Understanding Engineering Mathematics (eBook)
560 Seiten
Elsevier Science (Verlag)
978-0-08-048152-4 (ISBN)
* Unique interactive style enables students to diagnose their strengths and weaknesses and focus their efforts where needed
* Ideal for self-study and tutorial work, building from an initially supportive approach to the development of independent learning skills
* Free website includes solutions to all exercises, additional topics and applications, guide to learning mathematics, and practice material
Students today enter engineering courses with a wide range of mathematical skills, due to the many different pre-university qualifications studied. Bill Cox's aim is for students to gain a thorough understanding of the maths they are studying, by first strengthening their background in the essentials of each topic. His approach allows a unique self-paced study style, in which students Review their strengths and weaknesses through self-administered diagnostic tests, then focus on Revision where they need it, to finally Reinforce the skills required.
The book is structured around a highly successful 'transition' maths course at Aston University which has demonstrated a clear improvement in students' achievement in mathematics, and has been commended by QAA Subject Review and engineering accreditation reports.
A core undergraduate text with a unique interactive style that enables students to diagnose their strengths and weaknesses and focus their efforts where needed
Ideal for self-paced self-study and tutorial work, building from an initially supportive approach to the development of independent learning skills
Lots of targeted examples and exercises
Students today enter engineering courses with a wide range of mathematical skills, due to the many different pre-university qualifications studied. Bill Cox's aim is for students to gain a thorough understanding of the maths they are studying, by first strengthening their background in the essentials of each topic. His approach allows a unique self-paced study style, in which students Review their strengths and weaknesses through self-administered diagnostic tests, then focus on Revision where they need it, to finally Reinforce the skills required. Understanding Engineering Mathematics is structured around a highly successful 'transition' maths course at Aston University which has demonstrated a clear improvement in students' achievement in mathematics, and has been commended by QAA Subject Review and engineering accreditation reports. - A core undergraduate text with a unique interactive style that enables students to diagnose their strengths and weaknesses and focus their efforts where needed- Ideal for self-paced self-study and tutorial work, building from an initially supportive approach to the development of independent learning skills- Lots of targeted examples and exercises
Cover 1
Contents 4
Preface 8
To the Student 10
1 Number and Arithmetic 12
Prerequisites 12
Objectives 12
Motivation 12
A note about calculators 13
1.1 Review 13
1.1.1 Types of numbers 13
1.1.2 Use of inequality signs 13
1.1.3 Highest common factor and lowest common multiple 14
1.1.4 Manipulation of numbers 14
1.1.5 Handling fractions 14
1.1.6 Factorial and combinatorial notation – permutations and combinations 15
1.1.7 Powers and indices 15
1.1.8 Decimal notation 15
1.1.9 Estimation 16
1.2 Revision 16
1.2.1 Types of numbers 16
1.2.2 Use of inequality signs 18
1.2.3 Highest common factor and lowest common multiple 19
1.2.4 Manipulation of numbers 21
1.2.5 Handling fractions 23
An electrical example – resistances in parallel 24
1.2.6 Factorial and combinatorial notation – permutations and combinations 27
1.2.7 Powers and indices 29
Examples 30
1.2.8 Decimal notation 33
Examples 34
1.2.9 Estimation 36
1.3 Reinforcement 38
1.3.1 Types of numbers 38
1.3.2 Use of inequality signs 38
1.3.3 Highest common factor and lowest common multiple 39
1.3.4 Manipulation of numbers 39
1.3.5 Handling fractions 39
1.3.6 Factorial and combinatorial notation – permutations and 40
combinations 40
1.3.7 Powers and indices 40
