Paul Urban, John Owen, Robert Haese9781876543099, 1876543094
Table of contents :
Help……Page 0
Mathematics HL (Core)……Page 1
Foreword……Page 3
Using the interactive student cd……Page 5
Table of contents……Page 6
Symbols and notation used in this book……Page 11
Summary of circle properties……Page 13
Summary of measurement facts……Page 14
1. Functions……Page 17
A – Relations and functions……Page 18
Exercise 1A……Page 20
Answers……Page 763
B – Interval notation,domain and range……Page 21
Exercise 1B……Page 22
C – Function notation……Page 23
Exercise 1C……Page 24
Investigation – Fluid filling functions……Page 25
D – Composite functions, f o g……Page 26
Exercise 1D……Page 27
F – Inverse functions……Page 28
G – Functions which have inverses……Page 30
Exercise 1G……Page 32
Answers……Page 764
Review set 1A……Page 33
Review set 1B……Page 34
2. Sequences and series……Page 35
B – Sequences of numbers……Page 36
Answers……Page 765
Opening problem……Page 37
C – Arithmetic sequences……Page 38
Exercise 2C……Page 39
D – Geometric sequences……Page 41
Exercise 2D……Page 42
Exercise 2E.1……Page 47
Exercise 2E.2……Page 48
Exercise 2E.3……Page 50
Exercise 2F……Page 52
Exercise 2G……Page 53
Investigation – Von koch ’s snowflake curve……Page 54
Review set 2B……Page 55
Answers……Page 766
Review set 2C……Page 56
3. Exponents……Page 57
A – Index notation……Page 58
Exercise 3B……Page 59
Exercise 3C……Page 61
D – Rational indices……Page 65
Exercise 3D……Page 66
Exercise 3E……Page 67
Exercise 3F……Page 68
Answers……Page 767
G – Graphs of exponential functions……Page 69
Investigation – Exponential graphs……Page 70
Exercise 3G……Page 71
H – Growth……Page 72
Exercise 3H……Page 73
I – Decay……Page 74
Exercise 3I……Page 75
Review set 3B……Page 77
Answers……Page 768
Review set 3C……Page 78
4. Logarithms……Page 79
A – Introduction……Page 80
Exercise 4A……Page 81
B – Logarithms in base 10……Page 82
Exercise 4B……Page 83
C – Laws of logarithms……Page 84
Exercise 4C……Page 85
D – Exponential equations (using logarithms)……Page 87
E – Growth and decay revisited……Page 88
Answers……Page 769
Exercise 4E……Page 89
G – The change of base rule……Page 90
Exercise 4G……Page 91
H – Graphs of logarithmic functions……Page 92
Exercise 4H……Page 93
Review set 4B……Page 94
5. Natural logarithms……Page 95
Investigation 1 – e occurs naturally……Page 96
Investigation 2 – Continuous compound interest……Page 97
Exercise 5A……Page 98
Answers……Page 770
Exercise 5B……Page 99
Investigation 3 – The laws of natural logarithms……Page 100
Exercise 5C……Page 101
D – Exponential equations involving e……Page 102
E – Growth and decay revisited……Page 103
Exercise 5E……Page 104
Exercise 5F……Page 105
Review set 5B……Page 106
Answers……Page 771
6. Graphing and transforming functions……Page 107
Investigation – Function families……Page 108
Exercise 6A……Page 109
C – Transformations of graphs……Page 110
Exercise 6C.2……Page 111
Answers……Page 772
Exercise 6C.3……Page 112
Answers……Page 774
Exercise 6D……Page 113
E – Simple rational functions……Page 114
Exercise 6E……Page 115
Answers……Page 775
Exercise 6F……Page 118
Review set 6B……Page 120
Answers……Page 776
7. Quadratic equations and functions……Page 121
Exercise 7A……Page 123
Answers……Page 777
B – Graphs of quadratic functions……Page 124
Investigation 2 – Graphing y = a(x-h)²+k……Page 125
Exercise 7B.1……Page 126
Exercise 7B.2……Page 128
C – Completing the square……Page 129
Exercise 7C……Page 130
Answers……Page 778
D – Quadratic equations……Page 131
Exercise 7D.1……Page 132
Exercise 7D.2……Page 135
E – The quadratic formula……Page 136
Exercise 7E……Page 137
Answers……Page 779
Exercise 7F……Page 138
Exercise 7G……Page 139
H – Quadratic graphs (review)……Page 142
Exercise 7H……Page 144
