MathsQuest 12

Topic 1 — Sketching graphs 1

Exercise 1.2 — An introduction to the modulus function 1

Exercise 1.3 — Sketching graphs of reciprocal functions 2

Exercise 1.4 — Sketching graphs of rational functions 8

Exercise 1.5 — S ketching graphs of y = f (x) and y = f ( x ) from y = f (x) 21

Exercise 1.6 — Circles, ellipses and hyperbolas 26

Topic 2 — Trigonometry 33

Exercise 2.2 — Reciprocal trigonometric functions 33

Exercise 2.3 — Trigonometric identities using reciprocal trigonometric functions 36

Exercise 2.4 — Compound-angle formulas 38

Exercise 2.5 — Double-angle formulas 42

Exercise 2.6 — Inverse trigonometric functions 49

Exercise 2.7 — General solutions of trigonometric equations 59

Exercise 2.8 — Graphs of reciprocal trigonometric functions 65

Exercise 2.9 — Graphs of inverse trigonometric functions 74

Topic 3 — Complex numbers 83

Exercise 3.2 — Complex numbers in rectangular form 83

Exercise 3.3 — Complex numbers in polar form 88

Exercise 3.4 — Solving polynomial equations 100

Exercise 3.5 — Subsets of the complex plane: circles, lines and rays 105

Exercise 3.6 — Roots of complex numbers 114

Topic 4 — Kinematics 121

Exercise 4.2 — Constant acceleration 121

Exercise 4.3 — Motion under gravity 122

Exercise 4.4 — Velocity–time graphs 124

Exercise 4.5 — Variable acceleration 126

Topic 5 — Vectors in three dimensions 131

Exercise 5.2 — Vectors 131

Exercise 5.3 — i j k notation 136

Exercise 5.4 — Scalar product and applications 143

Exercise 5.5 — Vector proofs using the scalar product 152

Exercise 5.6 — Parametric equations 158

Topic 6 — Mechanics 167

Exercise 6.2 — Statics of a particle 167

Exercise 6.3 — Inclined planes and connected particles 175

Exercise 6.4 — Dynamics 180

Exercise 6.5 — Dynamics with connected particles 186

Topic 7 — Differential calculus 193

Exercise 7.2 — Review of differentiation techniques 193

Exercise 7.3 — Applications of differentiation 202

Exercise 7.4 — Implicit and parametric differentiation 209

Exercise 7.5 — Second derivatives 217

Exercise 7.6 — Curve sketching 225

Exercise 7.7 — Derivatives of inverse trigonometric functions 239

Exercise 7.8 — Related rate problems 246

Topic 8 — Integral calculus 253

Exercise 8.2 — Areas under and between curves 253

Exercise 8.3 — Linear substitutions 264

Exercise 8.4 — Other substitutions 275

Exercise 8.5 — Integrals of powers of trigonometric functions 284

Exercise 8.6 — Integrals involving inverse trigonometric functions 291

Exercise 8.7 — Integrals involving partial fractions 302

Topic 9 — Differential equations 315

Exercise 9.2 — Verifying solutions to a differential equation 315

Exercise 9.3 — Solving Type 1 differential equations, = dy dx f (x) 325

Exercise 9.4 — Solving Type 2 differential equations, = dy dx f ( y) 330

Exercise 9.5 — Solving Type 3 differential equations, = dy dx f (x)g(y) 338

Exercise 9.6 — Solving Type 4 differential equations, = d y dx f x ( ) 2 2 344

Topic 10 — Further applications of integration 353

Exercise 10.2 — Integration by recognition 353

Exercise 10.3 — Solids of revolution 365

Exercise 10.4 — Volumes 372

Exercise 10.5 — Arc length, numerical integration and graphs of antiderivatives 381

Exercise 10.6 — Water flow 389

Topic 11 — Applications of first-order differential equations 397

Exercise 11.2 — Growth and decay 397

Exercise 11.3 — Other applications of first-order differential equations 400

Exercise 11.4 — Bounded growth and Newton’s Law of Cooling 406

Exercise 11.5 — Chemical reactions and dilution problems 413

Exercise 11.6 — The logistic equation 425

Exercise 11.7 — Euler’s method 439

Exercise 11.8 — Slope fields 452

Topic 12 — Variable forces 455

Exercise 12.2 — Forces that depend on time 455

Exercise 12.3 — Forces that depend on velocity 461

Exercise 12.4 — Forces that depend on displacement 475

Topic 13 — Vector calculus 483

Exercise 13.2 — Position vectors as functions of time 483

Exercise 13.3 — Differentiation of vectors 489

Exercise 13.4 — Special parametric curves 500

Exercise 13.5 — Integration of vectors 512

Exercise 13.6 — Projectile motion 525

Topic 14 — Statistical inference 539

Exercise 14.2 — Linear combinations of random variables 539

Exercise 14.3 — Sample means 540

Exercise 14.4 — Confidence intervals 542

Exercise 14.5 — Hypothesis testing 543