Jacaranda Maths Quest 12 Specialist Mathematics VCE Units 3 and 4 2e learnON and Print

About This Resource vii

Acknowledgements xiv

1 Logic and proof 1

1.1 Overview 2

1.2 Logic 3

1.3 Direct proofs 13

1.4 Indirect proofs 20

1.5 Proof by mathematical induction 27

1.6 Review 40

Answers 43

2 Complex numbers 45

2.1 Overview 46

2.2 Complex numbers in Cartesian form 47

2.3 Complex numbers in polar form 59

2.4 Solving polynomial equations over C 75

2.5 Subsets of the complex plane 84

2.6 Roots of complex numbers 96

2.7 Review 105

Answers 112

3 Vectors 125

3.1 Overview 126

3.2 Vectors in two dimensions 127

3.3 Vectors in three dimensions 135

3.4 Scalar product and applications 149

3.5 Vector proofs using the scalar product 161

3.6 Parametric equations 165

3.7 Review 172

Answers 176

4 Vector equations of lines and planes 185

4.1 Overview 186

4.2 Vector cross product 187

4.3 Lines in three dimensions 195

4.4 Planes 207

4.5 Review 222

Answers 227

5 Differential calculus 231

5.1 Overview 232

5.2 Review of differentiation techniques 233

5.3 Applications of differentiation 245

5.4 Implicit and parametric differentiation 256

5.5 Second derivatives 265

5.6 Derivatives of inverse trigonometric functions 274

5.7 Related rates 282

5.8 Review 292

Answers 297

6 Functions and graphs 303

6.1 Overview 304

6.2 Sketching graphs of cubics and quartics 305

6.3 Sketching graphs of rational functions 314

6.4 Sketching graphs of product and quotient functions 328

6.5 Review 342

Answers 348

7 Integral calculus 363

7.1 Overview 364

7.2 Areas under and between curves 365

7.3 Linear substitutions 378

7.4 Non-linear substitutions 390

7.5 Integrals of powers of trigonometric functions 400

7.6 Integrals involving inverse trigonometric functions 408

7.7 Integrals involving partial fractions 421

7.8 Review 432

Answers 437

8 Differential equations 447

8.1 Overview 448

8.2 Verifying solutions to differential equations 449

8.3 Solving Type 1 differential equations,
dy/dx = f(x) 455

8.4 Solving Type 2 differential equations,
dy/dx = f(y) 465

8.5 Solving Type 3 differential equations,
dy/dx = f(x) g(y) 473

8.6 Solving Type 4 differential equations,
d2y/dx2 = f(x) 479

8.7 Review 488

Answers 491

9 Further integration techniques and applications 495

9.1 Overview 496

9.2 Integration by parts 497

9.3 Integration by recognition and graphs of anti-derivatives 509

9.4 Solids of revolution 516

9.5 Volumes 526

9.6 Arc length and surface area 538

9.7 Water flow 547

9.8 Review 558

Answers 563

10 Applications of first-order differential equations 569

10.1 Overview 570

10.2 Growth and decay 571

10.3 Other applications of first-order differential equations 579

10.4 Bounded growth and Newton’s law of cooling 586

10.5 Chemical reactions and dilution problems 593

10.6 The logistic equation 605

10.7 Euler’s method 616

10.8 Slope fields 626

10.9 Review 640

Answers 646

11 Kinematics: rectilinear motion 655

11.1 Overview 656

11.2 Differentiating position and velocity 657

11.3 Constant acceleration 663

11.4 Velocity–time graphs 671

11.5 Acceleration that depends on time 684

11.6 Acceleration that depends on velocity 690

11.7 Acceleration that depends on position 703

11.8 Review 710

Answers 715

12 Vector calculus 721

12.1 Overview 722

12.2 Position vectors as functions of time 723

12.3 Differentiation of vectors 729

12.4 Special parametric curves 741

12.5 Integration of vectors 756

12.6 Projectile motion 762

12.7 Review 778

Answers 786

13 Probability and statistics 795

13.1 Overview 796

13.2 Linear combinations of random variables 797

13.3 Sample means and simulations 810

13.4 Confidence intervals 821

13.5 Hypothesis testing 830

13.6 Review 844

Answers 850

Glossary 853

Index 855