1035

IndexPage numbers in italics refer to information boxes and hint boxes.

AAbel, Niels Henrik 121 absolute (global) maximum/minimum values 602,

603absolute value expressions 1367 function 70, 81, 824 inequalities 13942absolute values 1012Abu Jafar Muhammad ibn Musa al-Khwarizmi 26 AC circuits 429 acceleration 6069,829, 8334acute angles 350, 3514addition, vectors 4035, 4067, 62931addition rule (probabilities) 527, 528, 538, 540, 546additive identities 12additive inverses 12algebraic expressions 2634algebraic fractions (rational expressions) 314,

1334algebraic functions 90, 91, 126, 736, 739amplitude 305, 306, 307, 308analytical solutions, trigonometric equations 31921angle between axis and vector 632 line and plane 6823 plane and plane 681 vector and vector 421, 637angles 28084 notation 350 vectors 42122angles of depression 354angles of elevation 354, 355, 356angles of incidence/refraction 759anti-derivatives 7714apparent sizes 343arc length 2845areas

graphs 7957, 81118 triangles 2623, 3646, 389, 390, 639, 648arguments (complex numbers) 443arithmetic 952arithmetic means 156, 481arithmetic sequences 1556, 194arithmetic series 1667, 1945associative property 12, 407, 432asymptotes 53, 55, 812, 127, 128, 12930, 209augmented matrices 26973average values, functions 800, 801average velocities 571, 604, 605averages (see means, medians, modes)axioms 955, 956, 958, 960axis of symmetry, parabolas 99, 100, 101, 110, 111

BBabylonians 281 bar charts 466 barber paradox 954base vectors 631Bayes theorem 5527bearings (direction) 387, 414Bernoulli, Daniel 217 Bernoulli, Jakob 217 Bernoulli, Johann 91, 217 Bernoulli distribution 8701Bernoulli trials 870biconditional statements 114 bimodal distributions 501binomial coefficients 185, 872, 873binomial distribution 8718, 8823 cumulative distribution function 8778binomial experiments 8723binomial theorem 1878 proof 1979binomials 26, 183Black Death 9312Bolyai, Janos 959bounded intervals 9, 10box plots 4912, 4956

Index

Briggs, Henry 225 Buffon, Count 518

Ccalculus 571, 588carbon-14 212Cardano, Girolamo 121 Cartesian coordinate system 478Cartesian equations lines 379, 6578 planes 6712, 673, 676, 677, 679, 683, 684Cartesian products 6, 7 categorical (qualitative) data 466Cayley, Arthur 247 chain reasoning 956chain rule 7016, 712, 719, 745, 775, 776change of base formula (logarithms) 23031Chebyshevs rule 497circles 285circular functions (see trigonometric functions)classes (data) 468closed intervals 9, 10co-function identities 322, 333coefficients, polynomial functions 91collinear points 263, 264, 631column vectors 248combinations 17980common denominators 32common factors (algebraic expressions) 32common logarithmic function 227, 222commutative property 12, 2523, 403, 420, 432complementary angles 350, 354complements (probabilities) 527, 528, 531complements (sets) 6, 7completeness, axioms 960completing the square 1002, 103complex numbers 3, 4, 6, 12, 108, 118, 429, 4314 conjugates 4348, 446 division 4334, 4478 Eulers formula 4458 multiplication 4323, 4456 powers 431, 436, 44952 roots 4525 trigonometric form 4435complex plane 4403components, vectors 4012, 406, 420, 423, 628

composite functions 5760, 61, 62, 81 derivatives 7016compound angle identities 3269, 333, 341, 343compound fractions 33compound interest 15962, 1712, 213, 219, 220,

221, 234, 235, 2367concave graphs 609, 610conditional probabilities 54150, 554, 555conjectures 956conjugates 27, 29, 4348, 446 consistency, axioms 960constant functions 70, 91, 600constant multiple rule, derivatives 588constant rule, derivatives 586, 588constants of integration 773construction, triangles 36970, 3723constructivism 955continuous data 466continuous functions 600, 797continuous random variables 8556, 88998, 902convergence (see infinite series, limits)coplanar vectors 649, 650, 675, 678coordinate planes 37, 627coordinates 3cosecant/cosine/cotangent function (see

trigonometric functions)cosine rule (see law of cosines)counterexamples 331critical points, functions 5600, 601, 602, 603, 605,

