| Preface to the first edition |
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xix | |
| Preface to the second edition |
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xx | |
| Preface to the third edition |
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xxi | |
| Preface to the fourth edition |
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xxii | |
| Preface to the eighth edition |
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xxiii | |
| How to use this book |
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xxiv | |
| Useful background information |
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xxvi | |
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1 | (376) |
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3 | (60) |
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3 | (1) |
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4 | (2) |
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6 | (9) |
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Points on a line and order |
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Multiplication and division |
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Brackets and precedence rules |
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Factors and prime numbers |
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15 | (3) |
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Fundamental theorem of arithmetic |
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Highest common factor (HCF) |
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Lowest common multiple (LCM) |
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Fractions, ratios and percentages |
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18 | (9) |
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Equivalent fractions Dividing fractions |
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Adding and subtracting fractions |
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Fractions on a calculator |
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27 | (6) |
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Decimal places Trailing zeros |
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Unending decimals as fractions |
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Rational, irrational and real numbers |
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33 | (11) |
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Raising a number to a power |
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Fractional powers and roots |
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Multiplication and division by integer powers of 10 |
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Standard form Working in standard form |
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Preferred standard form Checking calculations |
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44 | (8) |
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Denary (or decimal) system |
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Octal system (base 8) Duodecimal system (base 12) |
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Hexadecimal system (base 16) An alternative method |
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Change of base from denary to a new base |
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52 | (4) |
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A denary decimal in octal form |
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Use of octals as an intermediate step |
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56 | (2) |
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58 | (1) |
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59 | (1) |
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60 | (3) |
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Programme F.2 Introduction to algebra |
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63 | (34) |
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63 | (1) |
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64 | (1) |
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65 | (7) |
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Symbols other than numerals |
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72 | (9) |
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Algebraic multiplication and division |
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81 | (4) |
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85 | (4) |
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Multiplication and division |
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Factorization of algebraic expressions |
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89 | (4) |
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Common factors by grouping |
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Useful products of two simple factors |
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Quadratic expressions as the product of two factors |
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93 | (1) |
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94 | (1) |
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95 | (2) |
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Programme F.3 Expressions and equations |
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97 | (26) |
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97 | (1) |
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98 | (1) |
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Expressions and equations |
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99 | (10) |
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Evaluating independent variables Transposition of formulas |
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109 | (11) |
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Evaluation of polynomials |
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Evaluation of a polynomial by nesting |
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Factor theorem The general quadratic equation-Factorization of fourth-order polynomials |
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120 | (1) |
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121 | (1) |
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121 | (2) |
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123 | (1) |
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124 | (1) |
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125 | (8) |
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133 | (9) |
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Formulas Clearing entries |
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Construction of a Cartesian graph |
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142 | (2) |
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Less than or greater than |
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144 | (9) |
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153 | (1) |
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154 | (1) |
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154 | (3) |
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Programme F.5 Linear equations |
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157 | (16) |
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157 | (1) |
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158 | (1) |
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159 | (10) |
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Solution of simple equations |
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Simultaneous linear equations with two unknowns |
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Simultaneous equations with three unknowns |
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169 | (1) |
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170 | (1) |
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170 | (3) |
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Programme F.6 Polynomial equations |
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173 | (14) |
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173 | (1) |
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174 | (1) |
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175 | (9) |
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Cubic equations having at least one simple linear factor |
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Fourth-order equations having at least two linear factors |
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184 | (1) |
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185 | (1) |
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185 | (2) |
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187 | (28) |
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187 | (1) |
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188 | (1) |
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Factorials and combinations |
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189 | (8) |
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Three properties of combinatorial coefficients |
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197 | (6) |
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The general term of the binomial expansion |
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203 | (9) |
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The sum of the first n natural numbers |
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Rules for manipulating sums |
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212 | (1) |
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213 | (1) |
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213 | (2) |
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Programme F.8 Partial fractions |
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215 | (20) |
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215 | (1) |
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216 | (1) |
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217 | (7) |
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Denominators with repeated and quadratic factors |
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224 | (8) |
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232 | (1) |
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232 | (1) |
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232 | (3) |
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Programme F.9 Trigonometry |
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235 | (24) |
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235 | (1) |
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236 | (1) |
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237 | (13) |
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250 | (6) |
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Identities for compound angles |
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256 | (1) |
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256 | (1) |
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257 | (2) |
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259 | (20) |
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259 | (1) |
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260 | (1) |
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261 | (11) |
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Functions are rules but not all rules are functions |
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Functions and the arithmetic operations |
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Graphs of inverses The graph of y = x3 |
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The graphs of y = x3 and y = x1/3 plotted together |
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272 | (4) |
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276 | (1) |
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277 | (1) |
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277 | (2) |
