Mathematics SL

Mathematlcs
for the international student

Mathematics SL
third edition

Editors
Robert Haese
Sandra Haese
Michael Haese
Mariut Maenpaa
Mark Humphries

for use with IE Diploma Programme

TABLE OF CONTENTS

BACKGROUND KNOWLEDGE
A Surds and radicals
B Scientific notation (standard form)
C Number systems and set notation
D Algebraic simplification
E Linear equations and inequalities
F Modulus or absolute value
G Product expansion
H Factorisation
I Formula rearrangement
J Adding and subtracting algebraic fractions
K Congruence and similarity
L Pythagoras’ theorem
M Coordinate geometry
N Right angled triangle trigonometry
Facts about number sets
Summary of circle properties
Summary of measurement facts

GRAPHICS CALCULATOR
INSTRUCTIONS
Casio fx-9860G PLUS
Casio fx-CG20
Texas Instruments TI-84 Plus
Texas Instruments TI-nspire

1 QUADRATICS 17
A Quadratic equations 19
B The discriminant of a quadratic 25
C Quadratic functions 28
D Finding a quadratic from its graph 38
E Where functions meet 42
F Problem solving with quadratics 44
G Quadratic optimisation 47
Review set 1A 49
Review set 1B 50
Review set 1C 51
F Miscellaneous transformations
Review set 5A
Review set 5B
Review set 5C

2 FUNCTION S 53
A Relations and functions 54
B Function notation 57
C Domain and range 59
D Composite functions 64
E Sign diagrams 66
F Rational functions 69
G Inverse functions 73
Review set 2A 77
Review set 2B 78
Review set 2C 80

3 EXPONENTIALS 81
A Exponents 82
B Laws of exponents 84
C Rational exponents 87
D Algebraic expansion and factorisation 90
E Exponential equations 92
F Exponential functions 94
G Growth and decay 98
H The natural exponential e33 101
Review set 3A 105
Review set 3B 106
Review set 3C 107

4 LOGARITHMS 109
A Logarithms in base 10 110
B Logarithms in base a 113
C Laws of logarithms 116
D Natural logarithms 120
E Exponential equations using logarithms 123
F The change of base rule 125
G Graphs of logarithmic functions 126
H Growth and decay 130
Review set 4A 132
Review set 4B 133
Review set 4C 134

5 TRANSFORMING FUNCTIONS 135
A Graphing functions 136
B Transformation of graphs 140
C Translations y = f(x) + b and y = f(x-a) 141
D Stretches y = pf(x), p > 0 and y=f(qx),q>0 143
E Reflections y = —f(x) and y = f(—x) 144

6 SEQUENCES AND SERIES
A Number sequences
B The general term of a number sequence
C Arithmetic sequences
D Geometric sequences
E Series
F Arithmetic series
G Geometric series
Review set 6A
Review set 6B
Review set 6C

7 THE BINOMIAL EXPANSION
A Binomial expansions
B The binomial coefficient (33)
C The binomial theorem
Review set 7A
Review set 7B

8 THE UNIT CIRCLE AND RADIAN MEASURE 189
A Radian measure 190
B Arc length and sector area 193
C The unit circle and the trigonometric ratios 196
D Applications of the unit circle 201
E Multiples of pi/6 and pi/4 205
F The equation of a straight line 209
Review set 8A 210
Review set 8B 211
Review set 8C 212

9 NON-RIGHT ANGLED TRIANGLE TRIGONOMETRY 213
A Areas of triangles 214
B The cosine rule 217
C The sine rule 220
D Using the sine and cosine rules 224
Review set 9A 228
Review set 9B 229
Review set 9C | 230

10 TRIGONOMETRIC FUNCTIONS 231
A Periodic behaviour 232
B The sine function 236
C Modelling using sine functions 243
D The cosine function 246
E The tangent function 248
F General trigonometric functions 251
Review set 10A 253
Review set 10B 253
Review set 10C 254

11 TRIGONOMETRIC EQUATIONS AND IDENTITIES 255
A Trigonometric equations 256
B Using trigonometric models 263
C Trigonometric relationships 265
D Double angle formulae 268
E Trigonometric equations in quadratic form 271
Review set 11A 272
Review set 11B 273
Review set 11C 274

12 VECTORS 275
A Vectors and scalars 276
B Geometric operations with vectors 279
C Vectors in the plane 286
D The magnitude of a vector 288
E Operations with plane vectors 290
F The vector between two points 293
G Vectors in space 296
H Operations with vectors in space 300
I Parallelism 304
J The scalar product of two vectors 307
Review set 12A 314
Review set 12B 315
Review set 12C 317

13 VECTOR APPLICATIONS 319
A Problems involving vector operations 320
B Lines in 2-D and 3-D 322
C The angle between two lines 326
D Constant velocity problems 328
E The shortest distance from a line to a point 331
F Intersecting lines 335
G Relationships between lines 337
Review set 13A 340
Review set 13B 341
Review set 13C 342

14 INTRODUCTION TO DIFFERENTIAL CALCULUS 343
A Limits 345
B Limits at infinity 347
C Rates of change 350
D The derivative function 353
E Differentiation from first principles 355
Review set 14A 357
Review set 14B 358
Review set 14C 358

15 RULES OF DIFFERENTIATION 359
A Simple rules of differentiation 360
B The chain rule 364
C The product rule 367
D The quotient rule 369
E Derivatives of exponential functions 371
F Derivatives of logarithmic functions 375
G Derivatives of trigonometric functions 378
H Second and higher derivatives 381
Review set 15A 383
Review set 15B 383
Review set 15C 384

16 PROPERTIES OF CURVES 385
A Tangents and normals 386
B Increasing and decreasing functions 392
C Stationary points 397
D Inflections and shape 401
Review set 16A 409
Review set 16B 410
Review set 16C 411

17 APPLICATIONS OF DIFFERENTIAL CALCULUS 413
A Kinematics 414
B Rates of change 423
C Optimisation 428
Review set 17A 437
Review set 17B 438
Review set 17C 439

18 INTEGRATION 441
A The area under a curve 442
B Antidifferentiation 448
C The fundamental theorem of calculus 449
D Integration 454
E Rules for integration 457
F Integrating f(a:r + b) 462
G Integration by substitution 465
H Definite integrals 468
Review set 18A 472
Review set 18B 473
Review set 18C 473

19 APPLICATIONS OF INTEGRATION 475
A The area under a curve 476
B The area between two functions 479
C Kinematics 483
D Solids of revolution 489
Review set 19A 494
Review set 19B 496
Review set 19C 497

20 DESCRIPTIVE STATISTICS 499
A Key statistical concepts 500
B Measuring the centre of data 505
C Measuring the spread of data 517
D Boxplots 521
E Cumulative frequency graphs 526
F Variance and standard deviation 531
Review set 20A 539
Review set 20B 541
Review set 20C 542

21 LINEAR MODELLING 545
A Correlation 546
B Pearson’s correlation coefficient 550
C Line of best fit 5 54
D The least squares regression line 557
E Interpolation and extrapolation 558
Review set 21A 562
Review set 21B 563
Review set 21C 565

22 PROBABILITY 567
A Experimental probability 569
B Sample space 574
C Theoretical probability 575
D Tables of outcomes 579
E Compound events 581
F Tree diagrams 585
G Sampling with and without replacement 588
H Sets and Venn diagrams 591
I Laws of probability 597
J Independent events 601
Review set 22A 604
Review set 22B 604
Review set 22C 606

23 DISCRETE RANDOM VARIABLES 607
A Discrete random variables 608
B Discrete probability distributions 610
C Expectation 6 14
D The binomial distribution 618
Review set 23A 626
Review set 23B 627
Review set 23C 628

24 THE NORMAL DISTRIBUTION 629
A The normal distribution 631
B Probabilities using a calculator 636
C The standard normal distribution
(Z-distribution) 639
D Quantiles or Ic-values 644
Review set 24A 648
Review set 24B 649
Review set 24C 650

25 MISCELLANEOUS QUESTIONS 651
A Non-calculator questions 652
B Calculator questions 665
ANSWERS 679
INDEX 755