608cube roots 15, 453, 455cubes (shape) 4950cubic equations 121cubic expressions 26 cubic functions 91, 118, 11920cubing function 70cumulative distribution functions 858, 8778, 8856,

893cumulative frequency graphs 47072, 500curve fitting 2734

Ddata sets 465de Moivre, A. 44950de Moivres theorem 45051, 452, 457de Morgans laws 521

1036

1037

decay factor 212, 214decimal representations 4decreasing functions 64, 600definite integrals 7979, 8013, 828, 830degrees (angles) 280, 282degrees (polynomial functions) 26, 91, 97, 119del Ferro, Scipione 121dependent variables 47, 49, 51derivatives 583-96, 599612 chain rule 7016 exponential functions 7237 higher derivatives 714 implicit differentiation 72934 inverse trigonometric functions 73942 logarithmic functions 7359 product rule 70710 quotient rule 71113 second derivatives 606, 60812, 71314 stationary points 6005 trigonometric functions 596, 71821Descartes, Ren 48, 91, 616descriptive statistics 465determinants 2601, 2623, 264, 645diagonal matrices 248difference of two squares 27differential equations 83645differentiation (see derivatives)dihedral angles 393Diogenes Laertius 351 Diophantus 35 Dirac, Paul 9612directed line segments 399direction angles/cosines (vectors) 41012, 63941discrete data 466discrete random variables 8556, 85766, 870discriminant, quadratic equations 1067, 139disjoint sets 6displacement 400, 604, 607, 654, 72021, 827, 828,

829, 830, 831distance 607, 827, 830, 831distance between ... line and line 6667, 680 line and plane 674, 67980 line and point 4223, 6646 origin and plane 673 plane and plane 674, 680 plane and point 699

point and point 3940, 409distancetime graphs 572distributions, data 483distributions, random variables (see probability

distributions)distributive property 12, 27, 407, 420, 432diverging series 169division 12 complex numbers 4334, 4478domains (functions) 48, 512, 53, 545 composite functions 57, 5960 exponential functions 209 inverse functions 62, 63, 65, 66, 678, 225 inverse trigonometric functions 336 logarithmic functions 225 one-to-one functions 64 polynomial functions 93 rational functions 127, 128 trigonometric functions 2923, 308dominoes 190 domino effect 190double angle identities 33032, 333, 342double inequalities 8

Ee (number) 2178, 21920, 2213eclipses 343 Einstein 961elementary operations 90elementary row operations (matrices) 270elements, sets 2elimination method, simultaneous equations 423ellipses (singular ellipsis) 2 elliptical geometry 959, 960empirical rule (standard deviation) 4979, 865, 903,

909empty set 6, 7 end behaviour (functions) 93, 97, 129equally likely outcomes 529, 53033equally likely outcomes theory 5256equations 359etymologies algebra 26 algorithm 26 quadratic 99 secant 583 trigonometry 279

Index

1038

Euclid 4 Euclids axioms/postulates 958, 959, 960Euclidean geometry 959Euler, Leonhard 91, 220, 429Eulers formula, complex numbers 455-8evaluation theorem 804even functions 93, 303events (probabilities) 520evolution 956exact solutions, trigonometric equations 31417expanding, polynomial expressions 27, 28expected values (means) 8613, 864, 893, 395 Bernoulli distribution 871 binomial distribution 8756, 877 normal distribution 903, 904 Poisson distribution 8834, 885, 887explicit definitions 152, 729exponential decay model 212, 214, 2212exponential equations 226, 2345exponential functions 20611, 21618 applications 21114, 220 derivatives 7237 natural exponential function 2212exponential growth model 212, 221, 236exponential inequalities 240, 241exponents (see powers)extraneous solutions 131, 1323, 33940extreme values (extrema) 601, 606, 753