1.3.8 Decimal notation 41
1.3.9 Estimation 42
1.4 Applications 42
Answers to reinforcement exercises 43
1.3.1 Types of numbers 43
1.3.2 Use of inequality signs 44
1.3.3 Highest common factor and lowest common multiple 44
1.3.4 Manipulation of numbers 45
1.3.5 Handling fractions 45
1.3.6 Factorial and combinatorial notation 46
1.3.7 Powers and indices 46
1.3.8 Decimal notation 46
1.3.9 Estimation 47
2 Algebra 48
Prerequisites 48
Objectives 48
Motivation 48
2.1 Review 49
2.1.1 Multiplication of linear expressions 49
2.1.2 Polynomials 49
2.1.3 Factorisation of polynomials by inspection 49
2.1.4 Simultaneous equations 49
2.1.5 Equalities and identities 49
2.1.6 Roots and factors of a polynomial 50
2.1.7 Rational functions 50
2.1.8 Algebra of rational functions 50
2.1.9 Division and the remainder theorem 50
2.1.10 Partial fractions 50
2.1.11 Properties of quadratic expressions and equations 50
2.1.12 Powers and indices for algebraic expressions 51
2.1.13 The binomial theorem 51
2.2 Revision 51
2.2.1 Multiplication of linear expressions 51
Examples 51
2.2.2 Polynomials 54
Examples 54
2.2.3 Factorisation of polynomials by inspection 56
Example 56
2.2.4 Simultaneous equations 59
Example 59
2.2.5 Equalities and identities 61
Example 61
2.2.6 Roots and factors of a polynomial 63
Example 64
2.2.7 Rational functions 65
Examples 66
2.2.8 Algebra of rational functions 67
Example 67
Examples 68
Example 68
Examples 69
2.2.9 Division and the remainder theorem 71
2.2.10 Partial fractions 73
2.2.11 Properties of quadratic expressions and equations 75
Example 76
a positive 79
a 79
0/ 79
a negative 79
a 79
0/ 79
Example 79
2.2.12 Powers and indices for algebraic expressions 81
2.2.13 Binomial theorem 82
Example 83
2.3 Reinforcement 84
2.3.1 Multiplication of linear expressions 84
2.3.2 Polynomials 85
2.3.3 Factorisation of polynomials by inspection 86
2.3.4 Simultaneous equations 86
2.3.5 Equalities and identities 87
2.3.6 Roots and factors of a polynomial 87
2.3.7 Rational functions 87
2.3.8 Algebra of rational functions 87
2.3.9 Division and the remainder theorem 88
2.3.10 Partial fractions 88
2.3.11 Properties of quadratic expressions and equations 89
2.3.12 Powers and indices for algebraic expressions 89
2.3.13 The binomial theorem 90
2.4 Applications 90
Answers to reinforcement exercises 92
2.3.1 Multiplication of linear expressions 92
2.3.2 Polynomials 93
2.3.3 Factorisation of polynomials by inspection 93
2.3.4 Simultaneous equations 94
2.3.5 Equalities and identities 94
2.3.6 Roots and factors of a polynomial 94
2.3.7 Rational functions 94
2.3.8 Algebra of rational functions 95
2.3.9 Division and the remainder theorem 95
2.3.10 Partial fractions 95
2.3.11 Properties of quadratic expressions and equations 96
2.3.12 Powers and indices for algebraic expressions 96
2.3.13 The binomial theorem 96
3 Functions and Series 98
Prerequisites 98
Objectives 98
Motivation 98
A note about rigour 99
3.1 Review 99
3.1.1 Definition of a function 99
3.1.2 Plotting the graph of a function 99
3.1.3 Formulae 99
3.1.4 Odd and even functions 99
3.1.5 Composition of functions 100
3.1.6 Inequalities 100
3.1.7 Inverse of a function 100
3.1.8 Series and sigma notation 100
3.1.9 Finite series 100
3.1.10 Infinite series 100
3.1.11 Infinite binomial series 100
3.2 Revision 101
3.2.1 Definition of a function 101
Example 101
Examples 101
3.2.2 Plotting the graph of a function 102
3.2.3 Formulae 104
3.2.4 Odd and even functions 105
3.2.5 Composition of functions 108
Example 108
3.2.6 Inequalities 108
Example 110
3.2.7 Inverse of a function 111
Example 111
3.2.8 Series and sigma notation 113
Example 114
3.2.9 Finite series 114
Example 115
3.2.10 Infinite series 116
Example 117
3.2.11 Infinite binomial series 117
Example 118
3.3 Reinforcement 118
3.3.1 Definition of a function 118
3.3.2 Plotting the graph of a function 119
3.3.3 Formulae 119
3.3.4 Odd and even functions 119
3.3.5 Composition of functions 119
3.3.6 Inequalities 120
3.3.7 Inverse of a function 120
3.3.8 Series and sigma notation 120
3.3.9 Finite series 120
3.3.10 Infinite series 121
