Editor/Author
Neill, Hugh

Publication Year: 2018

Publisher: Hodder & Stoughton

Price: Core Collection Only

ISBN: 978-1-4736-7844-6

Category: Mathematics & Statistics - Mathematics

Image Count:
87

Book Status: Available

Table of Contents

Calculus: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using calculus. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge.

- Introduction
- Functions
- 1.1 What is calculus?
- 1.2 Functions
- 1.3 Equations of functions
- 1.4 General notation for functions
- 1.5 Notation for increases in functions
- 1.6 Graphs of functions
- 1.7 Using calculators or computers for plotting functions
- 1.8 Inverse functions
- 1.9 Implicit functions
- 1.10 Functions of more than one variable
- Variations in functions; limits
- 2.1 Variations in functions
- 2.2 Limits
- 2.3 Limit of a function of the form
- 2.4 A trigonometric limit,
- 2.5 A geometric illustration of a limit
- 2.6 Theorems on limits
- Gradient
- 3.1 Gradient of the line joining two points
- 3.2 Equation of a straight line
- 3.3 Approximating to gradients of curves
- 3.4 Towards a definition of gradient
- 3.5 Definition of the gradient of a curve
- 3.6 Negative gradient
- Rate of change
- 4.1 The average change of a function over an interval
- 4.2 The average rate of change of a non-linear function
- 4.3 Motion of a body with non-constant velocity
- 4.4 Graphical interpretation
- 4.5 A definition of rate of change
- Differentiation
- 5.1 Algebraic approach to the rate of change of a function
- 5.2 The derived function
- 5.3 Notation for the derivative
- 5.4 Differentials
- 5.5 Sign of the derivative
- 5.6 Some examples of differentiation
- Some rules for differentiation
- 6.1 Differentiating a sum
- 6.2 Differentiating a product
- 6.3 Differentiating a quotient
- 6.4 Function of a function
- 6.5 Differentiating implicit functions
- 6.6 Successive differentiation
- 6.7 Alternative notation for derivatives
- 6.8 Graphs of derivatives
- Maxima, minima and points of inflexion
- 7.1 Sign of the derivative
- 7.2 Stationary values
- 7.3 Turning points
- 7.4 Maximum and minimum values
- 7.5 Which are maxima and which are minima?
- 7.6 A graphical illustration
- 7.7 Some worked examples
- 7.8 Points of inflexion
- Differentiating the trigonometric functions
- 8.1 Using radians
- 8.2 Differentiating sin x
- 8.3 Differentiating cos x
- 8.4 Differentiating tan x
- 8.5 Differentiating sec x, cosec x, cot x
- 8.6 Summary of results
- 8.7 Differentiating trigonometric functions
- 8.8 Successive derivatives
- 8.9 Graphs of the trigonometric functions
- 8.10 Inverse trigonometric functions
- 8.11 Differentiating sin−1 x and cos−1 x
- 8.12 Differentiating tan−1 x and cot−1 x
- 8.13 Differentiating sec−1 x and cosec−1 x
- 8.14 Summary of results
- Exponential and logarithmic functions
- 9.1 Compound Interest Law of growth
- 9.2 The value of
- 9.3 The Compound Interest Law
- 9.4 Differentiating ex
- 9.5 The exponential curve
- 9.6 Natural logarithms
- 9.7 Differentiating ln x
- 9.8 Differentiating general exponential functions
- 9.9 Summary of formulae
- 9.10 Worked examples
- Hyperbolic functions
- 10.1 Definitions of hyperbolic functions
- 10.2 Formulae connected with hyperbolic functions
- 10.3 Summary
- 10.4 Derivatives of the hyperbolic functions
- 10.5 Graphs of the hyperbolic functions
- 10.6 Differentiating the inverse hyperbolic functions
- 10.7 Logarithm equivalents of the inverse hyperbolic functions
- 10.8 Summary of inverse functions
- Integration; standard integrals
- 11.1 Meaning of integration
- 11.2 The constant of integration
- 11.3 The symbol for integration
- 11.4 Integrating a constant factor
- 11.5 Integrating xn
- 11.6 Integrating a sum
- 11.7 Integrating
- 11.8 A useful rule for integration
- 11.9 Integrals of standard forms
- 11.10 Additional standard integrals
- Methods of integration
- 12.1 Introduction
- 12.2 Trigonometric functions
- 12.3 Integration by substitution
- 12.4 Some trigonometrical substitutions
- 12.5 The substitution t = tan x
- 12.6 Worked examples
- 12.7 Algebraic substitutions
- 12.8 Integration by parts
- Integration of algebraic fractions
- 13.1 Rational fractions
- 13.2 Denominators of the form ax2 + bx + c
- 13.3 Denominator: a perfect square
- 13.4 Denominator: a difference of squares
- 13.5 Denominator: a sum of squares
- 13.6 Denominators of higher degree
- 13.7 Denominators with square roots
- Area and definite integrals
- 14.1 Areas by integration
- 14.2 Definite integrals
- 14.3 Characteristics of a definite integral
- 14.4 Some properties of definite integrals
- 14.5 Infinite limits and infinite integrals
- 14.6 Infinite limits
- 14.7 Functions with infinite values
- The integral as a sum; areas
- 15.1 Approximation to area by division into small elements
- 15.2 The definite integral as the limit of a sum
- 15.3 Examples of areas
- 15.4 Sign of an area
- 15.5 Polar coordinates
- 15.6 Plotting curves from their equations in polar coordinates
- 15.7 Areas in polar coordinates
- 15.8 Mean value
- Approximate integration
- 16.1 The need for approximate integration
- 16.2 The trapezoidal rule
- 16.3 Simpson's rule for area
- Volumes of revolution
- 17.1 Solids of revolution
- 17.2 Volume of a cone
- 17.3 General formula for volumes of solids of revolution
- 17.4 Volume of a sphere
- 17.5 Examples
- Lengths of curves
- 18.1 Lengths of arcs of curves
- 18.2 Length in polar coordinates
- Taylor's and Maclaurin's series
- 19.1 Infinite series
- 19.2 Convergent and divergent series
- 19.3 Taylor's expansion
- 19.4 Maclaurin's series
- 19.5 Expansion by the differentiation and integration of known series
- Differential equations
- 20.1 Introduction and definitions
- 20.2 Type I: one variable absent
- 20.3 Type II: variables separable
- 20.4 Type III: linear equations
- 20.5 Type IV: linear differential equations with constant coefficients
- 20.6 Type V: homogeneous equations
- Applications of differential equations
- 21.1 Introduction
- 21.2 Problems involving rates
- 21.3 Problems involving elements
- Answers