Mathematics: Applications and Interpretation HL 1 EXPONENTIALS 13 A Rational exponents 14 B Algebraic expansion and factorisation 16 C Exponential functions 19 D Graphing exponential functions from a table of values 20 E Graphs of exponential functions 21 F Exponential equations 25 G Growth and decay 27 H The natural exponential 34 I The logistic model 39 Review set 1A 40 Review set 1B 42 2 LOGARITHMS 45 A Logarithms in base 1010 46 B Laws of logarithms 49 C Natural logarithms 52 D Logarithmic functions 56 E Logarithmic scales 59 Review set 2A 64 Review set 2B 65 3 APPROXIMATIONS AND ERROR 67 A Errors in measurement 68 B Absolute and percentage error 71 Review set 3A 75 Review set 3B 75 4 LOANS AND ANNUITIES 77 A Loans 78 B Annuities 84 Review set 4A 88 Review set 4B 89 5 MODELLING 91 A The modelling cycle 92 B Linear models 98 C Piecewise models 101 D Systems of equations 108 Review set 5A 111 Review set 5B 114 6 DIRECT AND INVERSE VARIATION 117 A Direct variation 118 B Powers in direct variation 122 C Inverse variation 123 D Powers in inverse variation 126 E Determining the variation model 127 F Using technology to find variation models 129 Review set 6A 131 Review set 6B 133 7 BIVARIATE STATISTICS 135 A Association between numerical variables 136 B Pearson's product-moment correlation coefficient 141 C The coefficient of determination 146 D Line of best fit by eye 148 E The least squares regression line 152 F Statistical reliability and validity 160 G Spearman's rank correlation coefficient 164 Review set 7A 169 Review set 7B 172 8 NON-LINEAR MODELLING 175 A Logarithmic models 176 B Exponential models 178 C Power models 181 D Problem solving 184 E Non-linear regression 186 Review set 8A 189 Review set 8B 191 9 VECTORS 193 A Vectors and scalars 194 B Geometric operations with vectors 197 C Vectors in the plane 202 D The magnitude of a vector 204 E Operations with plane vectors 205 F Vectors in space 209 G Operations with vectors in space 211 H The vector between two points 212 I Parallelism 217 J The scalar product of two vectors 220 K The angle between two vectors 222 L The vector product of two vectors 227 M Vector components 234 Review set 9A 237 Review set 9B 239 10 VECTOR APPLICATIONS 241 A Lines in 22 and 33 dimensions 242 B The angle between two lines 246 C Constant velocity problems 248 D The shortest distance from a point to a line 251 E The shortest distance between two objects 254 F Intersecting lines 256 Review set 10A 261 Review set 10B 263 11 COMPLEX NUMBERS 265 A Real quadratics with \Delta < 0Δ<0 266 B Complex numbers 268 C Operations with complex numbers 269 D Equality of complex numbers 272 E The complex plane 273 F Modulus and argument 276 G Geometry in the complex plane 279 H Polar form 282 I Exponential form 290 J Frequency and phase 292 Review set 11A 293 Review set 11B 295 12 MATRICES 297 A Matrix structure 298 B Matrix equality 300 C Addition and subtraction 301 D Scalar multiplication 303 E Matrix algebra 305 F Matrix multiplication 307 G The inverse of a matrix 314 H Simultaneous linear equations 318 Review set 12A 323 Review set 12B 325 13 EIGENVALUES AND EIGENVECTORS 327 A Eigenvalues and eigenvectors 328 B Matrix diagonalisation 333 C Matrix powers 336 D Markov chains 338 Review set 13A 349 Review set 13B 351 14 AFFINE TRANSFORMATIONS 353 A Translations 355 B Rotations about the origin 356 C Reflections 359 D Stretches 361 E Enlargements 362 F Composite transformations 363 G Area 367 Review set 14A 372 Review set 14B 373 15 GRAPH THEORY 375 A Graphs 376 B Properties of graphs 380 C Routes on graphs 383 D Adjacency matrices 385 E Transition matrices for graphs 392 F Trees 394 G Minimum spanning trees 395 H Eulerian graphs 401 I The Chinese Postman Problem 404 J Hamiltonian graphs 406 K The Travelling Salesman Problem 409 Review set 15A 416 Review set 15B 419 16 VORONOI DIAGRAMS 423 A Voronoi diagrams 424 B Constructing Voronoi diagrams 428 C Adding a site to a Voronoi diagram 433 D Nearest neighbour interpolation 437 E The Largest Empty Circle problem 439 Review set 16A 443 Review set 16B 445 17 INTRODUCTION TO DIFFERENTIAL CALCULUS 447 A Rates of change 449 B Instantaneous