3.3.11 Infinite binomial series 121
3.4 Applications 121
Answers to reinforcement exercises 123
3.3.1 Definition of a function 123
3.3.2 Plotting the graph of a function 123
3.3.3 Formulae 125
3.3.4 Odd and even functions 126
3.3.5 Composition of functions 126
3.3.6 Inequalities 126
3.3.7 Inverse of a function 126
3.3.8 Series and sigma notation 126
3.3.9 Finite series 127
3.3.10 Infinite series 127
3.3.11 Infinite binomial series 127
4 Exponential and Logarithm Functions 129
Prerequisites 129
Objectives 129
Motivation 129
4.1 Review 130
4.1.1 y 130
an, n= an integer 130
4.1.2 The general exponential function ax 130
4.1.3 The natural exponential function ex 130
4.1.4 Manipulation of the exponential function 130
4.1.5 Logarithms to general base 131
4.1.6 Manipulation of logarithms 131
4.1.7 Some applications of logarithms 131
4.2 Revise 131
4.2.1 y 131
an, n= an integer 131
4.2.2 The general exponential function ax 132
4.2.3 The natural exponential function ex 135
Problem 1 135
Problem 2 136
Problem 3 137
Problem 4 137
4.2.4 Manipulation of the exponential function 140
4.2.5 Logarithms to general base 141
4.2.6 Manipulation of logarithms 142
4.2.7 Some applications of logarithms 145
4.3 Reinforcement 147
4.3.1 y 147
an, n= an integer 147
4.3.2 The general exponential function ax 147
4.3.3 The natural exponential function ex 147
4.3.4 Manipulation of the exponential function 147
4.3.5 Logarithms to general base 148
4.3.6 Manipulation of logarithms 148
4.3.7 Some applications of logarithms 149
4.4 Applications 149
Answers to reinforcement exercises 150
4.3.1 y 150
an, n= an integer 150
4.3.2 The general exponential function ax 151
4.3.3 The natural exponential function ex 151
4.3.4 Manipulation of the exponential function 151
4.3.5 Logarithms to general base 151
4.3.6 Manipulation of logarithms 151
4.3.7 Some applications of logarithms 152
5 Geometry of Lines, Triangles and Circles 153
Prerequisites 153
Objectives 153
Motivation 154
5.1 Review 154
5.1.1 Division of a line in a given ratio 154
5.1.2 Intersecting and parallel lines and angular measurement 154
5.1.3 Triangles and their elementary properties 155
5.1.4 Congruent triangles 155
5.1.5 Similar triangles 156
5.1.6 The intercept theorem 156
5.1.7 The angle bisector theorem 156
5.1.8 Pythagoras’ theorem 157
5.1.9 Lines and angles in a circle 157
5.1.10 Cyclic quadrilaterals 157
5.2 Revision 158
5.2.1 Division of a line in a given ratio 158
5.2.2 Intersecting and parallel lines and angular measurement 159
5.2.3 Triangles and their elementary properties 161
5.2.4 Congruent triangles 163
5.2.5 Similar triangles 163
5.2.6 The intercept theorem 164
5.2.7 The angle bisector theorem 164
5.2.8 Pythagoras’ theorem 165
5.2.9 Lines and angles in a circle 167
5.2.10 Cyclic quadrilaterals 170
5.3 Reinforcement 171
5.3.1 Division of a line in a given ratio 171
5.3.2 Intersecting and parallel lines and angular measurement 171
5.3.3 Triangles and their elementary properties 171
5.3.4 Congruent triangles 172
5.3.5 Similar triangles 173
5.3.6 The intercept theorem 173
5.3.7 The angle bisector theorem 174
5.3.8 Pythagoras’ theorem 174
5.3.9 Lines and angles in a circle 174
5.3.10 Cyclic quadrilaterals 176
5.4 Applications 176
Answers to reinforcement exercises 178
5.3.1 Division of a line in a given ratio 178
5.3.2 Intersecting and parallel lines and angular measurement 178
5.3.3 Triangles and their elementary properties 179
5.3.4 Congruent triangles 179
5.3.5 Similar triangles 179
5.3.6 The intercept theorem 179
5.3.7 The angle bisector theorem 179
5.3.8 Pythagoras’ theorem 179
5.3.9 Lines and angles in a circle 180
5.3.10 Cyclic quadrilaterals 180
6 Trigonometry 181
Prerequisites 181
Objectives 181
Motivation 182
6.1 Review 182
6.1.1 Radian measure and the circle 182
6.1.2 Definition of the trig ratios 182
6.1.3 Sine and cosine rules and solutions of triangles 183
6.1.4 Graphs of the trigonometric functions 183
6.1.5 Inverse trigonometric functions 183
6.1.6 The Pythagorean identities Ò cos2 183
1 183
6.1.7 Compound angle formulae 184
6.1.8 Trigonometric equations 184
6.1.9 The acos 184
