Maths in Focus 12 Advanced, 3rd Edition
By Margaret Grove
Contents:
Preface……………………………………………….Iii
Syllabus Reference Grid………………………. Vii
About This Book………………………………….Ix
Study Skills……………………………………….. Xiii
Mathematical Verbs…………………………… Xvi
1
Sequences And Series 2
M1.2 1.01 General sequences and series ….. 4
M1.2 1.02 Arithmetic sequences ….. 8
M1.2 1.03 Arithmetic series …….. 13
M1.3 1.04 Geometric sequences …….16
M1.3 1.05 Geometric series …….. 21
M1.3 1.06 Limiting sum of an infinite geometric series …… 25
Test yourself 1 ……..30
Challenge exercise 1 ……. 32
2
TRANSFORMATIONS OF FUNCTIONS 34
F2 2.01 Vertical translations of functions …….. 36
F2 2.02 Horizontal translations of functions ……. 40
F2 2.03 Vertical dilations of functions ……. 45
F2 2.04 Horizontal dilations of functions …….. 51
F2 2.05 Combinations of transformations ……. 58
F2 2.06 Graphs of functions with combined transformations ……. 66
F2 2.07 Equations and inequalities ……. 74
Test yourself 2 ………82
Challenge exercise 2 ……. 86
3
TRIGONOMETRIC FUNCTIONS 88
T3 3.01 Transformations of trigonometric functions ……. 90
T3 3.02 Combined transformations of trigonometric functions …… 100
T3 3.03 Trigonometric equations …… 104
Test yourself 3 …….. 110
Challenge exercise 3 ……… 113
PRACTICE SET 1 …………. 114
4
FURTHER DIFFERENTIATION 118
Revision 4.01 Differentiation review ……… 120
C2.1, 2.2 4.02 Derivative of exponential functions …….. 123
C2.1, 2.2 4.03 Derivative of logarithmic functions ……. 126
C2.1, 2.2 4.04 Derivative of trigonometric functions …….. 132
C3.1 4.05 Second derivatives …… 140
C4.1 4.06 Anti-derivative graphs ……. 143
C4.1 4.07 Anti-derivatives …………….. 147
C4.1 4.08 Further anti-derivatives ……. 153
Test yourself 4 ………. 159
Challenge exercise 4 ……. 162
5
GEOMETRICAL APPLICATIONS OF DIFFERENTIATION 164
C3.1 5.01 Increasing and decreasing curves …….166
C3.2 5.02 Stationary points …….. 169
C3.1 5.03 Concavity and points of inflection …….172
C3.2 5.04 Interpreting rates of change graphically ……. 176
C3.2 5.05 Stationary points and the second derivative ……. 179
C3.2 5.06 Curve sketching …… 183
C3.2 5.07 Global maxima and minima ……191
C3.2 5.08 Finding formulas for optimization problems …… 194
C3.2 5.09 Optimisation problems …… 198
Test yourself 5 …….. 207
Challenge exercise 5 ……. 209
6
INTEGRATION 210
C4.2 6.01 Approximating areas under a curve …… 212
C4.2 6.02 Trapezoidal rule …. 219
C4.2 6.03 Definite integrals ….. 225
C4.2 6.04 Indefinite integrals …….231
C4.1 6.05 Chain rule …… 233
C4.1 6.06 Integration involving exponential functions …… 237
C4.1 6.07 Integration involving logarithmic functions ….. 239
C4.1 6.08 Integration involving trigonometric functions …… 241
C4.2 6.09 Areas enclosed by the x-axis …… 244
C4.2 6.10 Areas enclosed by the y-axis ……. 251
C4.2 6.11 Sums and differences of areas ….. 254
Test yourself 6 …… 258
Challenge exercise 6 ….. 261
PRACTICE SET 2 ….. 262
7
STATISTICS 268
S2.1 7.01 Types of data …… 270
S2.1 7.02 Displaying numerical and categorical data ….. 273
S2.1 7.03 Measures of central tendency ….. 285
S2.1 7.04 Quartiles, deciles and percentiles ……. 298
S2.1 7.05 Range and interquartile range …… 306
S2.1 7.06 Variance and standard deviation …..311
S2.1 7.07 Shape and modality of data sets ….. 315
S2.1 7.08 Analysing data sets ….. 320
Test yourself 7 ….. 326
Challenge exercise 7 ….. 330
8
CORRELATION AND REGRESSION 332
S2.2 8.01 Bivariate data ….. 334
S2.2 8.02 Correlation ………. 340
S2.2 8.03 Line of best fit …… 349
S2.2 8.04 Least-squares regression line …… 353
Test yourself 8 …… 357
Challenge exercise 8 ….. 360
PRACTICE SET 3 ……. 361
9
INVESTMENTS, ANNUITIES AND LOANS 366
M1.2 9.01 Arithmetic growth and decay …… 368
M1.3 9.02 Geometric growth and decay …. 371
M1.1 9.03 Compound interest …… 378
M1.4 9.04 Compound interest formula …… 383
M1.1 9.05 Annuities ……….. 389
M1.4 9.06 Annuities and geometric series ….. 397
M1.1 9.07 Reducing balance loans ………. 400
M1.4 9.08 Loans and geometric series ….. 405
Test yourself 9 …………… 410
Challenge exercise 9 …… 413
10
CONTINUOUS PROBABILITY DISTRIBUTIONS 414
S3.1 10.01 Probability density functions ……. 416
S3.1 10.02 Calculating probabilities …….. 421
S3.1 10.03 Cumulative distribution function ………… 426
S3.1 10.04 Quantiles …………………. 433
S3.2 10.05 Normal distribution …….. 437
S3.2 10.06 Empirical rule …….. 449
S3.2 10.07 z-scores …….. 452
S3.2 10.08 Applications of the normal distribution ……….457
Test yourself 10 …………… 463
Challenge exercise 10 ….. 467
PRACTICE SET 4 ….. 468
ANSWERS ….472
INDEX XXX