rates of change 452 C Limits 455 D The gradient of a tangent 460 E The derivative function 462 F Differentiation from first principles 464 Review set 17A 466 Review set 17B 467 18 RULES OF DIFFERENTIATION 469 A Simple rules of differentiation 470 B The chain rule 474 C The product rule 477 D The quotient rule 479 E Derivatives of exponential functions 482 F Derivatives of logarithmic functions 485 G Derivatives of trigonometric functions 487 H Second derivatives 490 Review set 18A 492 Review set 18B 493 19 PROPERTIES OF CURVES 495 A Tangents 496 B Normals 502 C Increasing and decreasing 503 D Stationary points 508 E Shape 513 F Inflection points 515 Review set 19A 519 Review set 19B 521 20 APPLICATIONS OF DIFFERENTIATION 523 A Rates of change 524 B Optimisation 529 C Modelling with calculus 537 D Related rates 540 Review set 20A 543 Review set 20B 545 21 INTRODUCTION TO INTEGRATION 547 A Approximating the area under a curve 548 B The Riemann integral 552 C Antidifferentiation 556 D The Fundamental Theorem of Calculus 558 Review set 21A 563 Review set 21B 564 22 TECHNIQUES FOR INTEGRATION 565 A Discovering integrals 566 B Rules for integration 568 C Particular values 572 D Integrating f(ax+b)f(ax+b) 574 E Integration by substitution 577 Review set 22A 580 Review set 22B 581 23 DEFINITE INTEGRALS 583 A Definite integrals 584 B Definite integrals involving substitution 587 C The area under a curve 588 D The area above a curve 592 E The area between a curve and the yy-axis 596 F Solids of revolution 598 G Problem solving by integration 602 Review set 23A 604 Review set 23B 606 24 KINEMATICS 609 A Displacement 611 B Velocity 613 C Acceleration 619 D Speed 623 E Velocity and acceleration in terms of displacement 626 F Motion with variable velocity 628 G Projectile motion 634 Review set 24A 638 Review set 24B 640 25 DIFFERENTIAL EQUATIONS 643 A Differential equations 644 B Solutions of differential equations 646 C Differential equations of the form \frac{dy}{dx}=f(x) ​dx ​ ​dy ​​ =f(x) 649 D Separable differential equations 652 E Slope fields 658 F Euler's method for numerical integration 661 Review set 25A 666 Review set 25B 667 26 COUPLED DIFFERENTIAL EQUATIONS 669 A Phase portraits 671 B Coupled linear differential equations 678 C Second order differential equations 686 D Euler's method for coupled equations 688 Review set 26A 693 Review set 26B 694 27 DISCRETE RANDOM VARIABLES 697 A Random variables 698 B Discrete probability distributions 700 C Expectation 703 D Variance and standard deviation 709 E Properties of aX + baX+b 711 F The binomial distribution 714 G Using technology to find binomial probabilities 718 H The mean and standard deviation of a binomial distribution 721 I The Poisson distribution 722 Review set 27A 726 Review set 27B 728 28 THE NORMAL DISTRIBUTION 731 A Introduction to the normal distribution 733 B Calculating probabilities 736 C The standard normal distribution 743 D Quantiles 748 Review set 28A 751 Review set 28B 753 29 ESTIMATION AND CONFIDENCE INTERVALS 755 A Linear combinations of random variables 756 B The sum of two independent Poisson random variables 759 C Linear combinations of normal random variables 760 D The Central Limit Theorem 762 E Confidence intervals for a population mean with known variance 768 F Confidence intervals for a population mean with unknown variance 776 Review set 29A 779 Review set 29B 781 30 HYPOTHESIS TESTING 783 A Statistical hypotheses 785 B The ZZ-test 787 C Critical values and critical regions 794 D Student's tt-test 797 E Paired tt-tests 800 F The two-sample tt-test for comparing population means 803 G Hypothesis tests for the mean of a Poisson population 805 H Hypothesis tests for a population proportion 809 I Hypothesis tests for a population correlation coefficient 813 J Error probabilities and statistical power 817 Review set 30A 824 Review set 30B 826 31 \chi^2χ ​2 ​​ HYPOTHESIS TESTS 829 A The \chi^2χ ​2 ​​ goodness of fit test 830 B Estimating distribution parameters in a goodness of fit test 838 C Critical regions and critical values 842 D The \chi^2χ ​2 ​​ test for independence 844 Review set 31A 851 Review set 31B 853 ANSWERS 855 INDEX 967