Jacaranda Maths Quest 12 Specialist Mathematics

Title Page......Page 3
Copyright Page......Page 4
Contents......Page 5
Introduction......Page 7
About eBookPLUS and studyON......Page 10
Acknowledgements......Page 11
TOPIC 1 Sketching graphs......Page 12
Exploring the absolute value function with CAS......Page 13
Using modulus notation to describe domains......Page 14
Using the graph of y = |x|......Page 15
1.3 Sketching graphs of reciprocal functions......Page 17
Graphing the reciprocal functions for quadratic and cubic functions......Page 18
Graphing improper fractions......Page 23
Graphs that cross the horizontal asymptote......Page 28
Graphs without vertical asymptotes......Page 29
Oblique asymptotes......Page 32
Graphing y = |f(x)| from y = f(x)......Page 37
Graphing y = f(|x|) from y = f(x)......Page 38
Sketching graphs of y = |f(x)| and y = f(|x|) from y = f(x)......Page 39
Solving inequations involving the modulus function......Page 40
Circles......Page 44
Ellipses......Page 45
Hyperbolas......Page 46
1.7 Review......Page 53
Answers......Page 54
TOPIC 2 Trigonometry......Page 66
Exploring inverse trigonometric functions......Page 67
The unit circle......Page 68
Angles of any magnitude......Page 69
Reciprocal trigonometric functions......Page 72
Proving trigonometric identities......Page 76
Proof of the compound-angle formulas......Page 79
Using compound-angle formulas in problems......Page 81
Double-angle formulas......Page 86
Half-angle formulas......Page 91
Multiple-angle formulas......Page 92
Inverse functions......Page 95
2.7 General solutions of trigonometric equations......Page 107
General solutions involving sines......Page 108
General solutions involving tangents......Page 109
General solutions involving phase shifts......Page 111
Equations reducible to quadratics......Page 112
Trigonometric equations involving multiple angles......Page 113
Comparison of examples......Page 114
The graph of y = sec(x)......Page 116
The graph of y = cosec(x)......Page 117
The graph of y = cot(x)......Page 121
Transformations of reciprocal trigonometric graphs......Page 123
2.9 Graphs of inverse trigonometric functions......Page 125
2.10 Review......Page 132
Answers......Page 133
TOPIC 3 Complex numbers......Page 140
Exploring complex numbers with CAS......Page 141
Solving quadratic equations......Page 142
Operations on complex numbers in rectangular form......Page 143
Geometrical representation of operations on complex numbers......Page 144
Complex conjugates......Page 146
Solving equations involving complex numbers......Page 148
Multiplication by i......Page 149
Polar form......Page 152
Conversions......Page 153
Operations in polar form......Page 159
Powers in polar form......Page 160
Quadratic equations......Page 166
Solving cubic equations......Page 167
Quartic equations with real coefficients......Page 169
Solving quartic equations with real coefficients......Page 170
Circles......Page 172
Lines......Page 173
Intersection of lines and circles......Page 174
Rays......Page 176
Square roots of complex numbers......Page 180
Cube roots of complex numbers......Page 182
Fourth roots of complex numbers......Page 183
3.7 Review......Page 187
Answers......Page 188
TOPIC 4 Kinematics......Page 194
4.1 Kick off with CAS......Page 195
4.2 Constant acceleration......Page 196
4.3 Motion under gravity......Page 200
4.4 Velocity–time graphs......Page 203
4.5 Variable acceleration......Page 207
4.6 Review......Page 214
Answers......Page 215
TOPIC 5 Vectors in three dimensions......Page 218
Exploring the lengths of vectors with CAS......Page 219
Review of vectors......Page 220
Unit vectors......Page 225
Magnitude of a vector......Page 226
Addition and subtraction of vectors in three dimensions......Page 227
Scalar multiplication of vectors......Page 228
Direction cosines......Page 230
Column vector notation......Page 233
Multiplying vectors......Page 238
Projections......Page 241
Geometrical shapes......Page 247
Using vectors to prove geometrical theorems......Page 249
5.6 Parametric equations......Page 252
Eliminating the parameter......Page 253
Parametric representation......Page 254
5.7 Review......Page 258
Answers......Page 259
TOPIC 6 Mechanics......Page 266
Solving equations with CAS......Page 267
Statics......Page 268
Other types of forces......Page 271
Inclined planes......Page 282
Connected particles......Page 285
Newton’s laws of motion......Page 289
Constant acceleration formulas......Page 292
Two or more particles......Page 299
Particles connected by smooth pulleys......Page 300
Connected vehicles......Page 302
6.6 Review......Page 309
Answers......Page 310
TOPIC 7 Differential calculus......Page 312
Exploring differentiation with CAS......Page 313
