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}}}}±√a2−x21 and \displaystyle{\frac{a}{a^{2}+x^{2}}}a2+x2a | 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)}dxdy=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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