Cambridge Additional Mathematics

1	SETS AND VENN DIAGRAMS	11
 	A	Sets	12
 	B	Interval notation	15
 	C	Relations	17
 	D	Complements of sets	18
 	E	Properties of union and intersection	20
 	F	Venn diagrams	21
 	G	Numbers in regions	26
 	H	Problem solving with Venn diagrams	28
 	 	Review set 1A	31
 	 	Review set 1B	33
 	 	 	 
2	FUNCTIONS	35
A	Relations and functions	36
 	B	Function notation	40
 	C	Domain and range	43
 	D	The modulus function	46
 	E	Composite functions	49
 	F	Sign diagrams	51
 	G	Inverse functions	54
 	 	Review set 2A	60
 	 	Review set 2B	61
 	 	 	 
3	QUADRATICS	63
 	A	Quadratic equations	65
B	Quadratic inequalities	72
 	C	The discriminant of a quadratic	73
 	D	Quadratic functions	75
 	E	Finding a quadratic from its graph	87
 	F	Where functions meet	91
 	G	Problem solving with quadratics	93
 	H	Quadratic optimisation	95
 	 	Review set 3A	98
 	 	Review set 3B	99
 	 	 	 
4	SURDS, INDICES, AND EXPONENTIALS	101
 	A	Surds	102
 	B	Indices	107
 	C	Index laws	108
 	D	Rational indices	111
 	E	Algebraic expansion and factorisation	113
 	F	Exponential equations	116
 	G	Exponential functions	118
 	H	The natural exponential e^x	123
 	 	Review set 4A	125
 	 	Review set 4B	127
 	 	 	 
5	LOGARITHMS	129
 	A	Logarithms in base 10	130
 	B	Logarithms in base a	133
 	C	Laws of logarithms	135
 	D	Logarithmic equations	138
 	E	Natural logarithms	142
 	F	Solving exponential equations using logarithms	145
 	G	The change of base rule	147
 	H	Graphs of logarithmic functions	149
 	 	Review set 5A	152
 	 	Review set 5B	154
 	 	 	 
6	POLYNOMIALS	155
 	A	Real polynomials	156
 	B	Zeros, roots, and factors	162
 	C	The Remainder theorem	167
 	D	The Factor theorem	169
 	E	Cubic equations	171
 	 	Review set 6A	173
 	 	Review set 6B	173
 	 	 	 
7	STRAIGHT LINE GRAPHS	175
 	A	Equations of straight lines	177
 	B	Intersection of straight lines	183
 	C	Intersection of a straight line and a curve	186
 	D	Transforming relationships to straight line form	187
 	E	Finding relationships from data	192
 	 	Review set 7A	197
 	 	Review set 7B	199
 	 	 	 
8	THE UNIT CIRCLE AND RADIAN MEASURE	201
 	A	Radian measure	202
 	B	Arc length and sector area	205
 	C	The unit circle and the trigonometric ratios	208
 	D	Applications of the unit circle	213
 	E	Multiples of π/6 and π/4	217
 	F	Reciprocal trigonometric ratios	221
 	 	Review set 8A	221
 	 	Review set 8B	222
 	 	 	 
9	TRIGONOMETRIC FUNCTIONS	225
 	A	Periodic behaviour	226
 	B	The sine function	230
 	C	The cosine function	236
 	D	The tangent function	238
 	E	Trigonometric equations	240
 	F	Trigonometric relationships	246
 	G	Trigonometric equations in quadratic form	250
 	 	Review set 9A	251
 	 	Review set 9B	252
 	 	 	 
10	COUNTING AND THE BINOMIAL EXPANSION	255
 	A	The product principle	256
 	B	Counting paths	258
 	C	Factorial notation	259
 	D	Permutations	262
 	E	Combinations	267
 	F	Binomial expansions	270
 	G	The Binomial Theorem	273
 	 	Review set 10A	277
 	 	Review set 10B	278
 	 	 	 
11	VECTORS	279
 	A	Vectors and scalars	280
 	B	The magnitude of a vector	284
 	C	Operations with plane vectors	285
 	D	The vector between two points	289
 	E	Parallelism	292
 	F	Problems involving vector operations	294
 	G	Lines	296
 	H	Constant velocity problems	298
 	 	Review set 11A	302
 	 	Review set 11B	303
 	 	 	 
12	MATRICES	305
 	A	Matrix structure	307
 	B	Matrix operations and definitions	309
 	C	Matrix multiplication	315
 	D	The inverse of a 2 × 2 matrix	323
 	E	Simultaneous linear equations	328
 	 	Review set 12A	330
 	 	Review set 12B	331
 	 	 	 
13	INTRODUCTION TO DIFFERENTIAL CALCULUS	333
 	A	Limits	335
 	B	Rates of change	336
 	C	The derivative function	340
 	D	Differentiation from first principles	342
 	E	Simple rules of differentiation	344
 	F	The chain rule	348
 	G	The product rule	351
 	H	The quotient rule	353
 	I	Derivatives of exponential functions	355
 	J	Derivatives of logarithmic functions	359
 	K	Derivatives of trigonometric functions	361
 	L	Second derivatives	363
 	 	Review set 13A	365
 	 	Review set 13B	366
 	 	 	 
14	APPLICATIONS OF DIFFERENTIAL CALCULUS	367
 	A	Tangents and normals	369
 	B	Stationary points	375
 	C	Kinematics	380
 	D	Rates of change	388
 	E	Optimisation	393
 	F	Related rates	399
 	 	Review set 14A	402
 	 	Review set 14B	405
 	 	 	 
15	INTEGRATION	409
 	A	The area under a curve	410
 	B	Antidifferentiation	415
 	C	The fundamental theorem of calculus	417
 	D	Integration	422
 	E	Rules for integration	424
 	F	Integrating f(ax+b)	428
 	G	Definite integrals	431
 	 	Review set 15A	434
 	 	Review set 15B	435
 	 	 	 
16	APPLICATIONS OF INTEGRATION	437
 	A	The area under a curve	438
 	B	The area between two functions	440
 	C	Kinematics	444
 	 	Review set 16A	449
 	 	Review set 16B	450
 	 	 	 
 	ANSWERS	453
 	 	 	 
 	INDEX	503