Introduction......Page 314
Introduction......Page 324
Tangents and normal to curves......Page 325
Rates of change......Page 328
Introduction......Page 334
Parametric differentiation......Page 338
Introduction......Page 341
Using parametric differentiation to find second derivatives......Page 344
Curve sketching and derivatives......Page 348
Sketching the graphs of cubic functions......Page 351
Sketching the graphs of quartic functions......Page 352
Sketching the graphs of reciprocal functions using calculus......Page 354
Sketching rational functions......Page 356
Introduction......Page 361
Introduction......Page 370
7.9 Review......Page 378
Answers......Page 379
TOPIC 8 Integral calculus......Page 388
Exploring integration with CAS......Page 389
Area bounded by a curve and the x-axis......Page 390
Areas involving signed areas......Page 394
Area between curves......Page 396
8.3 Linear substitutions......Page 401
Evaluating definite integrals using a linear substitution......Page 404
Finding integrals using a back substitution......Page 405
Non-linear substitutions......Page 412
Definite integrals involving non-linear substitutions......Page 416
Introduction......Page 421
Integrals involving powers of tan (kx)......Page 424
Integrals involving the inverse sine function......Page 428
Integrals involving the inverse cosine function......Page 429
Definite integrals involving inverse trigonometric functions......Page 430
Integrals involving completing the square......Page 432
Integrals involving substitutions and inverse trigonometric functions......Page 434
Integration of rational functions......Page 438
8.8 Review......Page 449
Answers......Page 450
TOPIC 9 Differential equations......Page 456
Solving differential equations with CAS......Page 457
Verifying solutions to differential equations......Page 458
Type 1 differential equations, dy/dx = f(x)......Page 464
Invert, integrate and transpose......Page 471
Separation of variables......Page 478
Integrate twice......Page 482
Beam deflections......Page 485
9.7 Review......Page 490
Answers......Page 491
TOPIC 10 Further applications of integration......Page 494
Using CAS to solve area under curve problems......Page 495
Deducing an antiderivative......Page 496
Rotations around the x-axis......Page 504
Volumes of revolution......Page 514
Length of a curve......Page 522
Numerical integration......Page 524
Using numerical integration in differential equations......Page 525
Graphs of antiderivatives of functions......Page 528
Torricelli’s theorem......Page 534
10.7 Review......Page 543
Answers......Page 544
TOPIC 11 Applications of first-order differential equations......Page 548
11.1 Kick off with CAS......Page 549
The law of natural growth......Page 550
Miscellaneous types......Page 556
Other population models......Page 558
Bounded growth models......Page 563
Newton’s Law of Cooling......Page 565
Input–output mixing problems......Page 570
Chemical reaction rates......Page 574
Logistic growth......Page 582
Numerical solution of a differential equation......Page 594
Using Euler’s method to solve Type 3 differential equations......Page 598
Introduction......Page 603
11.9 Review......Page 614
Answers......Page 615
TOPIC 12 Variable forces......Page 622
Review of velocity and acceleration functions......Page 623
Setting up the equation of motion......Page 624
Setting up the equation of motion......Page 631
Vertical motion......Page 636
Setting up the equation of motion......Page 644
12.5 Review......Page 651
Answers......Page 652
TOPIC 13 Vector calculus......Page 654
Exploring parametric equations with CAS......Page 655
Closest approach......Page 656
Collision problems......Page 657
Vector functions......Page 661
Rules for differentiating vectors......Page 662
Applications of vector calculus......Page 667
Plane curves......Page 672
Rules for integrating vectors......Page 685
13.6 Projectile motion......Page 691
General theory of a projectile......Page 692
13.7 Review......Page 706
Answers......Page 707
TOPIC 14 Statistical inference......Page 714
Mean and spread of sample distributions......Page 715
14.2 Linear combinations of random variables......Page 716
Change of scale......Page 717
Change of origin and scale......Page 718
Linear combinations of random variables......Page 719
Linear combinations of normally distributed random variables......Page 722
Estimating population parameters......Page 723
The distribution of sample means......Page 725
Verifying the formulas......Page 726
Calculation of confidence intervals......Page 729
Hypotheses......Page 734
Type I and Type II errors......Page 735
Hypothesis testing......Page 736
P-value......Page 737
14.6 Review......Page 740
Answers......Page 741
Index......Page 743