C a b x c where A ϭ a12 b12 , X ϭ y΅ and C ϭ c21΅ ΅ ΄ ΄΅ ΄ ΅ and using the inverse: X ϭ AϪ1C SEQUENCES AND SERIES Factorial notation: n! ϭ n(nϪ1)(nϪ2)…(2)(1) Arithmetic sequences: an ϭ anϪ1ϩd nth term: an ϭ a1ϩ(nϪ1)d n Sum of n terms: Sn ϭ (a1ϩan) Geometric sequences: an ϭ ranϪ1 nth term: an ϭ a1r nϪ1 a (1Ϫrn) Sum of n terms: Sn ϭ where (r϶1) 1Ϫr a1 S Sum of geometric series: S ϭ nlim → ϱ n ϭ 1Ϫr where (ΈrΈϽ1) Binomial formula: (aϩb)n ϭ anϩnanϪ1bϩ n(nϪ1) anϪ2b2ϩ…ϩbn 2! Binomial series: (1ϩx)n ϭ 1ϩnxϩ n(nϪ1) x2ϩ 2! n(nϪ1)(nϪ2) x3ϩ… where (ΈxΈϽ1) 3! EXPANSION OF FUNCTIONS IN SERIES ϱ Infinite series: Α an ϭ a1ϩa2ϩa3ϩ…ϩanϩ… nϭ1 n→ ∞ n→ ∞ ∑ i =1 n Power series: f(x) ϭ a0ϩa1xϩa2x2ϩ…ϩanxnϩ… MacLaurin series: f(x) ϭ f(0)ϩf'(0)xϩ f "(0)x2 ϩ f "'(0)x3 ϩ…ϩ f (n)(0)xn ϩ… 2! 3! n! Taylor series: f(x) ϭ f(a)ϩf'(a)(xϪa)ϩ f "(a)(xϪa)2 ϩ… 2! Special series: ex ϭ 1ϩxϩ x ϩ x ϩ… 2! 3! sin x ϭ xϪ x ϩ x Ϫ x ϩ… 3! 5! 7! cos x ϭ 1Ϫ x ϩ x Ϫ x ϩ… 2! 4! 6! 1n(1ϩx)ϭ xϪ x ϩ x Ϫ x ϩ… where (ΈxΈϽ1) Fourier series: f(x)ϭa0ϩa1cosxϩa2cos2xϩ…ϩancosnxϩ…ϩ b1sinxϩb2sin2xϩ…ϩbnsinnxϩ… Fourier coefficients for period 2: a0 ϭ 2 f(x)dx - ͵ an ϭ ͵ f(x)cos nx dx bn ϭ ͵ f(x)sin nx dx - - Fourier coefficients for period 2L: L a0 ϭ 2L f(x)dx ͵ -L ͵ f(x)cos nx dx L bn ϭ L ͵ f(x)sin nx dx L L -L L -L STATISTICS BASICS Arithmetic mean ෆx ϭ x1f1ϩx2f2ϩ…ϩxnfn Αxf ϭ f1ϩf2ϩ…ϩfn Αf Standard deviation: හහහහ හහහහහහ Α(xϪxෆ)2 n(Αx2)Ϫ(Αx)2 sϭ ϭ nϪ1 n(nϪ1) Ί Ί SEQUENCE AND SERIES Sum of series: S = lim Sn = lim an ϭ L eϪ(xϪ) ր2 Ίහ 2 Standard normal distribution: y ϭ හ eϪx ր2 Ί 2 Standard (z) score: z ϭ xϪ Standard error of ෆx : x ϭ හ Ίn Standard error of s: s ϭ හ Ί 2n Least-squares line: y ϭ mxϩb nΑxyϪ(Αx)(Αy) mϭ nΑx2Ϫ(Αx)2 Normal distribution: y ϭ (Αx2)(Αy)Ϫ(Αxy)(Αx) nΑx2Ϫ(Αx)2 bϭ DERIVATIVES BASICS RULES The limiting value of the ratio ⌬x⌬y is known as the derivative of the function The derivative can be interpreted as the instantaneous rate of change of the dependent variable with respect to the independent variable dc Derivative of a constant: dx ϭ n Derivative of a polynomial: dx ϭ nxnϪ1 dx Derivative of a constant times a function: d(cu) du dx ϭ c dx Derivative of a sum: d(uϩv) ϭ du ϩ dv dx dx dx d(uv) dv du Product rule: dx ϭu ϩv dx dx v du Ϫu dv d u Quotient rule: dx v ϭ d(uv) ϭ dx dx dx v dy dy du Chain rule: dx ϭ du dx n du General power rule: du ϭ nunϪ1 dx and dx du pq ϭ p u(pq)Ϫ1 du q dx dx APPLICATIONS Newton’s Method: x2 ϭ x1Ϫ f(x1) f'(x1) Curvilinear motion: dx dy Velocity components: vx ϭ dt and vy ϭ dt dvx d2x ϭ and dt dt dvy d2y ϭ ay ϭ dt dt Acceleration components: ax ϭ හහහ Magnitude: v ϭΊහහහ vx2ϩvy2 and a ϭΊa2xϩay2 vy ay Direction: tanv ϭ v and tana ϭ a x x www.downloadslide.net Differential form of a function y ϭ f(x) is defined as dy ϭ f '(x)dx Linearization: L(x) ϭ f (a)ϩf '(a)(xϪa) Between two curves on y axis: Aϭ ͵ y2 dx ϭ ͵ ΄fx΅2 dx About y axis: V ϭ ͵ x2 dy ϭ ͵ ΄gy΅ dy About x axis: V ϭ a d c d c Radius of gyration: m1d12ϩ m2d22 ϩ … ϩ mnd n2 ϭ (m1 ϩ m2 ϩ … ϩmn)R2 Work: W ϭ a a ͵ f(x)dx Ϸ ᎏ⌬2ᎏx y0 ϩ 2y1 ϩ a ͵ f(x)dx Ϸ ᎏ⌬3ᎏx y0 ϩ 4y1 ϩ 2y2 ϩ b a velocity: v ϭ a dt ϭ at ϩ C1 and displacement: s = v dt dq electric current: i ϭ ᎏᎏ and dt electric charge: q ϭ i dt voltage across a capacitor: Vc ϭ ᎏᎏ i dt C Areas: Between a curve and the x axis: ͵ ͵ ͵ b a f(x)dx Between a curve and the y axis: Aϭ ͵ d c x dy ϭ ͵ d c g(y)dy Between two curves on x axis: Aϭ ͵ y2 Ϫ y1dx b a a a Average value: yavg ϭ ᎏ bϪa ͵ f(x)dx ϭ F(b) Ϫ F(a) y dx ϭ ͵ b APPLICATIONS OF INTEGRATION a a ͵ ydx where n is even ͵ ͵ f(x)dx b 4y3 ϩ 2y4 ϩ… ϩ 4ynϪ1 ϩ yn Aϭ c b Root-mean-square: yrms ϭ b b a d Force due to liquid pressure: F ϭ w lh dh 2y2 ϩ … ϩ 2ynϪ1 ϩ yn b ͵ x2 y2 Ϫ y1 dx and Ix ϭ k ͵ y2 x2 Ϫ x1 dy Moment of Inertia of area: Iy ϭ k ͵ ͵ ͵ c b Integral of sum: (du ϩ dv) ϭ u ϩ v ϩ C unϩ1 Power formula: undu ϭ ᎏᎏ ϩ C nϩ1 where (n Ϫ1) b Area under a curve: Aab ϭ ΄͐f(x)dx΅ ϭ F(b) Ϫ F(a) Simpson’s rule: c ͵ ͵ ͵ y(x2 Ϫ x1)dx yෆ ϭ ᎏᎏ ͵ (x2 Ϫ x1)dx ͵ b b a d x(y2 Ϫ y1)dx a Centroid of area: xෆ ϭ ᎏᎏ and b (y2 Ϫ y1) dx Indefinite integral: f (x)dx ϭ F(x) ϩ C Integral of a constant: ͐c du ϭ c͐du ϭ cu ϩ C Trapezoid rule: b a d Disk method: dV ϭ (radius)2 ϫ (thickness) Centre of mass: m1d1 ϩ m2d2 ϩ … ϩ mndn ϭ m1 ϩ m2 ϩ … ϩmndෆ INTEGRATION Definite integral: c Shell method: dV ϭ 2 (radius) ϫ (height) ϫ (thickness) d(cosϪ1 u) du ᎏᎏ ϭ Ϫ ᎏ2 ᎏᎏ dx ͙ෆ Ϫ u dx d(tanϪ1 u) du ᎏᎏ ϭ ᎏᎏ2 ᎏᎏ ϩ u dx dx d(logb u) du ᎏᎏ ϭ ᎏᎏ logb e ᎏᎏ dx u dx d(1n u) du ᎏᎏ ϭ ᎏᎏ ᎏ ᎏ dx u dx d(bu) du ᎏᎏ ϭ bu 1n b ᎏᎏ dx dx d(eu) du u ᎏᎏ ϭ e ᎏᎏ dx dx BASIC RULES d Volumes of rotation: DERIVATIVE OF TRANSCENDENTAL FUNCTIONS du d(sin u) ᎏᎏ ϭ cos u ᎏᎏ dx dx du d(cos u) ᎏᎏ ϭ Ϫsin u ᎏᎏ dx dx d(tan u) du ᎏᎏ ϭ sec2 u ᎏᎏ dx dx d(cot u) du ᎏᎏ ϭ Ϫcsc u ᎏᎏ dx dx d(sec u) du ᎏᎏ ϭ sec u tan u ᎏᎏ dx dx d(csc u) du ᎏᎏ ϭ Ϫcsc u cot u ᎏᎏ dx dx d(sinϪ1 u) du ᎏᎏ ϭ ᎏ2 ᎏᎏ dx ͙ෆ Ϫ u dx Cosine double angle substitutions: cos2 x ϭ ϩ cos2x sin2 x ϭ Ϫ cos2x ͵ x2 Ϫ x1dy ͵ ᎏ͵ y dx Ίᎏ T T INTEGRATION OF TRANSCENDENTAL FUNCTIONS ͵ᎏduuᎏ ϭ lnΈuΈϩ C ͵eudu ϭ eu ϩ C ͵sin u du ϭ Ϫcos u ϩ C ͵cos u du ϭ sin u ϩ C ͵sec2 u du ϭ tan u ϩ C ͵csc2 u du ϭ Ϫcot u ϩ C ͵sec u tan u du ϭ sec u ϩ C ͵csc u cot u du ϭ Ϫcsc u ϩ C ͵tan u du ϭ ϪlnΈcos uΈ ϩ C ͵cot u du ϭ lnΈsin uΈ ϩ C ͵sec u du ϭ lnΈsec u ϩ tan uΈ ϩ C ͵csc u du ϭ lnΈcsc u Ϫ cot uΈ ϩ C du u Ϫ1 ᎐ ᎏ ͵ ͙ළ ϩC Ϫ u2 ϭ sin a aෆ General form for nth order differential equation: dy dny dnϪ1y a0 n ϩa1 nϪ1 ϩ…ϩanϪ1 ϩany ϭ b dx dx dx alternately expressed using the differential operator notation D: a0Dny ϩ a1DnϪ1y ϩ…ϩ anϪ1Dy ϩ any ϭ b Solving first-order differential equations: From given form: M(x, y)dx ϩ N(x, y)dy ϭ Algebraically manipulate into general form: A(x)dx ϩ B(y)dy ϭ Using one of three methods: Separation of variables Integrable substitution combinations: d(xy) ϭ xdy ϩ ydx d(x2 ϩ y2) ϭ 2(x dx ϩ y dy) x dyϪy dx y d x ϭ x2 d x ϭ y y dxϪx dy y2 PQ Method for inseparable forms: dy ϩ Py dx ϭ Q dx where P,Q are functions of x with solution of the form: ye ͵Pdx ϭ Qe͵Pdx dx ϩ C Solving second-order differential equations: General form: a0D2y ϩ a1Dy ϩ a2y ϭ b Homogeneous linear form where (b ϭ 0) non-homogeneous form where (b ϶ 0) Three possible homogeneous solution forms (see quadratic discriminant) using auxiliary equation: a0m2 ϩ a1m ϩ a2 ϭ If discriminant Ͼ 0, then real roots, solution form: y ϭ c1em x ϩ c2em x If discriminant ϭ 0, then identical real roots, solution form: y ϭ emx (c1 ϩ c2x) If discriminant Ͻ 0, then complex roots, solution form: y ϭ e αx (c1sinxϩ c2 cosx) Non-homogeneous forms: y ϭ yc ϩ yp where: yc is homogeneous solution, yp is particular solution based on initial conditions ͵ APPLICATIONS Electric RLC circuits: L dq q d 2q ϩR ϩ ϭE dt C dt Motion in a resisting medium: m dv ϭ F Ϫ kv dt LAPLACE TRANSFORMS F(s) ϭ L(f) ϭ ͵ ͵ BASICS du u ᎏ ᎐᎐ᎏ ϭ ᎐ tanϪ1 ᎐ ϩ C a2 ϩ u2 a a INTEGRATION METHODS DIFFERENTIAL EQUATIONS ͵ Integration by parts: u dv ϭ uv Ϫ v du Trig substitutions: Ϫ x2 use x ϭ a sin For ͙ළaෆ ϩ x2 use x ϭ a tan For ͙ළaෆ Ϫ a2 use x ϭ a sec For ͙ළxෆ Square relation substitutions: cos2 x ϩ sin2 x ϭ 1 ϩ tan2 x ϭ sec2 x ϩ cot2 x ϭ csc2 x ͵ eϪstf(t)dt ϰ ᐆ΄af(t) ϩ bg(t)΅ ϭ aᐆ(f) ϩ bᐆ(g) ᐆ(f') ϭ sᐆ(f) Ϫ f(0) ᐆ(f") ϭ s2ᐆ(f) Ϫ sf(0) Ϫ f'(0) ᐆϪ1(F) ϭ f(t) www.pearsoned.ca Copyright © 2005 Pearson Canada Inc 10 11 13 12 11 ISBN 0-13-128739-7 ISBN 0-13-128739-7 ™xHSKBNBy287396z .. .Basic Technical Mathematics with Calculus SI Version OTHER PEARSON EDUCATION TITLES OF RELATED INTEREST Basic Technical Mathematics, Tenth Edition, by Allyn J Washington Basic Technical Mathematics. .. Mathematics with Calculus, Tenth Edition, by Allyn J Washington Introduction to Technical Mathematics, Fifth Edition, by Allyn J Washington, Mario F Triola, and Ellena Reda TENTH EDITION Basic Technical... Archives Canada Cataloguing in Publication Washington, Allyn J., author Basic technical mathematics with calculus : SI version / Allyn J Washington, Michelle Boué Tenth edition Includes