xii Contents
- 1 Algebra Syllabus guidance xv
- 1.1 Introduction
- 1.2 Revision of basic laws
- 1.3 Revision of equations
- 1.4 Polynomial division
- 1.5 The factor theorem
- 1.6 The remainder theorem
- 2 Partial fractions
- 2.1 Introduction to partial fractions
- linear factors 2.2 Worked problems on partial fractions with
- repeated linear factors 2.3 Worked problems on partial fractions with
- quadratic factors 2.4 Worked problems on partial fractions with
- 2.1 Introduction to partial fractions
- 3 Logarithms
- 3.1 Introduction to logarithms
- 3.2 Laws of logarithms
- 3.3 Indicial equations
- 3.4 Graphs of logarithmic functions
- 4 Exponential functions
- 4.1 Introduction to exponential functions
- 4.2 The power series for ex
- 4.3 Graphs of exponential functions
- 4.4 Napierian logarithms
- 4.5 Laws of growth and decay
- linear form 4.6 Reduction of exponential laws to
- Revision Test
- 5 Hyperbolic functions
- 5.1 Introduction to hyperbolic functions
- 5.2 Graphs of hyperbolic functions
- 5.3 Hyperbolic identities
- functions 5.4 Solving equations involving hyperbolic
- 5.5 Series expansions for coshxand sinhx
- 6 Arithmetic and geometric progressions
- 6.1 Arithmetic progressions
- progressions 6.2 Worked problems on arithmetic
- progressions 6.3 Further worked problems on arithmetic
- 6.4 Geometric progressions
- progressions 6.5 Worked problems on geometric
- progressions 6.6 Further worked problems on geometric
- 6.1 Arithmetic progressions
- 7 The binomial series
- 7.1 Pascal’s triangle
- 7.2 The binomial series
- 7.3 Worked problems on the binomial series
- series 7.4 Further worked problems on the binomial
- theorem 7.5 Practical problems involving the binomial
- Revision Test
- 8 Maclaurin’s series
- 8.1 Introduction
- 8.2 Derivation of Maclaurin’s theorem
- 8.3 Conditions of Maclaurin’s series
- 8.4 Worked problems on Maclaurin’s series
- series 8.5 Numerical integration using Maclaurin’s
- 8.6 Limiting values
- 9 Solving equations by iterative methods
- 9.1 Introduction to iterative methods
- 9.2 The bisection method
- approximations 9.3 An algebraic method of successive
- 9.4 The Newton-Raphson method
- 10 Binary, octal and hexadecimal
- 10.1 Introduction
- 10.2 Binary numbers
- 10.3 Octal numbers
- 10.4 Hexadecimal numbers
- Revision Test
- 6 Arithmetic and geometric progressions
- 11 Introduction to trigonometry vi Contents
- 11.1 Trigonometry
- 11.2 The theorem of Pythagoras
- 11.3 Trigonometric ratios of acute angles
- 11.4 Evaluating trigonometric ratios
- 11.5 Solution of right-angled triangles
- 11.6 Angles of elevation and depression
- 11.7 Sine and cosine rules
- 11.8 Area of any triangle
- triangles and finding their areas 11.9 Worked problems on the solution of
- triangles and finding their areas 11.10 Further worked problems on solving
- trigonometry 11.11 Practical situations involving
- trigonometry 11.12 Further practical situations involving
- 12 Cartesian and polar co-ordinates
- 12.1 Introduction
- co-ordinates 12.2 Changing from Cartesian into polar
- co-ordinates 12.3 Changing from polar into Cartesian
- 12.4 Use of Pol/Rec functions on calculators
- 12.1 Introduction
- 13 The circle and its properties
- 13.1 Introduction
- 13.2 Properties of circles
- 13.3 Radians and degrees
- 13.4 Arc length and area of circles and sectors
- 13.5 The equation of a circle
- 13.6 Linear and angular velocity
- 13.7 Centripetal force
- Revision Test
- 14 Trigonometric waveforms
- 14.1 Graphs of trigonometric functions
- 14.2 Angles of any magnitude
- 14.3 The production of a sine and cosine wave
- 14.4 Sine and cosine curves
- 14.5 Sinusoidal formAsin(ωt±α)
- waveforms 14.6 Harmonic synthesis with complex
- 15 Trigonometric identities and equations
- 15.1 Trigonometric identities
- identities 15.2 Worked problems on trigonometric
- 15.3 Trigonometric equations
- equations 15.4 Worked problems (i) on trigonometric
- equations 15.5 Worked problems (ii) on trigonometric
- equations 15.6 Worked problems (iii) on trigonometric
- equations 15.7 Worked problems (iv) on trigonometric
- hyperbolic functions 16 The relationship between trigonometric and
- and hyperbolic functions 16.1 The relationship between trigonometric
- 16.2 Hyperbolic identities
- 17 Compound angles
- 17.1 Compound angle formulae
- Rsin(ωt+α) 17.2 Conversion ofasinωt+bcosωtinto
- 17.3 Double angles
- into sums or differences 17.4 Changing products of sines and cosines
- cosines into products 17.5 Changing sums or differences of sines and
- 17.6 Power waveforms in a.c. circuits
- Revision Test
- 17.1 Compound angle formulae
- 18 Functions and their curves
- 18.1 Standard curves
- 18.2 Simple transformations
- 18.3 Periodic functions
- 18.4 Continuous and discontinuous functions
- 18.5 Even and odd functions
- 18.6 Inverse functions
- 18.7 Asymptotes
- 18.8 Brief guide to curve sketching