bsin 184
form 184
6.2 Revision 184
6.2.1 Radian measure and the circle 184
6.2.2 Definition of the trig ratios 185
6.2.3 Sine and cosine rules and solutions of triangles 189
6.2.4 Graphs of the trigonometric functions 191
6.2.5 Inverse trigonometric functions 195
6.2.6 The Pythagorean identities Ò cos2 196
1 196
6.2.7 Compound angle formulae 198
6.2.8 Trigonometric equations 202
6.2.9 The acos 203
bsin 203
form 203
6.3 Reinforcement 205
6.3.1 Radian measure and the circle 205
6.3.2 Definitions of the trig ratios 205
6.3.3 Sine and cosine rules and the solution of triangles 205
6.3.4 Graphs of trigonometric functions 206
6.3.5 Inverse trigonometric functions 206
6.3.6 The Pythagorean identities Ò cos2 206
1 206
6.3.7 Compound angle formulae 207
6.3.8 Trigonometric equations 207
6.3.9 The acos 208
bsin 208
form 208
6.4 Applications 208
Answers to reinforcement exercises 209
6.3.1 Radian measure and the circle 209
6.3.2 Definitions of the trig ratios 210
6.3.3 Sine and cosine rules and the solution of triangles 210
6.3.4 Graphs of trigonometric functions 211
6.3.5 Inverse trigonometric functions 211
6.3.6 The Pythagorean identities Ò cos2 212
1 212
6.3.7 Compound angle formulae 212
6.3.8 Trigonometric equations 213
6.3.9 The acos 213
bsin 213
form 213
7 Coordinate Geometry 214
Prerequisites 214
Objectives 214
Motivation 214
7.1 Review 215
7.1.1 Coordinate systems in a plane 215
7.1.2 Distance between two points 215
7.1.3 Midpoint and gradient of a line 215
7.1.4 Equation of a straight line 215
7.1.5 Parallel and perpendicular lines 216
7.1.6 Intersecting lines 216
7.1.7 Equation of a circle 216
7.1.8 Parametric representation of curves 216
7.2 Revision 216
7.2.1 Coordinate systems in a plane 216
Example 217
7.2.2 Distance between two points 219
7.2.3 Midpoint and gradient of a line 220
7.2.4 Equation of a straight line 223
7.2.5 Parallel and perpendicular lines 225
7.2.6 Intersecting lines 227
Example 227
7.2.7 Equation of a circle 228
Problem 228
7.2.8 Parametric representation of curves 230
7.3 Reinforcement 231
7.3.1 Coordinate systems in a plane 231
7.3.2 Distance between two points 232
7.3.3 Midpoint and gradient of a line 232
7.3.4 Equation of a straight line 232
7.3.5 Parallel and perpendicular lines 233
7.3.6 Intersecting lines 233
7.3.7 Equation of a circle 233
7.3.8 Parametric representation of curves 233
7.4 Applications 234
Answers to reinforcement exercises 235
7.3.1 Coordinate systems in a plane 235
7.3.2 Distance between two points 236
7.3.3 Midpoint and gradient of a line 236
7.3.4 Equation of a straight line 236
7.3.5 Parallel and perpendicular lines 236
7.3.6 Intersecting lines 236
7.3.7 Equation of a 236
circle 236
7.3.8 Parametric representation of curves 237
8 Techniques of Differentiation 238
Prerequisites 238
Objectives 238
Motivation 239
8.1 Review 239
8.1.1 Geometrical interpretation of differentiation 239
8.1.2 Differentiation from first principles 239
8.1.3 Standard derivatives 240
8.1.4 Rules of differentiation 240
8.1.5 Implicit differentiation 240
8.1.6 Parametric differentiation 240
8.1.7 Higher order derivatives 240
8.2 Revision 241
8.2.1 Geometrical interpretation of differentiation 241
8.2.2 Differentiation from first principles 241
8.2.3 Standard derivatives 243
8.2.4 Rules of differentiation 245
8.2.5 Implicit differentiation 249
Example 249
8.2.6 Parametric differentiation 251
8.2.7 Higher order derivatives 252
8.3 Reinforcement 254
8.3.1 Geometrical interpretation of differentiation 254
8.3.2 Differentiation from first principles 254
8.3.3 Standard derivatives 254
8.3.4 Rules of differentiation 255
8.3.5 Implicit differentiation 255
8.3.6 Parametric differentiation 256
8.3.7 Higher order derivatives 256
8.4 Applications 256
Answers to reinforcement exercises 258
8.3.1 Geometrical interpretation of differentiation 258
8.3.2 Differentiation from first principles 259
8.3.3 Standard derivatives 259
8.3.4 Rules of differentiation 259
8.3.5 Implicit differentiation 259
8.3.6 Parametric differentiation 260
8.3.7 Higher order derivatives 260
9 Techniques of Integration 261
Prerequisites 261
