Jacaranda Maths Quest 12 Specialist Mathematics

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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