Mathematics for the International Student: Mathematics HL (Core)

SYMBOLS AND NOTATION USED IN THIS BOOK 10 BACKGROUND KNOWLEDGE 16 - A Surds and radicals CD - B Scientific notation (standard form) CD - C Number systems and set notation CD - D Algebraic simplification CD - E Linear equations and inequalities CD - F Modulus or absolute value CD - G Product expansion CD - H Factorisation CD - I Formula rearrangement CD - J Adding and subtracting algebraic fractions CD - K Congruence and similarity CD - L Pythagoras’ theorem CD - M Coordinate geometry CD - N Right angled triangle trigonometry CD - Matrices CD - Statistics revision CD - Facts about number sets CD - Summary of circle properties CD - Summary of measurement facts CD GRAPHICS CALCULATOR INSTRUCTIONS 16 - Casio fx-9860G PLUS CD - Casio fx-CG20 CD - Texas Instruments TI-84 Plus CD - Texas Instruments TI-nspire CD 1 QUADRATICS 17 - A Quadratic equations 19 - B The discriminant of a quadratic 25 - C The sum and product of the roots 27 - D Quadratic functions 28 - E Finding a quadratic from its graph 37 - F Where functions meet 40 - G Problem solving with quadratics 43 - H Quadratic optimisation 45 - Review set 1A 47 - Review set 1B 48 - Review set 1C 49 2 FUNCTIONS 51 - A Relations and functions 52 - B Function notation 55 - C Domain and range 57 - D Composite functions 62 - E Even and odd functions 64 - F Sign diagrams 66 - G Inequalities (inequations) 70 - H The modulus function 73 - I Rational functions 78 - J Inverse functions 81 - K Graphing functions 87 - L Finding where graphs meet 89 - Review set 2A 90 - Review set 2B 91 - Review set 2C 93 3 EXPONENTIALS 95 - A Exponents 96 - B Laws of exponents 98 - C Rational exponents 101 - D Algebraic expansion and factorisation 104 - E Exponential equations 106 - F Exponential functions 108 - G Growth and decay 112 - H The natural exponential e^x 115 - Review set 3A 119 - Review set 3B 120 - Review set 3C 121 4 LOGARITHMS 123 - A Logarithms in base 10 124 - B Logarithms in base a 127 - C Laws of logarithms 130 - D Natural logarithms 134 - E Exponential equations using logarithms 137 - F The change of base rule 139 - G Graphs of logarithmic functions 141 - H Growth and decay 145 - Review set 4A 147 - Review set 4B 148 - Review set 4C 149 5 TRANSFORMING FUNCTIONS 151 - A Transformation of graphs 152 - B Translations 154 - C Stretches 156 - D Reflections 157 - E Miscellaneous transformations 159 - F Simple rational functions 161 - G The reciprocal of a function 165 - H Modulus functions 166 - Review set 5A 167 - Review set 5B 169 - Review set 5C 170 6 COMPLEX NUMBERS AND POLYNOMIALS 173 - A Real quadratics with ¢ < 0 174 - B Complex numbers 176 - C Real polynomials 183 - D Zeros, roots, and factors 189 - E Polynomial theorems 193 - F Graphing real polynomials 201 - Review set 6A 209 - Review set 6B 210 - Review set 6C 211 7 SEQUENCES AND SERIES 213 - A Number sequences 214 - B The general term of a number sequence 215 - C Arithmetic sequences 218 - D Geometric sequences 222 - E Series 229 - F Arithmetic series 231 - G Geometric series 233 - Review set 7A 240 - Review set 7B 240 - Review set 7C 242 8 COUNTING AND THE BINOMIAL EXPANSION 243 - A The product principle 244 - B Counting paths 246 - C Factorial notation 247 - D Permutations 250 - E Combinations 254 - F Binomial expansions 257 - G The binomial theorem 260 - Review set 8A 263 - Review set 8B 263 - Review set 8C 264 9 MATHEMATICAL INDUCTION 265 - A The process of induction 267 - B The principle of mathematical induction 269 - Review set 9A 277 - Review set 9B 277 - Review set 9C 278 10 THE UNIT CIRCLE AND RADIAN MEASURE 279 - A Radian measure 280 - B Arc length and sector area 283 - C The unit circle and the trigonometric ratios 286 - D Applications of the unit circle 292 - E Negative and complementary angle formulae 295 - F Multiples of π/6 and π/4 297 301 302 303 - Review set 10A 301 - Review set 10B 302 - Review set 10C 303 11 NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY 305 - A Areas of triangles 306 - B The cosine rule 309 - C The sine rule 313 - D Using the sine and cosine rules 317 - Review set 11A 321 - Review set 11B 323 - Review set 11C 323 12 TRIGONOMETRIC FUNCTIONS 325 - A Periodic behaviour 326 - B The sine function 330 - C Modelling using sine functions 336 - D The cosine function 339 - E The tangent function 341 - F General trigonometric functions 344 - G Reciprocal trigonometric functions 346 - H Inverse trigonometric functions 348 - Review set 12A 350 - Review set 12B 351 - Review set 12C 351 13 TRIGONOMETRIC EQUATIONS AND IDENTITIES 353 - A Trigonometric equations 354 - B Using