I – The discriminant, Delta……Page 145
Exercise 7I.1……Page 146
Answers……Page 780
J – Determining the quadratic from a graph……Page 148
Exercise 7J……Page 149
K – Where functions meet……Page 151
L – Quadratic modelling……Page 152
Exercise 7L……Page 153
Graphics calculator investigation – Tunnels and trucks……Page 154
Review set 7A……Page 155
Review set 7C, 7D, 7E……Page 156
8. Complex numbers and polynomials……Page 157
A – Solutions of real quadratics with Delta < 0……Page 158
Exercise 8A……Page 159
Answers……Page 781
B – Complex numbers……Page 160
Exercise 8B.1……Page 161
Exercise 8B.2……Page 163
Exercise 8B.3……Page 164
Exercise 8B.4……Page 165
C – Real polynomials……Page 167
Exercise 8C.1……Page 169
Exercise 8C.2……Page 171
Exercise 8C.3……Page 172
D – Roots, zeros and factors……Page 174
Answers……Page 782
Exercise 8D.1……Page 175
Exercise 8D.2……Page 177
Exercise 8D.3……Page 180
Exercise 8D.4……Page 181
Investigation 2 – Cubic graphing……Page 182
Exercise 8E.1……Page 184
Investigation 3 – Quartic graphing……Page 185
Exercise 8E.2……Page 186
Exercise 8E.3……Page 189
Exercise 8F.1……Page 190
Answers……Page 783
Exercise 8F.2……Page 191
G – Inequalities……Page 192
Exercise 8G.1……Page 193
Exercise 8G.2……Page 195
Exercise 8G.3……Page 196
Exercise 8G.4……Page 197
Review set 8B, 8C……Page 198
Answers……Page 784
9. Counting and binomial theorem……Page 199
A – The product principle……Page 200
Exercise 9A……Page 201
Exercise 9B……Page 202
Exercise 9C……Page 203
Exercise 9D……Page 205
E – Combinations……Page 208
Exercise 9E……Page 209
Investigation 2 – The binomial expansion of (a+b)^n, n >= 4……Page 211
Exercise 9F……Page 212
Exercise 9G……Page 214
Answers……Page 785
Review set 9B……Page 216
10. Mathematical induction……Page 217
A – The process of induction……Page 218
Exercise 10A……Page 219
B – The principle of mathematical induction……Page 220
Exercise 10B……Page 221
C – Indirect proof (extension)……Page 226
Review set 10A……Page 227
Review set 10C……Page 228
11. The unit circle and radian measure……Page 229
A – The unit quarter circle……Page 230
Exercise 11A……Page 231
Exercise 11B……Page 232
Exercise 11C……Page 234
Investigation – Parametric equations……Page 235
Exercise 11D.1……Page 236
Exercise 11D.2……Page 238
Exercise 11E.1……Page 239
Answers……Page 786
Exercise 11E.2……Page 242
F – Areas of triangles……Page 244
Exercise 11F……Page 245
G – Sectors and segments……Page 246
Exercise 11G……Page 247
Review set 11B……Page 249
Review set 11C……Page 250
12. Non right angled triangle trigonometry……Page 251
A – The cosine rule……Page 252
Exercise 12A……Page 253
B – The sine rule……Page 254
Investigation – The ambiguous case……Page 255
C – Using the sine and cosine rules……Page 258
Answers……Page 787
Exercise 12C……Page 259
Review set 12……Page 261
13. Periodic phenomena……Page 263
Opening problem……Page 264
A – Observing periodic behaviour……Page 265
Activity – Bicycle data……Page 266
Exercise 13A……Page 267
B – The sine function……Page 269
Investigation 2 – The family y = sin Bx, B > 0……Page 270
Exercise 13B.1……Page 271
Investigation 3 – The families y = sin(x-C) and y = sin x – D……Page 272
Exercise 13B.2……Page 273
Answers……Page 788
C – Modelling using sine functions……Page 274
Exercise 13C……Page 276
Exercise 13D.1……Page 277
Exercise 13D.2……Page 278
Exercise 13D.3……Page 280
Exercise 13D.4……Page 281
Answers……Page 789
E – The cosine function……Page 282
F – Solving cosine equations……Page 283
Exercise 13F……Page 284
Exercise 13G.1……Page 285
Answers……Page 790
Investigation 4 – Negative and complementary angle formulae……Page 288
Investigation 5 – Compound angle formulae……Page 289
Exercise 13H……Page 290
Exercise 13I……Page 293
J – The tangent function……Page 295
Exercise 13J.2……Page 296