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Programme F.11 Trigonometric and exponential functions |
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279 | (30) |
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279 | (1) |
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280 | (1) |
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281 | (1) |
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281 | (14) |
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Inverse trigonometric functions |
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Equations of the form a cos x + b sin x = c |
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Exponential and logarithmic functions |
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295 | (4) |
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299 | (5) |
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Odd and even parts of the exponential function Limits of functions |
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304 | (1) |
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305 | (1) |
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306 | (3) |
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Programme F.12 Differentiation |
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309 | (38) |
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309 | (1) |
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310 | (1) |
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311 | (13) |
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The gradient of a straight line |
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The gradient of a curve at a given point Algebraic determination of the gradient of a curve |
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Derivatives of powers of x |
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Differentiation of polynomials |
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Second and higher derivatives and an alternative notation |
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Standard derivatives and rules |
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324 | (14) |
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Limiting value of sin θ/θ as θ → 0 |
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Derivative of a product of functions |
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Derivative of a quotient of functions |
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Derivative of a function of a function |
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Newton-Raphson iterative method |
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338 | (6) |
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Tabular display of results |
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344 | (1) |
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345 | (1) |
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346 | (1) |
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Programme F.13 Integration |
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347 | (30) |
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347 | (1) |
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348 | (1) |
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349 | (3) |
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Integration of polynomial expressions |
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352 | (5) |
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Functions of a linear function of x |
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Integration by partial fractions |
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357 | (3) |
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360 | (4) |
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Integration as a summation |
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364 | (9) |
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The area between a curve and an intersecting line |
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373 | (1) |
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374 | (1) |
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375 | (2) |
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377 | (749) |
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Programme 1 Complex numbers 1 |
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379 | (27) |
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379 | (1) |
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380 | (1) |
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380 | (3) |
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383 | (1) |
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384 | (9) |
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Graphical representation of a complex number |
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393 | (2) |
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Graphical addition of complex numbers |
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Polar form of a complex number |
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395 | (5) |
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Exponential form of a complex number |
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400 | (2) |
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402 | (1) |
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403 | (1) |
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403 | (1) |
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404 | (2) |
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Programme 2 Complex numbers 2 |
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406 | (1) |
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407 | (8) |
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Roots of a complex number |
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415 | (6) |
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421 | (3) |
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Expansions of sin nθ and cos nθ |
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Expansions for cosnθ and sinnθ |
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424 | (3) |
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427 | (1) |
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428 | (1) |
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428 | (1) |
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429 | (2) |
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Programme 3 Hyperbolic functions |
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431 | (22) |
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431 | (1) |
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432 | (2) |
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Graphs of hyperbolic functions |
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434 | (5) |
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Evaluation of hyperbolic functions |
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439 | (1) |
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Inverse hyperbolic functions |
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440 | (2) |
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Log form of the inverse hyperbolic functions |
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442 | (3) |
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445 | (2) |
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Relationship between trigonometric and hyperbolic functions |
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447 | (3) |
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450 | (1) |
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451 | (1) |
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451 | (1) |
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452 | (1) |
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453 | (31) |
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453 | (1) |
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454 | (6) |
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Determinants of the third order |
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460 | (12) |
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Evaluation of a third-order determinant Simultaneous equations in three unknowns |
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464 | (8) |
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Consistency of a set of equations |
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472 | (3) |
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Properties of determinants |
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475 | (4) |
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479 | (1) |
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480 | (1) |
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480 | (1) |
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481 | (3) |
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484 | (35) |
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484 | (1) |
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485 | (1) |
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486 | (1) |
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487 | (1) |
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Addition and subtraction of matrices |
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487 | (1) |
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Multiplication of matrices |
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488 | (3) |
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Multiplication of two matrices |
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491 | (1) |
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492 | (2) |
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Determinant of a square matrix |
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494 | (2) |
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Adjoint of a square matrix |
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Inverse of a square matrix |
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496 | (3) |
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Product of a square matrix and its inverse |
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Solution of a set of linear equations |
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499 | (5) |
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Gaussian elimination method for solving a set of linear equations |
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Eigenvectors and eigenvalues |
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504 | (5) |
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509 | (4) |
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Raising a matrix to a whole number power |
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513 | (1) |
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514 | (1) |
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515 | (1) |
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516 | (3) |
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519 | (25) |
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519 | (1) |
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Introduction: scalar and vector quantities |
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520 | (1) |
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521 | (3) |
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The sum of a number of vectors |
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Components of a given vector |
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524 | (6) |
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Components of a vector in terms of unit vectors |
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530 | (2) |
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532 | (1) |
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Scalar product of two vectors |
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533 | (2) |
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Vector product of two vectors |
