Mathematics: Applications and Interpretation SL 2

Mathematics: Applications and Interpretation SL 2

This book has been written for the IB Diploma Programme course Mathematics: Applications and Interpretation SL, for first assessment in May 2021.

This book is designed to complete the course in conjunction with the Mathematics: Core Topics SL textbook. It is expected that students will start using this book approximately 6-7 months into the two-year course, upon the completion of the Mathematics: Core Topics SL textbook.

Table of Contents

Mathematics: Applications and Interpretation SL

1 APPROXIMATIONS AND ERROR 15
A Rounding numbers 16
B Approximations 20
C Errors in measurement 22
D Absolute and percentage error 25
Review set 1A 29
Review set 1B 30

2 LOANS AND ANNUITIES 31
A Loans 32
B Annuities 38
Review set 2A 43
Review set 2B 44

3 FUNCTIONS 45
A Relations and functions 46
B Function notation 49
C Domain and range 53
D Graphs of functions 57
E Sign diagrams 60
F Transformations of graphs 63
G Inverse functions 69
Review set 3A 73
Review set 3B 76

4 MODELLING 79
A The modelling cycle 80
B Linear models 86
C Piecewise linear models 89
D Systems of equations 94
Review set 4A 96
Review set 4B 98

5 BIVARIATE STATISTICS 101
A Association between numerical variables 102
B Pearson’s product-moment correlation coefficient 107
C Line of best fit by eye 112
D The least squares regression line 116
E Spearman’s rank correlation coefficient 123
Review set 5A 128
Review set 5B 130

6 QUADRATIC FUNCTIONS 133
A Quadratic functions 135
B Graphs from tables of values 137
C Axes intercepts 139
D Graphs of the form y = ax^2y=ax​2​​ 141
E Graphs of quadratic functions 143
F Axis of symmetry 144
G Vertex 147
H Finding a quadratic from its graph 149
I Intersection of graphs 152
J Quadratic models 153
Review set 6A 159
Review set 6B 161

7 DIRECT AND INVERSE VARIATION 163
A Direct variation 164
B Powers in direct variation 168
C Inverse variation 170
D Powers in inverse variation 172
E Determining the variation model 173
F Using technology to find variation models 175
Review set 7A 178
Review set 7B 180

8 EXPONENTIALS AND LOGARITHMS 183
A Exponential functions 185
B Graphing exponential functions from a table of values 186
C Graphs of exponential functions 187
D Exponential equations 191
E Growth and decay 192
F The natural exponential 199
G Logarithms in base 1010 204
H Natural logarithms 208
Review set 8A 211
Review set 8B 213

9 TRIGONOMETRIC FUNCTIONS 217
A The unit circle 218
B Periodic behaviour 221
C The sine and cosine functions 224
D General sine and cosine functions 226
E Modelling periodic behaviour 231
Review set 9A 236
Review set 9B 239

10 DIFFERENTIATION 241
A Rates of change 243
B Instantaneous rates of change 247
C Limits 251
D The gradient of a tangent 252
E The derivative function 254
F Differentiation 256
G Rules for differentiation 259
Review set 10A 265
Review set 10B 267

11 PROPERTIES OF CURVES 269
A Tangents 270
B Normals 273
C Increasing and decreasing 276
D Stationary points 280
Review set 11A 284
Review set 11B 285

12 APPLICATIONS OF DIFFERENTIATION 287
A Rates of change 288
B Optimisation 293
C Modelling with calculus 301
Review set 12A 303
Review set 12B 304

13 INTEGRATION 307
A Approximating the area under a curve 308
B The Riemann integral 313
C The Fundamental Theorem of Calculus 317
D Antidifferentiation and indefinite integrals 320
E Rules for integration 322
F Particular values 324
G Definite integrals 325
H The area under a curve 328
Review set 13A 331
Review set 13B 333

14 DISCRETE RANDOM VARIABLES 335
A Random variables 336
B Discrete probability distributions 338
C Expectation 342
D The binomial distribution 347
E Using technology to find binomial probabilities 352
F The mean and standard deviation of a binomial distribution 355
Review set 14A 357
Review set 14B 358

15 THE NORMAL DISTRIBUTION 361
A Introduction to the normal distribution 363
B Calculating probabilities 366
C Quantiles 373
Review set 15A 377
Review set 15B 378

16 HYPOTHESIS TESTING 381
A Statistical hypotheses 382
B Student’s tt-test 384
C The two-sample tt-test for comparing population means 393
D The χ​2​​ goodness of fit test 395
E The χ​2 ​ test for independence 405
Review set 16A 413
Review set 16B 415

17 VORONOI DIAGRAMS 417
A Voronoi diagrams 418
B Constructing Voronoi diagrams 422
C Adding a site to a Voronoi diagram 427
D Nearest neighbour interpolation 431
E The Largest Empty Circle problem 433
Review set 17A 437
Review set 17B 439

ANSWERS 441

INDEX 503

Michael Haese, Mark Humphries, Chris Sangwin, Ngoc Vo – Haese Mathematics