Ffactor theorem 11621factorial notation 177factorization (algebra) 278, 3031 quadratic functions 103, 108, 110Fermat, Pierre 35, 753Fermats last theorem 35 Fermats principle (optics) 459Ferrari, Lodovico 121 Fibonacci 153 Fibonacci sequences 153, 969finite sets 2first fundamental theorems of integral calculus 804first-order differential equations 836five-number summaries 491Fontana, Niccolo 121 forces 41214

formalism 955, 967formulae 36fractals 428frequency distributions 468, 856frequency tables 4689, 483function notation (derivatives) 703, 704functions 4755 antiderivatives 7714 average values 799, 800 composite functions 5760, 61 derivatives 5834, 599612 graphs 7084 integrals 77480 inverse functions 618 limits 57379fundamental principle of counting 1756fundamental theorem of algebra 121fundamental theorems of calculus 798, 8015

GGalileo 961Galois, Evariste 121Gauss, Carl Friedrich 429, 431GaussJordan elimination 26973, 679GDC (see graphical display calculator)general form, quadratic functions 100, 109general solutions 315 geometric probability distribution 870geometric sequences 15862geometric series 16872, 547geometry 952Gibbs, William 399 Gdel, Kurt 960golden ration 9679grade-point averages 860gradients (see slopes)graphical display calculator (GDC) data 470, 4889, 493 derivatives 58992 equation solver 136 functions 51, 545 inequalities 141 integrals 813 limitations 316 limits 573, 575, 5789 logarithms 227

1039

random variables 864, 866, 8734, 907, 910, 911 simultaneous equations 412 trigonometric functions 290, 298, 342graphical method simultaneous equations 412 trigonometric equations 31819graphs equations 369 functions 7084gravity 47, 832Greek letters 326 grouped data 4945groups 957growth factor 212

Hhalf-angle identities 335 half-lives 212half-open intervals 9, 10haphazard events 517health-adjusted life expectancy (HALE) 463, 483Hersch, Robert 9657higher derivatives 714Hilbert, David 960histograms 46970, 482, 483, 890horizontal asymptotes 53, 812, 127, 128, 130, 209,

211, 573horizontal shrinking 778, 79, 80, 84, 306, 310, 311horizontal stretching 789, 80, 84, 308horizontal translations 72, 73, 84, 306, 307, 308, 309,

311hyperbolic geometry 959, 960hypotenuses 18, 350

Iidentities 35, 314 proof 314, 332 identity function 62, 70identity matrices 252images (functions) 48, 53imaginary numbers 3, 29, 106, 119, 429 imaginary parts 431, 444, 457imaginary unit 430implicit definitions 729implicit differentiation 73034, 740, 746, 747, 748,

749, 750

incompleteness theorem (Gdel) 960increasing functions 64, 65, 600indefinite integrals (see also integrals) 773 independent events 539, 540, 5458independent variables 47, 49, 51indeterminate forms 575India 4 indices (singular index) 15, 20induction (see mathematical induction)inequalities 79, 13843 quadratic inequalities 138-9inferential statistics 516infinite series 16970, 572infinite sets 2inflexion points 61112, 737initial conditions 836instantaneous velocities 571, 581, 604, 605integers 3integrals 774801 areas in graphs 7957, 80915 definite integrals 7979, 8013, 825, 827 first fundamental theorem 8034 parts method 7816 second fundamental theorem 8045 substitution method 77680, 8067 trigonometric functions 78793 volumes 81925integrands 773, 797integration (by partial fractions) 80911interquartile ranges (IQRs) 491, 492, 493, 496, 896,

911intersections (geometry) line and line 3845, 65960, 661 line and plane 695 plane and plane 6836intersections (probabilities) 537, 546intersections (sets) 6, 7, 8 intervals (data) 4689intervals (real numbers) 910intuition 9634invariants 952inverse functions 618, 225 graph 656inverse matrices 25862, 2679inverse normal distribution 90912inverse operations (see also additive inverses,

multiplicative inverses) 62

1040

Index

inverse square function 70inverse trigonometric functions 335 derivatives 73941irrational exponents 208irrational numbers 3, 4, 16