trigonometric models 362 - C Trigonometric relationships 364 - D Double angle formulae 367 - E Compound angle formulae 370 - F Trigonometric equations in quadratic form 376 - G Trigonometric series and products 377 - Review set 13A 379 - Review set 13B 380 - Review set 13C 382 14 VECTORS 383 - A Vectors and scalars 384 - B Geometric operations with vectors 387 - C Vectors in the plane 394 - D The magnitude of a vector 396 - E Operations with plane vectors 398 - F The vector between two points 401 - G Vectors in space 404 - H Operations with vectors in space 408 - I Parallelism 412 - J The scalar product of two vectors 416 - K The vector product of two vectors 422 - Review set 14A 428 - Review set 14B 430 - Review set 14C 431 15 VECTOR APPLICATIONS 433 - A Problems involving vector operations 434 - B Area 436 - C Lines in 2-D and 3-D 437 - D The angle between two lines 442 - E Constant velocity problems 444 - F The shortest distance from a line to a point 447 - G Intersecting lines 451 - H Relationships between lines 453 - I Planes 460 - J Angles in space 465 - K Intersecting planes 467 - Review set 15A 472 - Review set 15B 474 - Review set 15C 476 16 COMPLEX NUMBERS 479 - A Complex numbers as 2-D vectors 480 - B Modulus 483 - C Argument and polar form 487 - D Euler’s form 495 - E De Moivre’s theorem 497 - F Roots of complex numbers 500 - G Miscellaneous problems 504 - Review set 16A 504 - Review set 16B 505 - Review set 16C 506 17 INTRODUCTION TO DIFFERENTIAL CALCULUS 507 - A Limits 509 - B Limits at infinity 512 - C Trigonometric limits 515 - D Rates of change 518 - E The derivative function 521 - F Differentiation from first principles 523 - Review set 17A 526 - Review set 17B 526 - Review set 17C 527 18 RULES OF DIFFERENTIATION 529 - A Simple rules of differentiation 530 - B The chain rule 534 - C The product rule 537 - D The quotient rule 540 - E Implicit differentiation 542 - F Derivatives of exponential functions 544 - G Derivatives of logarithmic functions 549 - H Derivatives of trigonometric functions 551 - I Derivatives of inverse trigonometric functions 555 - J Second and higher derivatives 557 - Review set 18A 559 - Review set 18B 560 - Review set 18C 561 19 PROPERTIES OF CURVES 563 - A Tangents and normals 564 - B Increasing and decreasing functions 570 - C Stationary points 575 - D Inflections and shape 579 - Review set 19A 587 - Review set 19B 588 - Review set 19C 589 20 APPLICATIONS OF DIFFERENTIAL CALCULUS 591 - A Kinematics 592 - B Rates of change 601 - C Optimisation 606 - D Related rates 617 - Review set 20A 621 - Review set 20B 623 - Review set 20C 625 21 INTEGRATION 627 - A The area under a curve 628 - B Antidifferentiation 634 - C The fundamental theorem of calculus 635 - D Integration 640 - E Rules for integration 643 - F Integrating f(ax + b) 648 - G Integration by substitution 653 - H Integration by parts 659 - I Miscellaneous integration 660 - J Definite integrals 661 - Review set 21A 667 - Review set 21B 668 - Review set 21C 669 22 APPLICATIONS OF INTEGRATION 671 - A The area under a curve 672 - B The area between two functions 675 - C Kinematics 681 - D Problem solving by integration 687 - E Solids of revolution 690 - Review set 22A 697 - Review set 22B 699 - Review set 22C 701 23 DESCRIPTIVE STATISTICS 703 - A Key statistical concepts 704 - B Measuring the centre of data 709 - C Variance and standard deviation 721 - Review set 23A 728 - Review set 23B 729 - Review set 23C 731 24 PROBABILITY 733 - A Experimental probability 735 - B Sample space 740 - C Theoretical probability 741 - D Tables of outcomes 745 - E Compound events 747 - F Tree diagrams 751 - G Sampling with and without replacement 754 - H Sets and Venn diagrams 757 - I Laws of probability 763 - J Independent events 767 - K Probabilities using permutations and combinations - L Bayes’ theorem - Review set 24A 774 - Review set 24B 775 - Review set 24C 776 25 DISCRETE RANDOM VARIABLES 779 - A Discrete random variables 780 - B Discrete probability distributions 782 - C Expectation 786 - D Variance and standard deviation 790 - E Properties of E(X) and Var(X) 792 - F The binomial distribution 795 - G The Poisson distribution 805 - Review set 25A 808 - Review set 25B 809 - Review set 25C 811 26 CONTINUOUS RANDOM VARIABLES 813 - A Continuous random variables 814 - B The normal distribution 818 - C Probabilities using a calculator 823 - D The standard normal distribution (Z-distribution) 826 - E Quantiles or k-values 831 - Review set 26A 835 - Review set 26B 836 - Review set 26C 837 27 MISCELLANEOUS QUESTIONS 839 - A Non-calculator questions 840 - B Calculator questions 852 ANSWERS 865 INDEX 958