Objectives 261
Motivation 261
9.1 Review 262
9.1.1 Definition of integration 262
9.1.2 Standard integrals 262
9.1.3 Addition of integrals 262
9.1.4 Simplifying the integrand 262
9.1.5 Linear substitution in integration 262
9.1.6 The du= f 263
x/ dx substitution 263
9.1.7 Integrating rational functions 263
9.1.8 Using trig identities in integration 263
9.1.9 Using trig substitutions in integration 263
9.1.10 Integration by parts 263
9.1.11 Choice of integration methods 264
9.1.12 The definite integral 264
9.2 Revision 264
9.2.1 Definition of integration 264
9.2.2 Standard integrals 266
9.2.3 Addition of integrals 268
9.2.4 Simplifying the integrand 269
9.2.5 Linear substitution in integration 271
9.2.6 The du= f 274
x/ dx substitution 274
9.2.7 Integrating rational functions 276
Example 277
9.2.8 Using trig identities in integration 280
9.2.9 Using trig substitutions in integration 283
9.2.10 Integration by parts 284
9.2.11 Choice of integration methods 287
9.2.12 The definite integral 289
9.3 Reinforcement 291
9.3.1 Definition of integration 291
9.3.2 Standard integrals 292
9.3.3 Addition of integrals 292
9.3.4 Simplifying the integrand 292
9.3.5 Linear substitution in integration 293
9.3.6 The du= f 293
x/ dx substitution 293
9.3.7 Integrating rational functions 294
9.3.8 Using trig identities in integration 294
9.3.9 Using trig substitutions in integration 294
9.3.10 Integration by parts 294
9.3.11 Choice of integration methods 295
9.3.12 The definite integral 295
9.4 Applications 296
Answers to reinforcement exercises 297
9.3.1 Definition of integration 297
9.3.2 Standard integrals 297
9.3.3 Addition of integrals 298
9.3.4 Simplifying the integrand 298
9.3.5 Linear substitution in integration 298
9.3.6 The du= f 298
x/ dx substitution 298
9.3.7 Integrating rational functions 299
9.3.8 Using trig 299
identities 299
in integration 299
9.3.9 Using trig substitutions in integration 299
9.3.10 299
Integration 299
by parts 299
9.3.11 Choice of integration methods 300
9.3.12 The definite integral 300
10 Applications of Differentiation and Integration 301
Prerequisites 301
Objectives 301
Motivation 302
10.1 Review 302
10.1.1 The derivative as a gradient and rate of change 302
10.1.2 Tangent and normal to a curve 302
10.1.3 Stationary points and points of inflection 302
10.1.4 Curve sketching in Cartesian coordinates 302
10.1.5 Applications of integration Ò area under a curve 302
10.1.6 Volume of a solid of revolution 303
10.2 Revise 303
10.2.1 The derivative as a gradient and rate of change 303
10.2.2 Tangent and normal to a curve 304
10.2.3 Stationary points and points of inflection 305
10.2.4 Curve sketching in Cartesian coordinates 310
10.2.5 Applications of integration Ò area under a curve 315
10.2.6 Volume of a solid of revolution 319
10.3 Reinforcement 320
10.3.1 The derivative as a gradient and rate of change 320
10.3.2 Tangent and normal to a curve 321
10.3.3 Stationary points and points of inflection 321
10.3.4 Curve sketching in Cartesian coordinates 321
10.3.5 Applications of integration Ò area under a curve 321
10.3.6 Volume of a solid of revolution 322
10.4 Applications 322
Answers to reinforcement exercises 325
10.3.1 The derivative as a gradient and rate of change 325
10.3.2 Tangent and normal to a curve 325
10.3.3 Stationary points and points of inflection 325
10.3.4 Curve sketching in Cartesian coordinates 326
10.3.5 Applications of integration Ò area under a curve 326
10.3.6 Volume of a solid of revolution 327
11 Vectors 328
Prerequisites 328
Objectives 328
Motivation 329
11.1 Introduction – representation of a vector quantity 329
Exercise on 11.1 330
Answers 330
11.2 Vectors as arrows 330
Exercise on 11.2 331
Answers 331
11.3 Addition and subtraction of vectors 332
Exercises on 11.3 333
Answers 334
11.4 Rectangular Cartesian coordinates in three dimensions 334
Exercise on 11.4 334
Answer 335
11.5 Distance in Cartesian coordinates 335
Problem 11.1 335
Exercise on 11.5 336
Answer 336
11.6 Direction cosines and ratios 336
Problem 11.2 337
Exercise on 11.6 338
Answer 338
11.7 Angle between two lines through the origin 338