JJorgensen, Palle 962

Kkinematics 826

LLaplace, Pierre-Simon 225 law of cosines 37880, 390, 421, 637, 682law of sines 37072, 3736, 387, 388laws of exponents 20, 207, 208, 228leading coefficient, polynomial functions 91Leibniz, Gottfried 91, 588, 707 Leibniz notation (derivatives) 588, 599, 606, 703lie detectors 556light year 24like terms 26limited growth 843limits 5729 trigonometric functions 595limits of integration 797line segments 350, 656linear equations 379, 2567, 26973 simultaneous equations 414linear expressions 26 linear functions 91lines of sight 354Lobachevsky, Nicolai 959logarithmic equations 226, 23840logarithmic functions 22530 common 227 derivatives 73539 natural logarithmic function 227, 231logarithmic inequalities 24041logical statements 520long division (polynomial functions) 11315, 116long-term relative frequencies 519lower bounds, functions 211lower fence (data) 492, 501

MMaclaurin, Colin 299 magnitude, vectors 399, 409, 416, 423, 628, 631, 632major arcs 281many-to-one functions 645mapping notation 53, 953mathematical induction 1919, 450mathematical truth 956matrices 24653 applications 258, 2624, 269-74 inverse matrices 25862, 2679 multiplication 25053matrix methods, simultaneous equations 258, 269

73maximum area 7534, 7567maximum values 99, 600, 6012, 605, 61112mean value theorem 773means, random variables (see expected values)medians data 480, 481, 484, 490, 498, 500501 random variables 896midpoints (lines) 40minimum length 7556minimum time 7578minimum values 99, 600, 6012, 61112minor arcs 281modes 480, 483, 484, 501, 894moduli (complex numbers) 4412, 443monomials 26 monotone functions 64 Monty Hall game 9645Moon 343multiplication complex numbers 4323, 4456 matrices 25053multiplication rule (probabilities) 531, 532, 53840,

542, 541, 545multiplicative identities 12, 433multiplicative inverses 12, 71 matrices 25860mutually exclusive (disjoint) events 527, 528, 531,

538, 539, 540, 553, 555, 556

NNapier, John 225 natural exponential function 22122

1041

natural logarithmic function 227, 736natural numbers 3negative binomial distribution 870negatively skewed distributions 483nested function notation 703Newton, Isaac 20, 299, 588, 700 Newton notation (derivatives) 599, 606Newtons Law of Cooling 727, 838non-Euclidean geometry 959normal distribution 902913normal lines 61519normal vectors 67071numerical analysis 121numerical (quantitative) data 466

Ooblique asymptotes 130odd functions 303, 323ogives 470, 494one-to-one correspondences 3one-to-one functions 634, 65open intervals 9, 10optimization 75359order relations 78ordered pairs 6, 37, 48oscilloscopes 279Oughtred, William 225 outcomes (probabilities) 517, 520outliers 482, 487, 492, 493

Pparabolas 99100, 1012, 107, 5845, 802, 910parallel lines 39, 659parallel vectors 676 parallelograms 404, 405, 6478, 676parameters (algebra) 36, 257parameters (data) 480, 481parametric equations 2567 lines 654, 6557, 658 planes 6756, 677, 684partial fractions 1448, 80911partial sums 164parts method, integration 7816Pascals rule 186Pascals triangle 1847, 196Pearson, Karl 518

