# Calculus: A Complete Introduction: Teach Yourself

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ISBN-13: 9781473678446 Quercus 09/25/2018 388 818,705 5.30(w) x 8.40(h) x 0.80(d)

Hugh Neill is a maths teacher who has also been an inspector and chief examiner. His books have helped over 100,000 people improve their mathematics.

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

3.1 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

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

8.2 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

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