Calculus : a complete introduction

Description

Cover Book title Contents Introduction 1 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 2 Variations in functions; limits 2.1 Variations in functions 2.2 Limits 2.3 Limit of a function of the form (omitted) 2.4 A trigonometric limit, (omitted) 2.5 A geometric illustration of a limit 2.6 Theorems on limits 3 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 4 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 5 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 6 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 7 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 8 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[sup(-1)] x and cos[sup(-1)] x 8.12 Differentiating tan[sup(-1)] x and cot[sup(-1)] x 8.13 Differentiating sec[sup(-1)] x and cosec[sup(-1)] x 8.14 Summary of results 9 Exponential and logarithmic functions 9.1 Compound Interest Law of growth 9.2 The value of (omitted) 9.3 The Compound Interest Law 9.4 Differentiating e[sup(x)] 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 10 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 11 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 x[sup(n)] 11.6 Integrating a sum 11.7 Integrating (omitted) 11.8 A useful rule for integration 11.9 Integrals of standard forms 11.10 Additional standard integrals 12 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 (omitted)x 12.6 Worked examples 12.7 Algebraic substitutions 12.8 Integration by parts 13 Integration of algebraic fractions 13.1 Rational fractions 13.2 Denominators of the form ax[sup(2)] + 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 14 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 15 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 16 Approximate integration 16.1 The need for approximate integration 16.2 The trapezoidal rule 16.3 Simpson’s rule for area 17 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 18 Lengths of curves 18.1 Lengths of arcs of curves 18.2 Length in polar coordinates 19 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 20 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 21 Applications of differential equations 21.1 Introduction 21.2 Problems involving rates 21.3 Problems involving elements Answers Index A C D E F G H I L M N O P Q R S T V Z