Exercise 13K.1……Page 297
Answers……Page 791
Exercise 13K.2……Page 298
Exercise 13M……Page 300
Exercise 13N……Page 301
Exercise 13O……Page 302
Review set 13A……Page 303
Review set 13C, 13D……Page 304
Answers……Page 792
14. Matrices……Page 305
A – Introduction……Page 306
B – Addition and subtraction of matrices……Page 308
Exercise 14B……Page 310
C – Multiples of matrices……Page 311
Exercise 14C……Page 312
D – Matrix algebra for addition……Page 313
E – Matrix multiplication……Page 314
Answers……Page 793
Exercise 14E.1……Page 315
Exercise 14E.2……Page 317
F – Using technology……Page 318
Exercise 14F……Page 319
Exercise 14G……Page 321
H – The inverse of a 2 x 2 matrix……Page 324
Exercise 14H……Page 325
I – Solving a pair of linear equations……Page 326
Exercise 14I……Page 327
Answers……Page 794
J – The 3 x 3 determinant……Page 329
Exercise 14J……Page 330
K – The inverse of a 3 x 3 matrix……Page 331
Exercise 14L……Page 332
Investigation – Using matrices in cryptography……Page 335
M – Solving using row operations……Page 336
Exercise 14M.1……Page 338
Answers……Page 795
Exercise 14M.2……Page 342
Exercise 14M.3……Page 343
Exercise 14M.4……Page 346
Review set 14A……Page 348
Answers……Page 796
Review set 14B……Page 349
Review set 14C, 14D, 14E……Page 350
15. Vectors in 2 and 3 dimensions……Page 351
Opening problem……Page 352
Exercise 15A.1……Page 354
B – Operations with vectors……Page 355
Exercise 15B.1……Page 356
Exercise 15B.2……Page 358
Answers……Page 797
C – 2-D vectors in component form……Page 362
Exercise 15C.1……Page 363
Exercise 15C.2……Page 364
Exercise 15C.4……Page 366
Answers……Page 798
D – 3-D coordinate geometry……Page 367
Exercise 15D……Page 368
E – 3-D vectors in component form……Page 369
Exercise 15E.1……Page 370
Exercise 15E.2……Page 371
F – Algebraic operations with vectors……Page 372
Exercise 15F……Page 373
Exercise 15G……Page 375
Exercise 15H……Page 378
Exercise 15I……Page 379
Answers……Page 799
J – The scalar product of two vectors……Page 381
Exercise 15J.1……Page 383
Exercise 15J.2……Page 386
K – The vector product of two vectors……Page 387
Exercise 15K.1……Page 388
Exercise 15K.2……Page 392
Exercise 15K.3……Page 394
Review set 15A……Page 395
Review set 15B……Page 397
Answers……Page 800
Review set 15D……Page 398
16. Complex numbers……Page 399
A – Complex numbers as 2-D vectors……Page 400
Exercise 16A.1……Page 401
Exercise 16A.2……Page 402
Answers……Page 801
Exercise 16B.1……Page 403
Exercise 16B.2……Page 406
Exercise 16B.3……Page 408
Exercise 16B.4……Page 410
Exercise 16B.5……Page 413
Exercise 16B.6……Page 414
C – De Moivre’s theorem……Page 415
Exercise 16C……Page 416
Exercise 16D……Page 418
Answers……Page 802
Exercise 16E……Page 420
Review set 16B……Page 421
Review set 16C……Page 422
17. Lines and planes in space……Page 423
A – Lines in a plane and in space……Page 425
Exercise 17A.1……Page 426
Exercise 17A.2……Page 428
Exercise 17A.3……Page 430
Exercise 17B.1……Page 431
Exercise 17B.2……Page 433
Answers……Page 803
Investigation – The two yachts problem……Page 434
Exercise 17B.3……Page 435
Exercise 17B.4……Page 438
C – Relationship between lines……Page 439
Exercise 17C……Page 441
D – Planes and distances……Page 444
Exercise 17D……Page 445
Answers……Page 804
E – Angles in space……Page 449
Exercise 17E……Page 450
F – The intersection of two or more planes……Page 451
Exercise 17F……Page 453
Review set 17A……Page 454
Review set 17B……Page 455
Review set 17C……Page 456
18. Descriptive statistics……Page 457
The pea problem……Page 458
A – Continuous numerical data and histograms……Page 459
Case study – Driving a golf ball……Page 460
Exercise 18A……Page 462
Answers……Page 805
B – Measuring the centre of data……Page 463
Investigation – Merits of the mean and median……Page 465