Exercise on 11.7 339
Answer 339
11.8 Basis vectors 339
Problem 11.3 340
Exercises on 11.8 341
Answers 341
11.9 Properties of vectors 341
Problem 11.4 342
Problem 11.5 342
Problem 11.6 343
Exercises on 11.9 343
Answers 343
11.10 The scalar product of two vectors 344
Problem 11.7 346
Problem 11.8 346
Exercises on 11.10 347
Answers 347
11.11 The vector product of two vectors 347
Problem 11.9 348
Exercises on 11.11 349
Answers 349
11.12 Vector functions 350
Exercise on 11.12 351
Answer 351
11.13 Differentiation of vector functions 351
Problem 11.10 351
Problem 11.11 352
Problem 11.12 353
Problem 11.13 354
Exercises on 11.13 354
11.14 Reinforcement 355
11.15 Applications 358
11.16 Answers to reinforcement exercises 359
12 Complex Numbers 362
Prerequisites 362
Objectives 362
Motivation 362
12.1 What are complex numbers? 363
Problem 12.1 363
Exercise on 12.1 364
Answer 364
12.2 The algebra of complex numbers 364
Problem 12.2 364
Problem 12.3 365
Problem 12.4 365
Problem 12.5 365
Problem 12.6 366
Exercise on 12.2 366
Answer 366
12.3 Complex variables and the Argand plane 366
Exercises on 12.3 367
Answers 368
12.4 Multiplication in polar form 368
Problem 12.7 368
Problem 12.8 369
Exercises on 12.4 370
Answers 370
12.5 Division in polar form 371
Problem 12.9 371
Exercise on 12.5 372
Answer 372
12.6 Exponential form of a complex number 372
Problem 12.10 372
Problem 12.11 373
Exercises on 12.6 373
12.7 De Moivre’s theorem for integer powers 373
Problem 12.12 373
Exercise on 12.7 374
Answer 374
12.8 De Moivre’s theorem for fractional powers 374
Problem 12.13 376
Exercises on 12.8 377
Answer 377
12.9 Reinforcement 378
12.10 Applications 381
12.11 Answers to reinforcement exercises 384
13 Matrices and Determinants 388
Prerequisites 388
Objectives 388
Motivation 388
13.1 An overview of matrices and determinants 389
13.2 Definition of a matrix and its elements 389
Problem 13.1 390
Problem 13.2 390
Exercise on 13.2 392
Answer 392
13.3 Adding and multiplying matrices 392
Problem 13.3 393
Problem 13.4 394
Problem 13.5 395
Problem 13.6 395
Problem 13.7 396
Exercises on 13.3 396
Answers 397
13.4 Determinants 397
Problem 13.8 397
Problem 13.9 400
Problem 13.10 401
Exercises on 13.4 402
Answers 402
13.5 CramerÌs rule for solving a system of linear equations 402
Problem 13.11 403
Exercise on 13.5 404
Answer 404
13.6 The inverse matrix 404
Problem 13.12 405
Problem 13.13 407
Exercise on 13.6 408
Answer 408
13.7 Eigenvalues and eigenvectors 408
Problem 13.14 409
Exercise on 13.7 411
Answer 411
13.8 Reinforcement 411
13.9 Applications 414
13.10 Answers to reinforcement exercises 416
14 Analysis for Engineers – Limits, Sequences, Iteration, Series and All That 420
Prerequisites 420
Objectives 420
Motivation 421
14.1 Continuity and irrational numbers 421
Problem 14.1 421
Exercises on 14.1 422
Answers 422
14.2 Limits 423
Problem 14.2 424
Problem 14.3 424
Problem 14.4 425
Problem 14.5 426
Exercises on 14.2 427
Answers 427
14.3 Some important limits 427
Exercise on 14.3 429
Answer 429
14.4 Continuity 429
Problem 14.6 430
Exercise on 14.4 432
Answer 432
14.5 The slope of a curve 432
Problem 14.7 433
Exercise on 14.5 433
Answer 433
14.6 Introduction to infinite series 433
Exercise on 14.6 435
14.7 Infinite sequences 435
Problem 14.8 435
Problem 14.9 436
Exercise on 14.7 437
Answer 437
14.8 Iteration 437
Exercise on 14.8 438
Answer 438
14.9 Infinite series 439
Problem 14.10 439
Problem 14.11 440
Exercise on 14.9 440
Answer 441
14.10 Tests for convergence 441
Problem 14.12 442
Problem 14.13 442
Problem 14.14 443
Exercise on 14.10 445
Answer 445
14.11 Infinite power series 445
Problem 14.15 447
Problem 14.16 447
Exercise on 14.11 448
Answer 448
14.12 Reinforcement 449
14.13 Applications 452
14.14 Answers to reinforcement exercises 453
15 Ordinary Differential Equations 456
Prerequisites 456
Objectives 457
Motivation 457
15.1 Introduction 457
Problem 15.1 458
Exercise on 15.1 459
Answer 459
15.2 Definitions 459
Problem 15.2 459
Problem 15.3 460
Problem 15.4 460
Problem 15.5 461
Exercises on 15.2 462
Answers 462
