1 Sets
1.1 What a set is
1.1.1 Set-builder notation
1.1.2 Kinds of set
1.1.3 Subsets and equality
1.1.4 Cardinality and the power set
1.2 Operations on sets
1.2.1 Venn diagrams
1.2.2 Counting with a Venn diagram
1.3 Laws of the algebra of sets
1.4 Practice problems
2 Sets of Numbers
2.1 From counting numbers to real numbers
2.2 Interval notation
2.3 Radicals and rational powers
2.3.1 Rationalising the denominator
2.4 Complex numbers
2.4.1 Arithmetic with complex numbers
2.5 Practice problems
3 Functions
3.1 Ordered pairs and relations
3.1.1 Classifying relations
3.2 Functions
3.3 Composite functions
3.4 Odd and even functions
3.5 Inverse of a function
3.5.1 Finding an inverse
3.6 Practice problems
4 Polynomial Functions
4.1 Linear equations
4.1.1 Equations with fractions
4.1.2 Cross-multiplying
4.2 Quadratic functions and equations
4.2.1 Completing the square
4.2.2 The quadratic formula and the nature of the roots
4.2.3 An application
4.3 Division of polynomials
4.3.1 The remainder and factor theorems
4.4 Rational functions
4.5 Partial fractions
4.6 Inequalities
4.7 The modulus function
4.8 Practice problems
5 Trigonometry
5.1 Radian measure
5.2 The trigonometric ratios
5.2.1 Angles beyond the first quadrant
5.2.2 The exact values
5.3 Identities
5.3.1 Compound and double angle formulae
5.4 Trigonometric equations
5.5 Graphs, amplitude and period
5.6 Practice problems
6 Exponential and Logarithmic Functions
6.1 Laws of indices
6.2 Exponential functions
6.2.1 Exponential equations
6.3 Logarithmic functions
6.3.1 Rules of logarithms
6.3.2 Logarithmic equations
6.4 Graphs of logarithmic functions
6.5 Practice problems
7 Differentiation
7.1 Limits
7.1.1 Evaluating limits
7.1.2 Properties of limits
7.1.3 When a limit does not exist
7.1.4 Limits at infinity
7.2 Continuity
7.3 The derivative
7.3.1 Differentiation from first principles
7.4 Rules for differentiation
7.5 Implicit differentiation
7.6 The second derivative
7.7 Applications
7.7.1 Gradient of a curve
7.7.2 Increasing and decreasing functions
7.7.3 Turning points
7.7.4 Points of inflexion
7.7.5 Greatest and least values on an interval
7.7.6 Optimisation
7.8 Practice problems
8 Integral Calculus
8.1 Reversing differentiation
8.2 Using the integral symbol
8.2.1 Simplifying before integrating
8.2.2 Standard integrals
8.3 Finding the constant of integration
8.3.1 Definite integrals
8.4 Area under a curve
8.4.1 Area above the \(x\)-axis
8.4.2 Area below the \(x\)-axis
8.4.3 Regions that cross the axis
8.4.4 Area between a curve and a line
8.5 Methods of integration
8.5.1 Integration by substitution
8.5.2 Integration by parts
8.5.3 Integration using partial fractions
8.6 Practice problems
9 Descriptive Statistics
9.1 Introduction
9.2 Some basic definitions
9.3 Frequency distribution
9.3.1 Class boundaries and the width of an interval
9.4 Histograms
9.4.1 Modal class
9.5 Frequency polygons
9.6 Cumulative frequency curves (ogives)
9.7 Measures of central tendency
9.7.1 The arithmetic mean
9.7.2 Assumed mean
9.7.3 Mean of a grouped distribution
9.8 The median
9.8.1 Median for grouped data
9.9 The mode
9.9.1 Mode for grouped data
9.10 Measures of dispersion
9.10.1 Range
9.10.2 Mean deviation
9.10.3 Standard deviation
9.10.4 Variance
10 Probability
10.1 Basic ideas
10.2 Probability of an event
10.2.1 Properties of the probability of an event
10.3 Mutually exclusive events
10.3.1 Rules of probability
10.4 Independent events
10.5 Bayes’ theorem