Understanding Pure Mathematics

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Description

Contents

Reviews

Language

English

ISBN

9781382018203

Front Cover

IFC

Title Page

Preface

Contents

Introductory work

The number line

Inequalities

Basic algebraic manipulation

Surds

Quadratics

Trigonometric ratios of acute and obtuse angles

Solution of triangles

Three-dimensional trigonometry

1 Functions

Basic concepts

Composite functions

The inverse of a function

Ordered pairs

The graph of a function

Some further considerations

2 Vectors

Basic concepts

Position vectors and unit vectors

The scalar product

Geometrical proofs by vector methods

The vector equation of a straight line

3 Coordinate geometry I

Loci

Distance between two points

Mid-point of a line joining two points

Gradient of a line joining two points

Parallel and perpendicular lines

The equation y = mx + c

Finding the cartesian equation of a straight line

Relationships shown as linear graphs

Intersection of lines

4 Trigonometry I

Trigonometric ratios of any angle

Trigonometric graphs

Graph sketching

Solving trigonometric equations

Pythagorean relationships

Secant, cosecant and cotangent

Compound angles

Circular measure

5 Algebra I

Indices

Variation

Elementary theory of logarithms

Quadratics

Polynomials in general

6 Matrices

Basic Concepts

Transformations

Further considerations

General reflections and rotations

7 Permutations and combinations

Successive operations

Permutations

Combinations

Selections of any size from a group

Division into groups

8 Series and the binomial theorem

Arithmetic progressions

Geometric progressions

Summation of other series and proof by induction

The binomial theorem for a positive integral index

The binomial theorem for any rational index

9 Probability

Simple probability

Sum and product laws

Further conditional probability

Probability involving permutations and combinations

10 Calculus I: Differentiation

The gradient of a curve

Differentiation of axn

Maximum, minimum and points of inflexion

Small changes

More about points of inflexion

11 Sketching functions I

General methods

Further considerations

Simple transformations

1/f(x)

12 Calculus II: Integration

The reverse of differentiation

The area under a curve

Volume of revolution

Two applications

Mean value of a function

13 Calculus III: Further techniques

Function of a function

Integration of f’(x)[f(x)]n

Related rates of change

Parameters

Product rule

Quotient rule

Implicit functions

14 Sketching functions II

Rational functions

Inequalities

15 Trigonometry II

Sum and product formulae

a cos x + b sin x

Inverse trigonometric functions

General solutions

Small angles

Differentiation of trigonometric functions

Integration of trigonometric functions

Differentiation of inverse trigonometric functions

16 Coordinate geometry II

Angle between two straight lines

Distance of a point from a line

The circle

The parabola

The ellipse and the hyperbola

Polar coordinates

17 Three-dimensional work: vectors and matrices

Three-dimensional work

Differentiation and integration of vectors

Equations of a straight line in 3-D

Equations of a plane

3 x 3 Matrices

Sets of linear equations

18 Algebra II

Partial fractions

Partial fractions and series

Complex numbers

19 Exponential and logarithmic functions

Differentiating the exponential function

The logarithmic function

The exponential series

The logarithmic series

20 Calculus IV: Further integration

Change of variable

Integration by parts

General methods

Differential equations

21 Numerical methods

The trapezium rule

Simpson’s rule

Taylor’s theorem

Maclaurin’s theorem

Solution of equations

ez, cos z and sin z for complex z

Answers

Index

IBC

Back Cover

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