- 18.9 Worked problems on curve sketching
- waveforms 19 Irregular areas, volumes and mean values of
- 19.1 Areas of irregular figures
- 19.2 Volumes of irregular solids
- 19.3 The mean or average value of a waveform
- Revision Test
- 20 Complex numbers
- 20.1 Cartesian complex numbers
- 20.2 The Argand diagram
- numbers 20.3 Addition and subtraction of complex
- numbers 20.4 Multiplication and division of complex
- 20.5 Complex equations Contents vii
- 20.6 The polar form of a complex number
- 20.7 Multiplication and division in polar form
- 20.8 Applications of complex numbers
- equations 15.4 Worked problems (i) on trigonometric
- 15.1 Trigonometric identities
- 21 De Moivre’s theorem
- 21.1 Introduction
- 21.2 Powers of complex numbers
- 21.3 Roots of complex numbers
- number 21.4 The exponential form of a complex
- 22 The theory of matrices and determinants
- 22.1 Matrix notation
- of matrices 22.2 Addition, subtraction and multiplication
- 22.3 The unit matrix
- 22.4 The determinant of a 2 by 2 matrix
- 22.5 The inverse or reciprocal of a 2 by 2 matrix
- 22.6 The determinant of a 3 by 3 matrix
- 22.7 The inverse or reciprocal of a 3 by 3 matrix
- matrices and determinants 23 The solution of simultaneous equations by
- matrices 23.1 Solution of simultaneous equations by
- determinants 23.2 Solution of simultaneous equations by
- Cramers rule 23.3 Solution of simultaneous equations using
- the Gaussian elimination method 23.4 Solution of simultaneous equations using
- Revision Test
- 22.1 Matrix notation
- 24 Vectors
- 24.1 Introduction
- 24.2 Scalars and vectors
- 24.3 Drawing a vector
- 24.4 Addition of vectors by drawing
- vertical components 24.5 Resolving vectors into horizontal and
- 24.6 Addition of vectors by calculation
- 24.7 Vector subtraction
- 24.8 Relative velocity
- 24.9 i,jandknotation
- 25 Methods of adding alternating waveforms
- 25.1 Combination of two periodic functions
- 25.2 Plotting periodic functions
- 25.3 Determining resultant phasors by drawing
- and cosine rules 25.4 Determining resultant phasors by the sine
- horizontal and vertical components 25.5 Determining resultant phasors by
- numbers 25.6 Determining resultant phasors by complex
- 26 Scalar and vector products
- 26.1 The unit triad
- 26.2 The scalar product of two vectors
- 26.3 Vector products
- 26.4 Vector equation of a line
- Revision Test
- 27 Methods of differentiation
- 27.1 Introduction to calculus
- 27.2 The gradient of a curve
- 27.3 Differentiation from first principles
- 27.4 Differentiation of common functions
- 27.5 Differentiation of a product
- 27.6 Differentiation of a quotient
- 27.7 Function of a function
- 27.8 Successive differentiation
- 28 Some applications of differentiation
- 28.1 Rates of change
- 28.2 Velocity and acceleration
- 28.3 Turning points
- and minimum values 28.4 Practical problems involving maximum
- 28.5 Tangents and normals
- 28.6 Small changes
- 29 Differentiation of parametric equations
- 29.1 Introduction to parametric equations
- 29.2 Some common parametric equations
- 29.3 Differentiation in parameters
- differentiation of parametric equations 29.4 Further worked problems on
- 30 Differentiation of implicit functions
- 30.1 Implicit functions
- 30.2 Differentiating implicit functions
- containing products and quotients 30.3 Differentiating implicit functions
- 30.4 Further implicit differentiation
- 31 Logarithmic differentiation
- 31.1 Introduction to logarithmic differentiation
- 31.2 Laws of logarithms
- 31.3 Differentiation of logarithmic functions
- 72 Inequalities Website Chapters
- 72.1 Introduction to inequalities
- 72.2 Simple inequalities
- 72.3 Inequalities involving a modulus
- 72.4 Inequalities involvingquotients
- 72.5 Inequalities involving square functions
- 72.6 Quadratic inequalities
- 73 Boolean algebra and logic circuits
- 73.1 Boolean algebra and switching circuits
- 73.2 Simplifying Boolean expressions
- 73.3 Laws and rules of Boolean algebra
- 73.4 De Morgan’s laws
- 73.5 Karnaugh maps
- 73.6 Logic circuits
- 73.7 Universal logic gates
- Revision Test
- 74 Sampling and estimation theories
- 74.1 Introduction
- 74.2 Sampling distributions
- 74.3 The sampling distribution of the means
- based on a large sample size 74.4 The estimation of population parameters
- on a small sample size 74.5 Estimating the mean of a population based
- 75 Significance testing
- 75.1 Hypotheses
- 75.2 Type I and Type II errors
- 75.3 Significance tests for population means
- 75.4 Comparing two sample means
- 76 Chi-square and distribution-free tests
- 76.1 Chi-square values
- 76.2 Fitting data to theoreticaldistributions
- 76.3 Introduction to distribution-free tests
- 76.4 The sign test
- 76.5 Wilcoxon signed-rank test
- 76.6 The Mann-Whitney test
- Revision Test
- 74.3 The sampling distribution of the means