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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. The book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and integration. Everything you will need to know is here in one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions. Cover 1 Book title 2 Contents 4 Introduction 10 1 Functions 12 1.1 What is calculus? 13 1.2 Functions 13 1.3 Equations of functions 15 1.4 General notation for functions 16 1.5 Notation for increases in functions 18 1.6 Graphs of functions 19 1.7 Using calculators or computers for plotting functions 21 1.8 Inverse functions 22 1.9 Implicit functions 22 1.10 Functions of more than one variable 22 2 Variations in functions; limits 25 2.1 Variations in functions 26 2.2 Limits 29 2.3 Limit of a function of the form (omitted) 29 2.4 A trigonometric limit, (omitted) 31 2.5 A geometric illustration of a limit 33 2.6 Theorems on limits 33 3 Gradient 38 3.1 Gradient of the line joining two points 39 3.2 Equation of a straight line 41 3.3 Approximating to gradients of curves 42 3.4 Towards a definition of gradient 43 3.5 Definition of the gradient of a curve 45 3.6 Negative gradient 46 4 Rate of change 49 4.1 The average change of a function over an interval 50 4.2 The average rate of change of a non-linear function 53 4.3 Motion of a body with non-constant velocity 55 4.4 Graphical interpretation 58 4.5 A definition of rate of change 59 5 Differentiation 62 5.1 Algebraic approach to the rate of change of a function 63 5.2 The derived function 66 5.3 Notation for the derivative 67 5.4 Differentials 67 5.5 Sign of the derivative 68 5.6 Some examples of differentiation 68 6 Some rules for differentiation 76 6.1 Differentiating a sum 77 6.2 Differentiating a product 79 6.3 Differentiating a quotient 83 6.4 Function of a function 86 6.5 Differentiating implicit functions 92 6.6 Successive differentiation 94 6.7 Alternative notation for derivatives 96 6.8 Graphs of derivatives 96 7 Maxima, minima and points of inflexion 100 7.1 Sign of the derivative 101 7.2 Stationary values 103 7.3 Turning points 105 7.4 Maximum and minimum values 107 7.5 Which are maxima and which are minima? 109 7.6 A graphical illustration 113 7.7 Some worked examples 114 7.8 Points of inflexion 118 8 Differentiating the trigonometric functions 124 8.1 Using radians 125 8.2 Differentiating sin x 125 8.3 Differentiating cos x 127 8.4 Differentiating tan x 128 8.5 Differentiating sec x, cosec x, cot x 129 8.6 Summary of results 130 8.7 Differentiating trigonometric functions 131 8.8 Successive derivatives 134 8.9 Graphs of the trigonometric functions 135 8.10 Inverse trigonometric functions 140 8.11 Differentiating sin[sup(-1)] x and cos[sup(-1)] x 141 8.12 Differentiating tan[sup(-1)] x and cot[sup(-1)] x 142 8.13 Differentiating sec[sup(-1)] x and cosec[sup(-1)] x 143 8.14 Summary of results 145 9 Exponential and logarithmic functions 148 9.1 Compound Interest Law of growth 149 9.2 The value of (omitted) 152 9.3 The Compound Interest Law 153 9.4 Differentiating e[sup(x)] 154 9.5 The exponential curve 156 9.6 Natural logarithms 157 9.7 Differentiating ln x 158 9.8 Differentiating general exponential functions 159 9.9 Summary of formulae 159 9.10 Worked examples 160 10 Hyperbolic functions 163 10.1 Definitions of hyperbolic functions 164 10.2 Formulae connected with hyperbolic functions 167 10.3 Summary 169 10.4 Derivatives of the hyperbolic functions 170 10.5 Graphs of the hyperbolic functions 171 10.6 Differentiating the inverse hyperbolic functions 172 10.7 Logarithm equivalents of the inverse hyperbolic functions 174 10.8 Summary of inverse functions 176 11 Integration; standard integrals 179 11.1 Meaning of integration 180 11.2 The constant of integration 182 11.3 The symbol for integration 183 11.4 Integrating a constant factor 184 11.5 Integrating x[sup(n)] 185 11.6 Integrating a sum 187 11.7 Integrating (omitted) 188 11.8 A useful rule for integration 189 11.9 Integrals of standard forms 193 11.10 Additional standard integrals 195 12 Methods of integration 200 12.1 Introduction 201 12.2 Trigonometric functions 201 12.3 Integration by substitution 204 12.4 Some trigonometrical substitutions 205 12.5 The substitution t = tan (omitted)x 207 12.6 Worked examples 209 12.7 Algebraic substitutions 211 12.8 Integration by parts 214 13 Integration of algebraic fractions 219 13.1 Rational fractions 220 13.2 Denominators of the form ax[sup(2)] + bx + c 221 13.3 Denominator: a perfect square 223 13.4 Denominator: a difference of squares 223 13.5 Denominator: a sum of squares 226 13.6 Denominators of higher degree 228 13.7 Denominators with square roots 230 14 Area and definite integrals 234 14.1 Areas by integration 235 14.2 Definite integrals 237 14.3 Characteristics of a definite integral 240 14.4 Some properties of definite integrals 244 14.5 Infinite limits and infinite integrals 246 14.6 Infinite limits 247 14.7 Functions with infinite values 249 15 The integral as a sum; areas 254 15.1 Approximation to area by division into small elements 255 15.2 The definite integral as the limit of a sum 257 15.3 Examples of areas 259 15.4 Sign of an area 268 15.5 Polar coordinates 277 15.6 Plotting curves from their equations in polar coordinates 279 15.7 Areas in polar coordinates 281 15.8 Mean value 283 16 Approximate integration 287 16.1 The need for approximate integration 288 16.2 The trapezoidal rule 288 16.3 Simpson’s rule for area 290 17 Volumes of revolution 294 17.1 Solids of revolution 295 17.2 Volume of a cone 295 17.3 General formula for volumes of solids of revolution 297 17.4 Volume of a sphere 300 17.5 Examples 300 18 Lengths of curves 304 18.1 Lengths of arcs of curves 305 18.2 Length in polar coordinates 308 19 Taylor’s and Maclaurin’s series 312 19.1 Infinite series 313 19.2 Convergent and divergent series 313 19.3 Taylor’s expansion 314 19.4 Maclaurin’s series 316 19.5 Expansion by the differentiation and integration of known series 320 20 Differential equations 322 20.1 Introduction and definitions 323 20.2 Type I: one variable absent 326 20.3 Type II: variables separable 327 20.4 Type III: linear equations 329 20.5 Type IV: linear differential equations with constant coefficients 332 20.6 Type V: homogeneous equations 340 21 Applications of differential equations 344 21.1 Introduction 345 21.2 Problems involving rates 346 21.3 Problems involving elements 348 Answers 356 Index 380 A 380 C 380 D 380 E 381 F 381 G 381 H 382 I 382 L 382 M 382 N 383 O 383 P 383 Q 383 R 383 S 383 T 384 V 384 Z 384 "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. The book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and integration...Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions"--Publisher's description "Calculus--A Complete Introduction provides everything you need to broaden your skills and gain confidence. Assuming only a basic level of arithmetic, this carefully graded and progressive book guides you through the basic principles of the subject with the help of exercises and fully worked examples."--Publisher A bestselling introductory course, this book covers all areas of calculus, including functions, gradients, rates of change, differentiation, exponential and logarithmic functions and integration