Exercise 18B.1……Page 466
Exercise 18B.2……Page 470
Exercise 18B.3……Page 473
C – Cumulative data……Page 474
Exercise 18C……Page 475
D – Measuring the spread of data……Page 476
Exercise 18D.1……Page 478
Answers……Page 806
Exercise 18D.2……Page 480
E – Statistics using technology……Page 483
Exercise 18E.2……Page 484
F – Variance and standard deviation……Page 485
Exercise 18F……Page 486
Answers……Page 807
G – The significance of standard deviation……Page 490
Review set 18A……Page 491
Review set 18B……Page 492
19. Probability……Page 493
Opening problem……Page 495
Investigation 1 – Tossing drawing pins……Page 496
Investigation 2 – Coin tossing experiments……Page 497
Answers……Page 808
Investigation 3 – Dice rolling experiments……Page 498
B – Sample space……Page 500
C – Theoretical probability……Page 501
Exercise 19C……Page 503
Exercise 19D……Page 504
Investigation 5 – Revisiting drawing pins……Page 505
Exercise 19E.1……Page 506
Exercise 19E.2……Page 508
F – Using tree diagrams……Page 509
Exercise 19F……Page 510
Answers……Page 809
G – Sampling with and without replacement……Page 511
Exercise 19G……Page 512
Investigation 6 – Sampling simulation……Page 513
H – Binomial probabilities……Page 514
Exercise 19H……Page 515
I – Sets and venn diagrams……Page 516
Exercise 19I……Page 518
J – Laws of probability……Page 521
Exercise 19J……Page 523
K – Independent events revisited……Page 525
L – Probabilities using permutations and combinations……Page 526
Answers……Page 810
Exercise 19L……Page 527
M – Bayes’ theorem……Page 528
Exercise 19M……Page 529
Review set 19B……Page 530
20. Introduction to calculus……Page 531
Investigation 1 – The speed of falling objects……Page 532
A – Rate of change……Page 533
Exercise 20A.1……Page 534
Investigation 2 – Instantaneous speed……Page 536
Exercise 20B.1……Page 538
Exercise 20B.2……Page 541
Review set 20……Page 542
21. Differential calculus……Page 543
Investigation 1 – The slope of a tangent……Page 544
B – Derivatives at a given x-value……Page 547
Exercise 21B……Page 549
C – The derivative function……Page 552
Exercise 21C……Page 553
Investigation 3 – Simple rules of differentiation……Page 555
Exercise 21D……Page 557
Answers……Page 811
Investigation 4 – Differentiating composites……Page 559
Exercise 21E.2……Page 561
F – Product and quotient rules……Page 562
Exercise 21F.1……Page 564
Exercise 21F.2……Page 565
G – Tangents and normals……Page 566
Exercise 21G……Page 567
H – The second derivative……Page 570
Exercise 21H……Page 571
Review set 21B, 21C……Page 572
Answers……Page 812
22. Applications of differential calculus……Page 573
Exercise 22A……Page 575
B – General rates of change……Page 576
Exercise 22B……Page 577
C – Motion in a straight line……Page 579
Exercise 22C.1……Page 580
Investigation – Displacement,velocity and acceleration graphs……Page 583
Exercise 22C.2……Page 585
D – Some curve properties……Page 586
Exercise 22D.1……Page 588
Answers……Page 813
Exercise 22D.2……Page 589
Exercise 22D.3……Page 592
E – Rational functions……Page 594
Exercise 22E.2……Page 595
Answers……Page 814
F – Inflections and shape type……Page 598
Exercise 22F……Page 600
Answers……Page 815
G – Optimisation……Page 601
Exercise 22G……Page 605
Answers……Page 816
H – Economic models……Page 610
I – Implicit differentiation……Page 613
Exercise 22I……Page 615
Review set 22B……Page 616
23. Derivatives of exponential and logarithmic functions……Page 617
Investigation 1 – The derivative of y = a^x……Page 618
Investigation 2 – Finding a when y = a^x and dy/dx = a^x……Page 619
Exercise 23A……Page 621
Answers……Page 817
B – Using natural logarithms……Page 623
Exercise 23B……Page 624
Investigation 3 – The derivative of ln x……Page 626
Exercise 23C……Page 627
Exercise 23D……Page 629
Review set 23B……Page 632
