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.

Table of Contents

• 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