Mathematics HL for the IB (Core) 3rd ed, 2012
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David Martin
Robert Haese
core math
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Mathematics HL for the IB (Core) 3rd ed, 2012
Preparing book pages
1
Entries
298
Symbols and notation used in this book
Theory
Page 10
Background knowledge
Theory
Page 16
Graphics calculator instructions
Theory
Page 16
1. Quadratics
Theory
Page 17
1A Quadratic equations
Theory
Page 19
1B The discriminant of a quadratic
Theory
Page 25
1C The sum and product of the roots
Theory
Page 27
1D Quadratic functions
Theory
Page 28
1E Finding a quadratic from its graph
Theory
Page 37
1F Where functions meet
Theory
Page 40
1G Problem solving with quadratics
Theory
Page 43
1H Quadratic optimisation
Theory
Page 45
Review set 1A
Problem
Page 47
Review set 1B
Problem
Page 48
Review set 1C
Problem
Page 49
2. Functions
Theory
Page 51
2A Relations and functions
Theory
Page 52
2B Function notation
Theory
Page 55
2C Domain and range
Theory
Page 57
2D Composite functions
Theory
Page 62
2E Even and odd functions
Theory
Page 64
2F Sign diagrams
Theory
Page 66
2G Inequalities (inequations)
Theory
Page 70
2H The modulus function
Theory
Page 73
2I Rational functions
Theory
Page 78
2J Inverse functions
Theory
Page 81
2K Graphing functions
Theory
Page 87
2L Finding where graphs meet
Theory
Page 89
Review set 2A
Problem
Page 90
Review set 2B
Problem
Page 91
Review set 2C
Problem
Page 93
3. Exponentials
Theory
Page 95
3A Exponents
Theory
Page 96
3B Laws of exponents
Theory
Page 98
3C Rational exponents
Theory
Page 101
3D Algebraic expansion and factorisation
Theory
Page 104
3E Exponential equations
Theory
Page 106
3F Exponential functions
Theory
Page 108
3G Growth and decay
Theory
Page 112
3H The natural exponential $e^x$
Theory
Page 115
Review set 3A
Problem
Page 119
Review set 3B
Problem
Page 120
Review set 3C
Problem
Page 121
4. Logarithms
Theory
Page 123
4A Logarithms in base 10
Theory
Page 124
4B Logarithms in base $a$
Theory
Page 127
4C Laws of logarithms
Theory
Page 130
4D Natural logarithms
Theory
Page 134
4E Exponential equations using logarithms
Theory
Page 137
4F The change of base rule
Theory
Page 139
4G Graphs of logarithmic functions
Theory
Page 141
4H Growth and decay
Theory
Page 145
Review set 4A
Problem
Page 147
Review set 4B
Problem
Page 148
Review set 4C
Problem
Page 149
5. Transforming functions
Theory
Page 151
5A Transformation of graphs
Theory
Page 152
5B Translations
Theory
Page 154
5C Stretches
Theory
Page 156
5D Reflections
Theory
Page 157
5E Miscellaneous transformations
Theory
Page 159
5F Simple rational functions
Theory
Page 161
5G The reciprocal of a function
Theory
Page 165
5H Modulus functions
Theory
Page 166
Review set 5A
Problem
Page 167
Review set 5B
Problem
Page 169
Review set 5C
Problem
Page 170
6. Complex numbers and polynomials
Theory
Page 173
6A Real quadratics with $\Delta<0$
Theory
Page 174
6B Complex numbers
Theory
Page 176
6C Real polynomials
Theory
Page 183
6D Zeros, roots, and factors
Theory
Page 189
6E Polynomial theorems
Theory
Page 193
6F Graphing real polynomials
Theory
Page 201
Review set 6A
Problem
Page 209
Review set 6B
Problem
Page 210
Review set 6C
Problem
Page 211
7. Sequences and series
Theory
Page 213
7A Number sequences
Theory
Page 214
7B The general term of a number sequence
Theory
Page 215
7C Arithmetic sequences
Theory
Page 218
7D Geometric sequences
Theory
Page 222
7E Series
Theory
Page 229
7F Arithmetic series
Theory
Page 231
7G Geometric series
Theory
Page 233
Review set 7A
Problem
Page 240
Review set 7B
Problem
Page 240
Review set 7C
Problem
Page 242
8. Counting and the binomial expansion
Theory
Page 243
8A The product principle
Theory
Page 244
8B Counting paths
Theory
Page 246
8C Factorial notation
Theory
Page 247
8D Permutations
Theory
Page 250
8E Combinations
Theory
Page 254
8F Binomial expansions
Theory
Page 257
8G The binomial theorem
Theory
Page 260
Review set 8A
Problem
Page 263
Review set 8B
Problem
Page 263
Review set 8C
Problem
Page 264
9. Mathematical induction
Theory
Page 265
9A The process of induction
Theory
Page 267
