JAIN CORPORATION AUSTRALIA

Mathematics for Australia 12 Specialist Mathematics

 1 MATHEMATICAL INDUCTION 9 A The process of induction 10 B The principle of mathematical induction 11 Review set 1A 17 Review set 1B 18 2 REAL POLYNOMIALS 19 A Operations with polynomials 21 B Zeros, roots, and factors 23 C Polynomial equality 26 D Polynomial division 29 E The Remainder theorem 36 F The Factor theorem 38 G The Fundamental Theorem of Algebra 39 H Sum and product of roots theorem (Extension) 41 I Graphing real polynomials 43 J Polynomial equations 50 Review set 2A 51 Review set 2B 53 3 FUNCTIONS 55 A Composite functions 56 B Inverse functions 58 C Reciprocal functions 65 D The reciprocals of other functions 66 E Rational functions 68 F Absolute value functions 71 Review set 3A 78 Review set 3B 80 4 COMPLEX NUMBERS 83 A The complex plane 84 B Modulus and argument 86 C Polar form 90 D Euler’s form 97 E De Moivre’s theorem 100 F Roots of complex numbers 103 Review set 4A 107 Review set 4B 108 5 VECTORS 111 A Vectors in space 113 B Operations with vectors in space 115 C Vector algebra 117 D The vector between two points 118 E Parallelism 122 F The scalar product of two vectors 125 G The angle between two vectors 127 H Proof using vector geometry 131 I The vector product of two vectors 132 Review set 5A 138 Review set 5B 139 6 VECTOR APPLICATIONS 141 A Area 143 B Lines in 2 and 3 dimensions 144 C The angle between two lines 148 D Constant velocity problems 150 E The shortest distance from a point to a line 154 F Intersecting lines 158 G Relationships between lines 160 H Planes 168 I Angles in space 176 J Solving 3 × 3 linear systems 178 K Intersecting planes 180 Review set 6A 184 Review set 6B 187 7 INTEGRATION 191 A Rules for integration 192 B Integrating \displaystyle{\pm\frac{1}{\sqrt{a^{2}-x^{2}}}}±​√​a​2​​−x​2​​​​​​​1​​ and \displaystyle{\frac{a}{a^{2}+x^{2}}}​a​2​​+x​2​​​​a​​ 198 C Integration by substitution 201 D Integration by parts 208 E The area between two functions 210 F Solids of revolution 216 Review set 7A 223 Review set 7B 224 8 RATES OF CHANGE AND DIFFERENTIAL EQUATIONS 227 A Implicit differentiation 228 B Related rates 232 C Differential equations 237 D Differential equations of the form \displaystyle{\frac{dy}{dx}=f(x)}​dx​​dy​​=f(x) 240 E Separable differential equations 242 F Slope fields 244 G Problem solving 248 H Logistic growth 252 I Equations of motion 258 J Simple harmonic motion 261 Review set 8A 265 Review set 8B 267 9 VECTOR CALCULUS 271 A Parametric equations 272 B Pairs of uniformly varying quantities 274 C Pairs of non-uniformly varying quantities 276 D Bezier curves 283 E Trigonometric parameterisation 286 F Arc lengths of parametric curves 294 Review set 9A 297 Review set 9B 299 ANSWERS 301 INDEX 335