Answers……Page 818
24. Derivatives of circular functions and related rates……Page 633
Investigation – Examining sin(theta)/theta near theta = 0……Page 635
Exercise 24A……Page 638
Answers……Page 819
B – The derivatives of reciprocal circular functions……Page 640
C – The derivatives of inverse circular functions……Page 641
Exercise 24C.2……Page 642
D – Maxima/minima with trigonometry……Page 643
Exercise 24D……Page 644
E – Related rates……Page 645
Exercise 24E……Page 647
Answers……Page 820
Review set 24A……Page 649
Review set 24B……Page 650
25. Integration……Page 651
A – Antidifferentiation……Page 652
B – Integration……Page 653
Exercise 25B.1……Page 654
Exercise 25B.2……Page 656
Exercise 25C……Page 659
Answers……Page 821
D – Integrating f(u)u'(x) by substitution……Page 661
Exercise 25D……Page 662
E – Definite integrals……Page 663
Exercise 25E.1……Page 664
Exercise 25E.2……Page 665
Review set 25A……Page 666
Review set 25B……Page 667
Review set 25C……Page 668
26. Integration (areas and other applications)……Page 669
Exercise 26A.1……Page 670
Answers……Page 822
Investigation 1 – Finding areas using rectangles……Page 672
Exercise 26A.2……Page 673
B – The fundamental theorem of calculus……Page 675
Exercise 26B……Page 677
Investigation 2 – int(f(x), x, b) and areas……Page 678
Exercise 26C……Page 679
D – Distances from velocity functions……Page 683
Exercise 26D.1……Page 685
Exercise 26D.2……Page 687
Exercise 26E……Page 688
Answers……Page 823
Review set 26A……Page 690
Review set 26B……Page 691
Review set 26C……Page 692
27. Circular function integration……Page 693
Exercise 27A……Page 694
B – Integrals of f(ax+b) circular functions……Page 696
Exercise 27B……Page 697
Answers……Page 824
Exercise 27C……Page 699
D – Area determination……Page 700
Exercise 27D……Page 701
Review set 27B……Page 702
28. Volumes of revolution……Page 703
A – Solids of revolution……Page 704
Exercise 28A.1……Page 706
B – Volumes for two defining functions……Page 707
Exercise 28B……Page 708
Review set 28……Page 710
29. Further integration and differential equations……Page 711
A – The integrals of 1/(sqr(a²-x²)) and 1/(x²+a²)……Page 712
Exercise 29B……Page 713
Answers……Page 825
Exercise 29C……Page 715
Investigation – Functions which cannot be integrated……Page 716
D – Separable differential equations……Page 717
Exercise 29D.1……Page 719
Exercise 29D.2……Page 720
Review set 29A……Page 725
Review set 29B……Page 726
30. Statistical distributions……Page 727
Exercise 30A……Page 728
Exercise 30B……Page 730
Answers……Page 826
C – Expectation……Page 732
Exercise 30C……Page 733
D – The mean and standard deviation of a discrete random variable……Page 735
Exercise 30D……Page 737
E – Expected values (discrete RV)……Page 738
Investigation 1 – E(aX+b) and Var(aX+b)……Page 739
Exercise 30E.2……Page 740
F – The binomial distribution……Page 741
Exercise 30F……Page 742
Answers……Page 827
G – Mean and standard deviation of a binomial random variable……Page 743
Investigation 2 – The mean and standard deviation of a binomial random variable……Page 744
Exercise 30G……Page 745
Investigation 3 – Poisson mean and variance……Page 746
Exercise 30H……Page 747
I – Continuous probability density functions……Page 748
Exercise 30I……Page 749
J – Normal distributions……Page 750
Investigation 4 – Standard deviation significance……Page 752
Exercise 30J.1……Page 753
Investigation 5 – Mean and standard deviation of Z = (x-mu)/sigma……Page 754
K – The standard normal distribution (Z-distribution)……Page 755
Exercise 30K.1……Page 757
Exercise 30K.2……Page 759
Index……Page 828
L – Applications of the normal distribution……Page 760
Exercise 30L……Page 761
Review set 30B……Page 762
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