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535 | (2) |
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Angle between two vectors |
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537 | (3) |
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540 | (1) |
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540 | (1) |
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541 | (1) |
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541 | (1) |
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542 | (2) |
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Programme 7 Differentiation |
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544 | (19) |
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544 | (1) |
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545 | (1) |
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546 | (6) |
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Logarithmic differentiation |
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552 | (3) |
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555 | (2) |
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557 | (2) |
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559 | (1) |
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560 | (1) |
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560 | (1) |
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561 | (2) |
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Programme 8 Differentiation applications |
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563 | (22) |
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563 | (1) |
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Differentiation of inverse trigonometric functions |
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564 | (2) |
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Derivatives of inverse hyperbolic functions |
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566 | (5) |
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Maximum and minimum values |
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571 | (4) |
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575 | (6) |
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581 | (1) |
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581 | (1) |
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582 | (1) |
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582 | (3) |
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Programme 9 Tangents, normals and curvature |
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585 | (22) |
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585 | (1) |
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Equation of a straight line |
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586 | (3) |
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Tangents and normals to a curve at a given point |
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589 | (5) |
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594 | (9) |
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603 | (1) |
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604 | (1) |
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604 | (1) |
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605 | (2) |
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607 | (235) |
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607 | (1) |
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Functions with integer input |
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608 | (12) |
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620 | (9) |
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Solving difference equations |
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Second-order homogeneous equations Equal roots of the characteristic equation |
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629 | (8) |
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Indeterminate limits Review summary |
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637 | (1) |
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638 | (1) |
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639 | (203) |
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842 | (24) |
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642 | (1) |
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643 | (5) |
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Series of powers of the natural numbers |
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648 | (3) |
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651 | (2) |
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653 | (9) |
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Convergent and divergent series |
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662 | (1) |
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663 | (1) |
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663 | (1) |
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664 | (1) |
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666 | (1) |
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666 | (1) |
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667 | (10) |
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677 | (3) |
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680 | (7) |
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L'Hopital's rule for finding limiting values |
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687 | (1) |
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688 | (1) |
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689 | (1) |
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689 | (3) |
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Programme 13 Curves and curve fitting |
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692 | (44) |
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692 | (1) |
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693 | (1) |
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693 | (9) |
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Second-degree curves-Third-degree curves |
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Exponential curves Hyperbolic curves |
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702 | (5) |
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Determination of an asymptote |
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Asymptotes parallel to the x- and y-axes |
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Systematic curve sketching, given the equation of the curve |
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707 | (5) |
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Intersection with the axes |
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Asymptotes Large and small values of x and y |
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712 | (5) |
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Graphs of the form y = axn, where a and n are constants |
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Graphs of the form y = aenx |
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717 | (6) |
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Fitting a straight-line graph |
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723 | (7) |
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The Pearson product-moment correlation coefficient |
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Spearman's rank correlation coefficient |
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730 | (1) |
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731 | (1) |
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732 | (1) |
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733 | (3) |
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Programme 14 Partial differentiation 1 |
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736 | (21) |
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736 | (1) |
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737 | (12) |
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749 | (5) |
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754 | (1) |
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754 | (1) |
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755 | (2) |
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Programme 15 Partial differentiation 2 |
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757 | (16) |
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757 | (1) |
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758 | (2) |
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760 | (8) |
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768 | (2) |
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770 | (1) |
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770 | (1) |
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771 | (1) |
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771 | (2) |
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Programme 16 Integration 1 |
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773 | (27) |
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773 | (1) |
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774 | (3) |
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Functions of a linear function of x |
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777 | (2) |
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Integrals of the forms ∫f'(x)/f(x)dx and ∫f{x)f'(x)dx |
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779 | (4) |
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783 | (4) |
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Integration by partial fractions |
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787 | (5) |
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Integration of trigonometric functions |
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792 | (4) |
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796 | (1) |
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796 | (1) |
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797 | (1) |
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797 | (3) |
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Programme 17 Integration 2 |
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800 | (28) |
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800 | (24) |
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824 | (1) |
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825 | (1) |
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826 | (1) |
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826 | (2) |
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Programme 18 Reduction formulas |
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828 | (13) |
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828 | (10) |
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838 | (1) |
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838 | (1) |
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838 | (1) |
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839 | (2) |
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Programme 19 Integration applications 1 |
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841 | (18) |
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841 | (1) |
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842 | (8) |
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850 | (1) |
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851 | (2) |
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Root mean square (rms) value |
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853 | (2) |
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855 | (1) |
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856 | (1) |
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856 | (1) |
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857 | (2) |
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Programme 20 Integration applications 2 |
