Teach Yourself: Calculus: A Complete Introduction

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.

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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