Table of Contents

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

 

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