15.3 First order equations Ò direct integration and separation of variables 463
Problem 15.6 464
Problem 15.7 465
Problem 15.8 468
Exercise on 15.3 468
Answer 468
15.4 Linear equations and integrating factors 469
Problem 15.9 470
Problem 15.10 471
Problem 15.11 471
Problem 15.12 471
Problem 15.13 472
Exercise on 15.4 472
Answer 473
15.5 Second order linear homogeneous differential equations 473
Problem 15.14 476
Problem 15.15 477
Problem 15.16 477
Exercises on 15.5 479
Answers 479
15.6 The inhomogeneous equation 479
Problem 15.17 480
Problem 15.18 481
Problem 15.19 481
Problem 15.20 482
Problem 15.21 483
Problem 15.22 483
Exercises on 15.6 485
Answers 486
15.7 Reinforcement 486
15.8 Applications 487
15.9 Answers to reinforcement exercises 491
16 Functions of More than One Variable – Partial Differentiation 494
Prerequisites 494
Objectives 494
Motivation 494
16.1 Introduction 495
Exercises on 16.1 495
Answers 495
16.2 Function of two variables 495
Exercise 16.2 497
Answer 497
16.3 Partial differentiation 498
Problem 16.1 499
Exercise on 16.3 499
Answer 499
16.4 Higher order derivatives 500
Problem 16.2 500
Exercises on 16.4 500
Answers 501
16.5 The total differential 501
Problem 16.3 501
Problem 16.4 503
Problem 16.5 504
Problem 16.6 505
Exercises on 16.5 505
Answers 505
16.6 Reinforcement 505
16.7 Applications 506
16.8 Answers to reinforcement exercises 507
17 An Appreciation of Transform Methods 511
17.1 Introduction 511
Prerequisites 511
Objectives 512
Motivation 512
17.2 The Laplace transform 512
Problem 17.1 512
Exercises on 17.2 515
Answers 515
17.3 Laplace transforms of the elementary functions 515
Problem 17.2 515
Problem 17.3 515
Problem 17.4 518
Exercises on 17.3 519
Answers 519
17.4 Properties of the Laplace transform 520
1. The Laplace transformis linear 520
2. The first shift theorem 520
3. The Laplace transformof the derivative 520
Problem 17.5 521
Problem 17.6 521
Exercises on 17.4 523
Answers 523
17.5 The inverse Laplace transform 523
Problem 17.7 523
Exercise on 17.5 523
Answer 524
17.6 Solution of initial value problems by Laplace transform 524
Problem 17.8 524
Problem 17.9 525
Exercises on 17.6 525
Answers 526
17.7 Linear systems and the principle of superposition 526
Exercise on 17.7 527
Answer 527
17.8 Orthogonality relations for trigonometric functions 527
Exercise on 17.8 528
17.9 The Fourier series expansion 528
Exercise on 17.9 530
Answer 531
17.10 The Fourier coefficients 531
Problem 17.10 532
Exercise on 17.10 533
17.11 Reinforcement 534
17.12 Applications 535
17.13 Answers to reinforcement exercises 538
Index 540
Number and Arithmetic
In this chapter we review the key features of elementary numbers and arithmetic. The topics covered are those found to be most useful later on.
Prerequisites
It will be helpful if you know something about:
• simple types of numbers such as integers, fractions, negative numbers, decimals
• the concepts of ‘greater than’ and ‘less than’
• elementary arithmetic: addition, subtraction, multiplication and division
• powers and indices notation, 23 = 2 × 2 × 2, for example
• how to convert a simple fraction to a decimal and vice versa
Objectives
In this chapter you will find:
• different types of numbers and their properties (particularly zero)
• the use of inequality signs
• highest common factors and lowest common denominators
• manipulation of numbers (BODMAS)
• handling fractions
• factorial (n!) and combinatorial nCror(rn)) notation
• powers and indices
• decimal notation
• estimation of numerical expressions
Motivation
You may need the material of this chapter for:
• numerical manipulation and calculation in engineering applications
• checking and using scientific formulae
• illustrating and checking results used later in mathematics
• statistical calculations
• numerical estimation and ‘back of an envelope’ calculations
A note about calculators
Calculators obviously have their place, particularly in applied mathematics, numerical methods and statistics. However, they are very rarely needed in this chapter, and the skills it aims to develop are better learnt without them.