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859 | (22) |
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859 | (1) |
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860 | (1) |
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Volume of a solid of revolution |
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860 | (4) |
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Centroid of a plane figure |
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864 | (3) |
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Centre of gravity of a solid of revolution |
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867 | (1) |
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868 | (4) |
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872 | (3) |
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875 | (1) |
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876 | (1) |
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877 | (1) |
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878 | (1) |
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878 | (3) |
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Programme 21 Integration applications 3 |
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881 | (29) |
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881 | (1) |
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882 | (14) |
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Perpendicular axes theorem (for thin plates) |
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896 | (4) |
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900 | (6) |
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Pressure at a point P, depth z below the surface |
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Total thrust on a vertical plate immersed in liquid |
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Depth of the centre of pressure |
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906 | (1) |
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906 | (1) |
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907 | (1) |
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908 | (2) |
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Programme 21 Approximate integration |
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910 | (19) |
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910 | (1) |
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911 | (1) |
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912 | (12) |
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924 | (1) |
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925 | (1) |
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926 | (1) |
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926 | (1) |
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927 | (2) |
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Programme 23 Polar coordinate systems |
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929 | (22) |
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929 | (1) |
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Introduction to polar coordinates |
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930 | (2) |
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932 | (2) |
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934 | (3) |
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937 | (11) |
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948 | (1) |
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949 | (1) |
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949 | (1) |
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950 | (1) |
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Programme 24 Multiple integrals |
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951 | (26) |
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951 | (1) |
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Summation in two directions |
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952 | (3) |
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955 | (2) |
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957 | (2) |
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959 | (5) |
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964 | (4) |
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Determination of areas by multiple integrals |
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968 | (2) |
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Determination of volumes by multiple integrals |
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970 | (3) |
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973 | (1) |
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974 | (1) |
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974 | (1) |
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975 | (2) |
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Programme 25 First-order differential equations |
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977 | (36) |
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977 | (1) |
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978 | (1) |
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Formation of differential equations |
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979 | (2) |
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Solution of differential equations |
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981 | (7) |
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By separating the variables |
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Solution of differential equations |
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988 | (6) |
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Solution of differential equations |
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994 | (8) |
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Use of integrating factor |
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1002 | (5) |
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1007 | (1) |
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1008 | (1) |
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1009 | (1) |
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1009 | (4) |
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Programme 26 Second-order differential equations |
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1013 | (23) |
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1013 | (1) |
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1014 | (8) |
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1022 | (7) |
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1029 | (4) |
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1033 | (1) |
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1033 | (1) |
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1034 | (1) |
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1034 | (2) |
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Programme 27 Introduction to Laplace transforms |
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1036 | (18) |
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1036 | (1) |
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1037 | (4) |
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The inverse Laplace transform |
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Table of Laplace transforms |
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1041 | (4) |
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Laplace transform of a derivative |
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Two properties of Laplace transforms |
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Table of Laplace transforms |
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1045 | (6) |
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Generating new transforms |
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Laplace transforms of higher derivatives Table of Laplace transforms |
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Linear, constant-coefficient, inhomogeneous differential equations |
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1051 | (1) |
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1052 | (1) |
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1052 | (2) |
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Programme 28 Data handling and statistics |
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1054 | (31) |
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1054 | (1) |
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1055 | (1) |
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1055 | (7) |
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Grouping with continuous data Relative frequency |
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1062 | (2) |
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Relative frequency histogram |
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Measures of central tendency |
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1064 | (8) |
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Coding for calculating the mean |
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Coding with a grouped frequency distribution |
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Mode with grouped data Median |
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1072 | (3) |
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Alternative formula for the standard deviation |
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1075 | (3) |
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Normal distribution curve |
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Standardized normal curve |
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1078 | (1) |
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1079 | (1) |
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1080 | (1) |
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1081 | (1) |
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1082 | (3) |
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1085 | (41) |
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1085 | (1) |
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1086 | (2) |
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Sequences of random experiments |
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1088 | (3) |
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Probabilities of combined events |
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1091 | (5) |
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Non-mutually exclusive events |
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Dependent events Independent events |
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1096 | (3) |
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Probability distributions |
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1099 | (16) |
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Variance and standard deviation Bernoulli trials |
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Binomial probability distribution |
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Expectation and standard deviation |
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The Poisson probability distribution |
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Binomial and Poisson compared |
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Continuous probability distributions |
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1115 | (1) |
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Normal distribution curve |
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1115 | (6) |
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1121 | (1) |
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1122 | (1) |
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1123 | (3) |
| Answers |
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1126 | (27) |
| Index |
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1153 | |
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