pendulums 467percentiles 490, 896, 897, 909periodic functions 289, 293periods 46, 47, 289, 301permutations 1769perpendicular lines 38, 617, 660perpendicular vectors 420, 638phase shifts 307, 308, 309 (pi) (number) 220pie charts 466planes 67086Plato 954, 955Platonist view 954, 955, 967Poisson distribution 8815 cumulative distribution function 8857polynomial equations 3, 91, 1224, 4378polynomial expressions 2631polynomial functions 917, 11221 complex zeros 108, 119, 121, 4378 derivatives 7023, 709 factor theorem 104, 11621, 122 graphs 924 long division 11315 remainder theorem 11516 synthetic substitution 947, 116point-slope form, linear equations 37population growth 162, 727, 838, 841populations (statistics) 4645, 480position 827, 828position functions 831, 832position vectors 401, 414, 41516, 628, 664positively skewed distributions 483power functions 207power rule, derivatives 5867, 588, 712, 723 power series 2989powers (exponents) 203, 24 complex numbers 431, 436, 44952primitive concepts 953Principia Mathematica 700, 953, 960Principle of Least Time 753principal square roots 15, 431probabilities 517, 519, 52533, 5378 Bayes theorem 5528 conditional probabilities 54044 independent events 5458probability density functions 89092, 902probability distribution functions 857, 858, 897

1042

Index

probability distributions 85661, 890probability models 519probability tables 522product rule, derivatives 70710, 713, 781projections 420proof 9568proof by contradiction 4 proof by induction 1919, 450proper classes 954proper subsets 5, 7 pure imaginary numbers 430Pythagoras 4 Pythagoras theorem 16, 35, 39, 40, 378Pythagorean identities 3235, 333

QQ.E.D. 324 quadratic equations 37, 103, 4289quadratic expressions 26 quadratic form 1346 exponential equations 2378quadratic formula 103, 1068, 42930quadratic functions 91, 99103, 10610 completing the square 1002 curve fitting 2734 graph 99100, 10810 zeros 1024, 1068quadratic inequalities 1389qualitative data (see categorical data)quantitative data (see numerical data)quartic equations 121, 122, 135quartic expressions 26quartiles 491, 4945, 896, 897, 910quintic equations 121quintic expressions 26 quotient rule, derivatives 71113, 718

Rradians 2824, 298, 314radical expressions 1323radicals (roots) 1619radioactive materials 212, 237, 727, 859radon 859random events 517, 518random experiments 519random variables 8456 expected value 8613, 864

standard deviation 863, 865 variance 863, 864, 866ranges (data) 487, 4912ranges (functions) 48, 51, 52, 53, 545 composite functions 5960 exponential functions 209 inverse functions 62, 63, 65, 66, 678, 225 inverse trigonometric functions 336, 338, 339, 340 logarithmic functions 225 rational functions 128, 129 trigonometric functions 293, 308rates of change (see also derivatives) 21718, 5923,

701, 74551rational exponents 213, 207rational expressions (see algebraic fractions)rational functions 12630rational inequalities 1423rational numbers 3, 4, 5, 16rationalizing a denominator 1718 algebraic fractions 334rationalizing a numerator 1819real numbers 3, 4, 6, 12, 15 real parts 431, 444, 457reciprocal function 70, 81reciprocals (see multiplicative inverses)recursive definitions 152, 153reduced row echelon form (matrices) 273reductio ad absurdum 4 reflections 745, 84, 210, 446 relations 49, 731relative cumulative frequency distributions 470relative frequency distributions 525, 526, 861relative maximum/minimum values 601, 603, 604,

61112remainder theorem 11516resolving (vectors) 412resultant forces 412, 414, 415right triangles 35052, 3547, 386, 388, 389right-hand rule 646rigid transformations 76roots (equations) (see solutions)roots (numbers) 1415, 16, 21 complex numbers 4525rotational symmetry 3023, 310row vectors 248Russell, Bertrand 953, 954Russells paradox 954

1043

Ssample spaces (probabilities) 519, 520, 521, 526samples (statistics) 464, 48081scalar multiplication (ku) 4023scalar product (u v) 41923, 6379scalar triple product (u ? (v w)) 64850scalars 250, 399scale factors 76, 77, 78, 79scientific notation 245secant function (see trigonometric functions)secant lines 5823, 592second derivatives 606, 60812, 71314second fundamental theorems of integral

calculus 8045sectors (circles) 2856, 366segments (circles) 365semantic method 957separable differential equations 83845separation of variables 840sequences 1514 arithmetic sequences 1556 geometric sequences 15862 sums (see series)series 16472 arithmetic series 1667 geometric series 16872sets 27, 520, 953, 954sexagesimal number system 281 sigma notation 1646significant figures 24similar triangles 351simple events 520simultaneous equations (systems of equations) 40

