Table of Contents
Introduction xi
1 Functions 1
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 13
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
3 Gradient 27
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 37
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 49
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 63
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 85
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 107
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-1x and cos-1x
8.12 Differentiating tarn-1x and cot-1 x
8.13 Differentiating sec-1x and cosec-1x
8.14 Summary of results
9 Exponential and logarithmic functions 129
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
10 Hyperbolic functions 143
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 159
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 1/x
11.8 A useful rule for integration
11.9 Integrals of standard forms
11.10 Additional standard integrals
12 Methods of integration 179
12.1 Introduction
12.2 Trigonometrie functions
12.3 Integration by substitution
12.4 Some trigonometrical substitutions
12.5 The substitution t=tan ½x
12.6 Worked examples
12.7 Algebraic substitutions
12.8 Integration by parts
13 Integration of algebraic fractions 197
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
14 Area and definite integrals 211
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 229
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 259
16.1 The need for approximate integration
16.2 The trapezoidal rule
16.3 Simpson's rule for area
17 Volumes of revolution 267
17.1 Solids of revolution
17.2 Volume of a cone
17.3 General formula for volumes of solids of revolution
17.6 Volume of a sphere
17.5 Examples
18 Lengths of curves 277
18.1 Lengths of arcs of curves
18.2 Length in polar coordinates
19 Taylor's and Maclaurin's series 285
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 295
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 315
21.1 Introduction
21.2 Problems involving rates
21.3 Problems involving elements
Answers 325
Index 347
