UNDERSTANDING ENGINEERING MATHEMATICS JOHN BIRD WORKED SOLUTIONS TO EXERCISES 1
INTRODUCTION In Understanding Engineering Mathematic there are over 750 further problems arranged regularly throughout the text within 370 Exercises. The solutions for all 750 of these further problems has been prepared in this document. CONTENTS Page Chapter 1 Basic arithmetic (Exercises 1 to 4) 1 Chapter Fractions (Exercises 5 to 7) 6 Chapter 3 Decimals (Exercises 8 to 11) 1 Chapter 4 Using a calculator (Exercises 1 to ) 33 Chapter 5 Percentages (Exercises 3 to 5) 47 Chapter 6 Ratio and proportion (Exercises 6 to 30) 74 Chapter 7 Powers, roots and laws of indices (Exercises 31 to 33) 88 Chapter 8 Units, prefixes and engineering notation (Exercises 34 to 36) 10 Chapter 9 Basic algebra (Exercises 37 to 40) 11 Chapter 10 Further algebra (Exercises 41 to 43) 1 Chapter 11 Solving simple equations (Exercises 44 to 47) 133 Chapter 1 Transposing formulae (Exercises 48 to 50) 143 Chapter 13 Solving simultaneous equations (Exercises 51 to 55) 163 Chapter 14 Solving quadratic equations (Exercises 56 to 60) 180 Chapter 15 Logarithms (Exercises 61 to 63) 10 Chapter 16 Exponential functions (Exercises 64 to 68) 37 Chapter 17 Inequalities (Exercises 69 to 73) 49 Chapter 18 Polynomial division and the factor and remainder theorems (Exercises 74 to 76) 73 Chapter 19 Number sequences (Exercises 77 to 33) 84 Chapter 0 Binary, octal and hexadecimal (Exercises 84 to 89) 93 Chapter 1 Partial fractions (Exercises 90 to 9) 311 Chapter The binomial series (Exercises 93 to 96) 335 Chapter 3 Maclaurin s series (Exercises 97 to 99) 345 Chapter 4 Hyperbolic functions (Exercises 100 to 103) 359 Chapter 5 Solving equations by iterative methods (Exercises 104 to 106) 370 Chapter 6 Boolean algebra and logic circuits (Exercises 107 to 11) 385 Chapter 7 Areas of common shapes (Exercises 113 to 117) 410 Chapter 8 The circle and its properties (Exercises 118 to 13) 438 Chapter 9 Volumes and surface areas of common solids (Exercises 14 to 130) 451 Chapter 30 Irregular areas and volumes and mean values (Exercises 131 to 133) 468 Chapter 31 Straight line graphs (Exercises 134 to 136) 503
Chapter 3 Reduction of non-linear laws to linear form (Exercises 137 to 138) 513 Chapter 33 Graphs with logarithmic scales (Exercises 139 to 141) 541 Chapter 34 Polar curves (Exercise 14) 558 Chapter 35 Graphical solution of equations (Exercises 143 to 146) 568 Chapter 36 Functions and their curves (Exercises 147 to 149) 571 Chapter 37 Angles and triangles (Exercises 150 to 155) 591 Chapter 38 Introduction to trigonometry (Exercises 156 to 16) 615 Chapter 39 Trigonometric waveforms (Exercises 163 to 166) 634 Chapter 40 Cartesian and polar coordinates (Exercises 167 to 168) 661 Chapter 41 Non-right-angled triangles and some practical applications (Exercises 169 to 17) 679 Chapter 4 Trigonometric identities and equations Chapter 43 (Exercises 173 to 177) 687 The relationship between trigonometric and hyperbolic functions (Exercises 178 to 179) 706 Chapter 44 Compound angles (Exercises 180 to 184) 73 Chapter 45 Complex numbers (Exercises 185 to 189) 79 Chapter 46 De Moivre s theorem (Exercises 190 to 193) 757 Chapter 47 The theory of matrices and determinants (Exercises 194 to 198) 779 Chapter 48 Applications of matrices and determinants (Exercises 199 to 03) 800 Chapter 49 Vectors (Exercises 04 to 08) 816 Chapter 50 Methods of adding alternating waveforms (Exercises 09 to 13) 849 Chapter 51 Scalar and vector products (Exercises 14 to 16) 866 Chapter 5 Introduction to differentiation (Exercises 17 to ) 886 Chapter 53 Methods of differentiation (Exercises 3 to 7) 897 Chapter 54 Some applications of differentiation (Exercises 8 to 34) 911 Chapter 55 Differentiation of parametric equations (Exercises 35 to 36) 98 Chapter 56 Differentiation of implicit functions (Exercises 