9B The principle of mathematical induction
Theory
Page 269
Review set 9A
Problem
Page 277
Review set 9B
Problem
Page 277
Review set 9C
Problem
Page 278
10. The unit circle and radian measure
Theory
Page 279
10A Radian measure
Theory
Page 280
10B Arc length and sector area
Theory
Page 283
10C The unit circle and the trigonometric ratios
Theory
Page 286
10D Applications of the unit circle
Theory
Page 292
10E Negative and complementary angle formulae
Theory
Page 295
10F Multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$
Theory
Page 297
Review set 10A
Problem
Page 301
Review set 10B
Problem
Page 302
Review set 10C
Problem
Page 303
11. Non-right angled triangle trigonometry
Theory
Page 305
11A Areas of triangles
Theory
Page 306
11B The cosine rule
Theory
Page 309
11C The sine rule
Theory
Page 313
11D Using the sine and cosine rules
Theory
Page 317
Review set 11A
Problem
Page 321
Review set 11B
Problem
Page 323
Review set 11C
Problem
Page 323
12. Trigonometric functions
Theory
Page 325
12A Periodic behaviour
Theory
Page 326
12B The sine function
Theory
Page 330
12C Modelling using sine functions
Theory
Page 336
12D The cosine function
Theory
Page 339
12E The tangent function
Theory
Page 341
12F General trigonometric functions
Theory
Page 344
12G Reciprocal trigonometric functions
Theory
Page 346
12H Inverse trigonometric functions
Theory
Page 348
Review set 12A
Problem
Page 350
Review set 12B
Problem
Page 351
Review set 12C
Problem
Page 351
13. Trigonometric equations and identities
Theory
Page 353
13A Trigonometric equations
Theory
Page 354
13B Using trigonometric models
Theory
Page 362
13C Trigonometric relationships
Theory
Page 364
13D Double angle formulae
Theory
Page 367
13E Compound angle formulae
Theory
Page 370
13F Trigonometric equations in quadratic form
Theory
Page 376
13G Trigonometric series and products
Theory
Page 377
Review set 13A
Problem
Page 379
Review set 13B
Problem
Page 380
Review set 13C
Problem
Page 382
14. Vectors
Theory
Page 383
14A Vectors and scalars
Theory
Page 384
14B Geometric operations with vectors
Theory
Page 387
14C Vectors in the plane
Theory
Page 394
14D The magnitude of a vector
Theory
Page 396
14E Operations with plane vectors
Theory
Page 398
14F The vector between two points
Theory
Page 401
14G Vectors in space
Theory
Page 404
14H Operations with vectors in space
Theory
Page 408
14I Parallelism
Theory
Page 412
14J The scalar product of two vectors
Theory
Page 416
14K The vector product of two vectors
Theory
Page 422
Review set 14A
Problem
Page 428
Review set 14B
Problem
Page 430
Review set 14C
Problem
Page 431
15. Vector applications
Theory
Page 433
15A Problems involving vector operations
Theory
Page 434
15B Area
Theory
Page 436
15C Lines in 2-D and 3-D
Theory
Page 437
15D The angle between two lines
Theory
Page 442
15E Constant velocity problems
Theory
Page 444
15F The shortest distance from a line to a point
Theory
Page 447
15G Intersecting lines
Theory
Page 451
15H Relationships between lines
Theory
Page 453
15I Planes
Theory
Page 460
15J Angles in space
Theory
Page 465
15K Intersecting planes
Theory
Page 467
Review set 15A
Problem
Page 472
Review set 15B
Problem
Page 474
Review set 15C
Problem
Page 476
16. Complex numbers
Theory
Page 479
16A Complex numbers as 2-D vectors
Theory
Page 480
16B Modulus
Theory
Page 483
16C Argument and polar form
Theory
Page 487
16D Euler’s form
Theory
Page 495
16E De Moivre’s theorem
Theory
Page 497
16F Roots of complex numbers
Theory
Page 500
16G Miscellaneous problems
Theory
Page 504
Review set 16A
Problem
Page 504
Review set 16B
Problem
Page 505
Review set 16C
Problem
Page 506
17. Introduction to differential calculus
Theory
Page 507
17A Limits
Theory
Page 509
17B Limits at infinity
Theory
Page 512
17C Trigonometric limits
Theory
Page 515
17D Rates of change
Theory
Page 518
17E The derivative function
Theory
Page 521
17F Differentiation from first principles
Theory
Page 523
Review set 17A
Problem
Page 526
Review set 17B
Problem
Page 526
Review set 17C
Problem
Page 527
18. Rules of differentiation
Theory
Page 529
18A Simple rules of differentiation
Theory
Page 530