1.1 Review
1.1.1 Types of numbers
A. For each number choose one or more descriptions from the following: (a) integer, (b) negative, (c) rational number (fraction), (d) real, (e) irrational, (f) decimal, (g) prime.
(i) is done as an example
(i) −1(a, b, c, d)
(ii) 2
(iii) 0
(iv) 7
(v) 5
(vi) 34
(vii) 0.73
(viii) 11
(ix) 8
(x)
(xi) −0.49
(xii) π
B. Which of the following descriptions apply to the expressions in (i)−(x) below?
(a) infinite
(b) does not exist
(c) negative
(d) zero
(e) finite
(f) non-zero
(i) 0 × 1(d, e)
(ii) 0+1
(iii) 0
(iv) 2−0
(v) 02
(vi) 0 − 1
(vii) 0
(viii) ×0+30
(ix) 30
(x) 2
1.1.2 Use of inequality signs
Express symbolically:
(i) x is a positive, non-zero, number (x > 0)
(ii) x lies strictly between 1 and 2
(iii) x lies strictly between −1 and 3
(iv) x is equal to or greater than −2 and is less than 2
(v) The absolute value of x is less than 2.
1.1.3 Highest common factor and lowest common multiple
A. Express in terms of prime factors
(i) 15 (= 3 × 5)
(ii) 21
(iii) 60
(iv) 121
(v) 405
(vi) 1024
(vii) 221
B. Find the highest common factor (HCF) of each of the following sets of numbers
(i) 24, 30 (6)
(ii) 27, 99
(iii) 28, 98
(iv) 12, 54, 78
(v) 3, 6, 15, 27
C. Find the lowest common multiple (LCM) of each of the following sets of numbers
(i) 3, 7 (21)
(ii) 3, 9
(iii) 12, 18
(iv) 3, 5, 9
(v) 2, 4, 6
1.1.4 Manipulation of numbers
Evaluate
(i) 2 + 3 − 7 (= −2)
(ii) 4 × 3 ÷ 2
(iii) 3 + 2 × 5
(iv) (3 + 2) × 5
(v) 3 + (2 × 5)
(vi) 18 ÷ 2 × 3
(vii) 18 ÷ (2 ×3)
(viii) −2 − (4 − 5)
(ix) (4 ÷ (−2)) × 3 − 4
(x) (3 + 7) ÷ 5 + (7−3) × (2 − 4)
1.1.5 Handling fractions
A. Simplify
(i) 6(=23)
(ii) 9
(iii) 3×47
(iv) 5×314
(v) 4÷45
(vi) 2+13
(vii) 2−13
(viii) 15−73
(ix) +12+13
(x) 3−34+18
B. If the numbers a and b are in the ratio a : b = 3 : 2 and a = 6, what is b?
1.1.6 Factorial and combinatorial notation – permutations and combinations
A. Evaluate
(i) 3! (=6)
(ii) 6!
(iii) !23!
(iv) !9!?3!
B.
(i) Evaluate (a) 3C2 (= 3) (b) 6C4 (c) 6P3
(ii) In how many ways can two distinct letters be chosen from ABCD?
(iii) How many permutations of the letters ABCDE are there?
1.1.7 Powers and indices
A. Reduce to simplest power form.
(i) 2324(=27)
(ii) 34/33
(iii) (52)3
(iv) (3 × 4)4/(9 × 23)
(v) 162/44
(vi) −6)2(−32)3
(vii) (−ab2)3/a2b
(viii) 2(12)−3
B. Express in terms of simple surds such as...
| Erscheint lt. Verlag | 19.10.2001 |
|---|---|
| Sprache | englisch |
| Themenwelt | Literatur |
| Kinder- / Jugendbuch | |
| Mathematik / Informatik ► Informatik ► Datenbanken | |
| Mathematik / Informatik ► Mathematik ► Algebra | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften | |
| Technik ► Bauwesen | |
| ISBN-10 | 0-08-048152-3 / 0080481523 |
| ISBN-13 | 978-0-08-048152-4 / 9780080481524 |
| Haben Sie eine Frage zum Produkt? |
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