37 to 39) 961 Chapter 57 Logarithmic differentiation (Exercises 40 to 4) 970 Chapter 58 Differentiation of hyperbolic functions (Exercise 43) 978 Chapter 59 Differentiation of inverse trigonometric and hyperbolic functions (Exercises 44 to 46) 987 Chapter 60 Partial differentiation (Exercises 47 to 48) 989 Chapter 61 Total differential, rates of change and small changes (Exercises 49 to 51) 100 Chapter 6 Maxima, minima and saddle points for functions of two variables (Exercises 5 to 53) 101 Chapter 63 Standard integration (Exercises 54 to 55) 101 Chapter 64 Chapter 65 Integration using algebraic substitutions (Exercises 56 to 57) 1034 Integration using trigonometric and hyperbolic substitutions (Exercises 58 to 64) 1043 Chapter 66 Integration using partial fractions (Exercises 65 to 67) 1053 Chapter 67 The t = tan θ/ substitution (Exercises 68 to 69) 1068 Chapter 68 Integration by parts (Exercises 70 to 71) 1079 Chapter 69 Reduction formulae (Exercises 7 to 75) 1086 Chapter 70 Double and triple integrals (Exercises 76 to 77) 1099 Chapter 71 Numerical integration (Exercises 78 to 80) 1110 3
Chapter 7 Areas under and between curves (Exercises 81 to 83) 110 Chapter 73 Mean and root mean square values (Exercises 84 to 85) 1133 Chapter 74 Volumes of solids of revolution (Exercises 86 to 87) 1146 Chapter 75 Centroids of simple shapes (Exercises 88 to 90) 1154 Chapter 76 Second moments of area (Exercises 91 to 9) 1166 Chapter 77 Solution of first-order differential equations by separation of variables (Exercises 93 to 96) 118 Chapter 78 Homogeneous first-order differential equations (Exercises 97 to 98) 1191 Chapter 79 Linear first-order differential equations (Exercises 99 to 300) 107 Chapter 80 Numerical methods for first-order differential equations (Exercises 301 to 303) 118 Chapter 81 d y dy Second-order differential equations of the form a b cy 0 dx + dx (Exercises 304 to 305) 130 Chapter 8 d y dy Second-order differential equations of the form a b cy f(x) dx + dx (Exercises 306 to 309) 147 Chapter 83 Power series methods of solving ordinary differential equations (Exercises 310 to 315) 157 Chapter 84 An introduction to partial differential equations (Exercises 316 to 30) 185 Chapter 85 Presentation of statistical data (Exercises 31 to 33) 131 Chapter 86 Measures of central tendency and dispersion (Exercises 34 to 37) 1333 Chapter 87 Probability (Exercises 38 to 330) 1350 Chapter 88 The binomial and Poisson distributions (Exercises 331 to 33) 136 Chapter 89 The normal distribution (Exercises 333 to 334) 1370 Chapter 90 Linear correlation (Exercise 335) 1378 Chapter 91 Linear regression (Exercise 336 ) 1393 Chapter 9 Sampling and estimation theories (Exercises 337 to 339) 1399 Chapter 93 Significance testing (Exercises 340 to 34) 1408 Chapter 94 Chi-square and distribution-free tests (Exercises 343 to 347) 141 Chapter 95 Introduction to Laplace transforms (Exercise 348) 1435 Chapter 96 Properties of Laplace transforms (Exercises 349 to 351) 145 Chapter 97 Inverse Laplace transforms (Exercises 35 to 354) 1455 Chapter 98 The Laplace transform of the Heaviside function (Exercises 355 to 357) 1464 Chapter 99 The solution of differential equations using Laplace transforms (Exercise 358) 1475 Chapter 100 The solution of simultaneous differential equations using Laplace 1487 transforms (Exercise 359) 1500 Chapter 101 Fourier series for periodic functions of period π (Exercise 360) 1507 Chapter 10 Fourier series for a non-periodic functions over period π (Exercise 361) 1516 Chapter 103 Even and odd functions and half-range Fourier series (Exercises 36 to 363) 156 Chapter 104 Fourier series over any range (Exercises 364 to 365) 1538 4
Chapter 105 Chapter 106 A numerical method of harmonic analysis (Exercises 366 to 367) 1549 The complex or exponential form of a Fourier series (Exercises 368 to 370) 1558 5