18B The chain rule
Theory
Page 534
18C The product rule
Theory
Page 537
18D The quotient rule
Theory
Page 540
18E Implicit differentiation
Theory
Page 542
18F Derivatives of exponential functions
Theory
Page 544
18G Derivatives of logarithmic functions
Theory
Page 549
18H Derivatives of trigonometric functions
Theory
Page 551
18I Derivatives of inverse trigonometric functions
Theory
Page 555
18J Second and higher derivatives
Theory
Page 557
Review set 18A
Problem
Page 559
Review set 18B
Problem
Page 560
Review set 18C
Problem
Page 561
19. Properties of curves
Theory
Page 563
19A Tangents and normals
Theory
Page 564
19B Increasing and decreasing functions
Theory
Page 570
19C Stationary points
Theory
Page 575
19D Inflections and shape
Theory
Page 579
Review set 19A
Problem
Page 587
Review set 19B
Problem
Page 588
Review set 19C
Problem
Page 589
20. Applications of differential calculus
Theory
Page 591
20A Kinematics
Theory
Page 592
20B Rates of change
Theory
Page 601
20C Optimisation
Theory
Page 606
20D Related rates
Theory
Page 617
Review set 20A
Problem
Page 621
Review set 20B
Problem
Page 623
Review set 20C
Problem
Page 625
21. Integration
Theory
Page 627
21A The area under a curve
Theory
Page 628
21B Antidifferentiation
Theory
Page 634
21C The fundamental theorem of calculus
Theory
Page 635
21D Integration
Theory
Page 640
21E Rules for integration
Theory
Page 643
21F Integrating $f(ax+b)$
Theory
Page 648
21G Integration by substitution
Theory
Page 653
21H Integration by parts
Theory
Page 659
21I Miscellaneous integration
Theory
Page 660
21J Definite integrals
Theory
Page 661
Review set 21A
Problem
Page 667
Review set 21B
Problem
Page 668
Review set 21C
Problem
Page 669
22. Applications of integration
Theory
Page 671
22A The area under a curve
Theory
Page 672
22B The area between two functions
Theory
Page 675
22C Kinematics
Theory
Page 681
22D Problem solving by integration
Theory
Page 687
22E Solids of revolution
Theory
Page 690
Review set 22A
Problem
Page 697
Review set 22B
Problem
Page 699
Review set 22C
Problem
Page 701
23. Descriptive statistics
Theory
Page 703
23A Key statistical concepts
Theory
Page 704
23B Measuring the centre of data
Theory
Page 709
23C Variance and standard deviation
Theory
Page 721
Review set 23A
Problem
Page 728
Review set 23B
Problem
Page 729
Review set 23C
Problem
Page 731
24. Probability
Theory
Page 733
24A Experimental probability
Theory
Page 735
24B Sample space
Theory
Page 740
24C Theoretical probability
Theory
Page 741
24D Tables of outcomes
Theory
Page 745
24E Compound events
Theory
Page 747
24F Tree diagrams
Theory
Page 751
24G Sampling with and without replacement
Theory
Page 754
24H Sets and Venn diagrams
Theory
Page 757
24I Laws of probability
Theory
Page 763
24J Independent events
Theory
Page 767
24K Probabilities using permutations and combinations
Theory
Page 769
24L Bayes’ theorem
Theory
Page 770
Review set 24A
Problem
Page 774
Review set 24B
Problem
Page 775
Review set 24C
Problem
Page 776
25. Discrete random variables
Theory
Page 779
25A Discrete random variables
Theory
Page 780
25B Discrete probability distributions
Theory
Page 782
25C Expectation
Theory
Page 786
25D Variance and standard deviation
Theory
Page 790
25E Properties of $\mathrm{E}(X)$ and $\operatorname{Var}(X)$
Theory
Page 792
25F The binomial distribution
Theory
Page 795
25G The Poisson distribution
Theory
Page 805
Review set 25A
Problem
Page 808
Review set 25B
Problem
Page 809
Review set 25C
Problem
Page 811
26. Continuous random variables
Theory
Page 813
26A Continuous random variables
Theory
Page 814
26B The normal distribution
Theory
Page 818
26C Probabilities using a calculator
Theory
Page 823
26D The standard normal distribution (Z-distribution)
Theory
Page 826
26E Quantiles or $k$-values
Theory
Page 831
Review set 26A
Problem
Page 835
Review set 26B
Problem
Page 836
Review set 26C
Problem
Page 837
27. Miscellaneous questions
Problem
Page 839
27A Non-calculator questions
Problem
Page 840
27B Calculator questions
Problem
Page 852
Answers
Answer
Page 865
Index
Other
Page 958
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