4, 2578, 26973sine function (see trigonometric functions)sine rule (see law of sines)singular matrices 260skew lines 65961, 6667skewed distributions 4834, 501, 896slide rules 225 slope-intercept form, linear equations 37, 580slopes (gradients) (see also derivatives, rates of

change) 37, 38, 383, 58081Snells law 759socially constructed facts 9656solids of revolution 82021solution sets 35, 36, 48, 256

solutions (roots), equations 35, 92something has to happen rule 526, 527speed 399, 41516, 423, 607, 662, 663, 828spheres 8212square matrices 247square root function 70square roots 1415, 16, 21, 103, 431, 455squaring function 70squeeze theorem 5945standard deviation data 48090, 496, 497, 4989 random variables 863, 865, 887, 897, 898, 903,

909, 912standard form (see scientific notation)standard normal distribution 904, 9056standard position, angles 280standardization, random variables 904, 905, 908stationary points 600605, 608, 610statistics 480, 481, stem-and-leaf diagrams 467Stewart, Ian 969strep throat 236subsets 5, 7 substitution method 41 integration 77680, 8067 simultaneous equations 434subtraction 12sum rule, derivatives 587, 588Sun 343supplementary angles 363 surds (see radicals)symmetric distributions 483 matrices 249symmetry 483, 961syntactic method 957synthetic division 116, 117 substitution 947, 116systems of equations (see simultaneous equations)

Ttangent function (see trigonometric functions)tangent lines 58084, 615, 716Taylor, Brook 299, 782terminal velocities 832Thales of Miletus 351

1044

Index

theorems 956three-dimensional structures 38890total change 804transcendental functions 736 numbers 736transformations, graphs of functions 70, 7184,

20911transitive property 8translations 714, 99tree diagrams 522, 8712triangles area 2623, 3646 construction 361, 366, 36970 law of cosines 37680 law of sines 36976 similar triangles 351 right triangles 280, 350, 3547triangular matrices 249trigonometric equations 31421trigonometric form (complex numbers) 432, 4436trigonometric functions 28899, 3514, 3616 derivatives 594, 719, 73943 graphs 301311 integration 78793 inverse 335344, 73943 limits 594trigonometric identities 32233, 787trinomials 26two-point equation, lines 264

Uunbounded intervals 9, 10uniformly accelerated motion 8324unions (probabilities) 537, 546unions (sets) 6, 7, 9unit circle 2824, 28899unit vectors 405, 40916, 6313universal quantifier 197 universal sets 5, 6, 7, 96upper bounds, functions 211upper fence (data) 492, 501

Vvariables 465 classification 4668

random 85466variance 4878, 8636, 871, 8978vector equations lines 633, 6546, 660, 661 planes 672, 673vectors 398425, 62688 addition 398, 4035, 406 applications 408, 41012 direction angles/cosines 40912, 63941 products (u v) 644-50 scalar multiplication (ku) 4023 scalar product (u v) 41923, 445, 6714 scalar triple product (u ? (v w)) 64850 unit vectors 405, 40916, 6313velocity 399, 41216, 5712, 5812, 6049, 828, 829,

832velocitytime graphs 829Venn diagrams 6, 520, 526, 546vertex form, quadratic functions 100vertical asymptotes 53, 55, 812, 127, 575vertical shrinking 77, 305vertical stretching 77, 305vertical translations 71, 73, 81, 84, 303, 587vertices (singular vertex), graphs 99, 100, 101, 108,

109, 583volumes 649, 81925

Wwashers 8235water molecules 24wheat and chessboard problem 25 Whitehead, A.N. 953Wiles, Andrew 35 work (forces) 424 World Health Organization (WHO) 4634wrapping functions 2889

Zz-scores 905, 910, 91113zero matrices 248 vector 401, 628zeros polynomial functions 92, 96, 99, 1023, 106, 112

21, 4378 complex 108, 4323 multiplicity 106, 107