Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Acknowledgements......Page 15
Dedication......Page 16
Introduction......Page 17
Introduction to the second edition......Page 19
1.A.(a) Where do you start? Self-test 1......Page 21
1.A.(b) A mind-reading explained......Page 22
1.A.(c) Some basic rules......Page 23
1.A.(d) Working out in the right order......Page 25
1.A.(e) Using negative numbers......Page 26
1.B.(a) Self-test 2......Page 27
1.B.(b) Multiplying out two brackets......Page 28
1.B.(c) More factorisation: putting things back into brackets......Page 30
1.C.(a) Equivalent fractions and cancelling down......Page 32
1.C.(b) Tidying up more complicated fractions......Page 34
1.C.(c) Adding fractions in arithmetic and algebra......Page 36
1.C.(d) Repeated factors in adding fractions......Page 38
1.C.(e) Subtracting fractions......Page 40
1.C.(f) Multiplying fractions......Page 41
1.D.(a) Handling powers which are whole numbers......Page 42
1.D.(b) Some special cases......Page 44
1.E.(c) Including fractions: the set of rational numbers......Page 46
1.E.(d) Including everything on the number line: the set of real numbers......Page 47
1.F.(a) Other number bases: the binary system......Page 49
1.F.(b) Prime numbers and factors......Page 51
1.F.(d) Simplifying fractions with…signs underneath......Page 52
2.A.(a) Do you need help with this? Self-test 3......Page 54
2.A.(b) Rules for solving simple equations......Page 55
2.A.(c) Solving equations involving fractions......Page 56
2.A.(d) A practical application – rearranging formulas to fit different situations......Page 59
2.B Introducing graphs......Page 61
2.B.(b) A reminder on plotting graphs......Page 62
2.B.(c) The midpoint of the straight line joining two points......Page 63
2.B.(d) Steepness or gradient......Page 65
2.B.(e) Sketching straight lines......Page 66
2.B.(f) Finding equations of straight lines......Page 68
2.B.(g) The distance between two points......Page 69
2.B.(i) Dividing a straight line in a given ratio......Page 70
2.C.(a) What do simultaneous equations mean?......Page 72
2.C.(b) Methods of solving simultaneous equations......Page 73
2.D.(a) What do the graphs which show quadratic equations look like?......Page 76
2.D.(b) The method of completing the square......Page 79
2.D.(c) Sketching the curves which give quadratic equations......Page 80
2.D.(d) The ‘formula’ for quadratic equations......Page 81
2.D.(e) Special properties of the roots of quadratic equations......Page 83
2.D.(f) Getting useful information from ‘b2 – 4ac’......Page 84
2.D.(g) A practical example of using quadratic equations......Page 86
2.D.(h) All equations are equal – but are some more equal than others?......Page 88
2.E.(a) Cubic expressions and equations......Page 92
2.E.(b) Doing long division in algebra......Page 95
2.E.(c) Avoiding long division – the Remainder and Factor Theorems......Page 96
2.E.(d) Three examples of using these theorems, and a red herring......Page 97
3.A.(a) Direct proportion......Page 100
3.A.(b) Some physical examples of direct proportion......Page 101
3.A.(c) More exotic examples......Page 103
3.A.(d) Partial direct proportion – lines not through the origin......Page 105
3.A.(e) Inverse proportion......Page 106
3.B.(a) What are functions? Some relationships examined......Page 108
3.B.(b) y = f(x) – a useful new shorthand......Page 111
3.B.(d) Stretching and shifting – new functions from old......Page 112
3.B.(e) Two practical examples of shifting and stretching......Page 118
3.B.(f) Finding functions of functions......Page 120
3.B.(g) Can we go back the other way? Inverse functions......Page 122
3.B.(h) Finding inverses of more complicated functions......Page 125
3.B.(i) Sketching the particular case of f(x ) = (x + 3)/(x – 2), and its inverse......Page 127
3.B.(j) Odd and even functions......Page 131
3.C.(a) Exponential functions – describing population growth......Page 132
3.C.(b) The inverse of a growth function: log functions......Page 134
3.C.(c) Finding the logs of some particular numbers......Page 135
3.C.(d) The three laws or rules for logs......Page 136
3.C.(e) What are ‘e’ and ‘exp’? A brief introduction......Page 138
3.C.(f) Negative exponential functions – describing population decay......Page 140
3.D.(a) Relationships of the form y = axn......Page 142
3.D.(b) Relationships of the form y = anx......Page 145
3.D.(c) What can we do if logs are no help?......Page 146
4.A.(a) Why use trig ratios?......Page 148
4.A.(b) Pythagoras’ Theorem......Page 153
4.A.(d) Triangles with particular shapes......Page 155
4.A.(e) Congruent triangles – what are they, and when?......Page 156
4.A.(f) Matching ratios given by parallel lines......Page 158
4.A.(g) Special cases – the sin, cos and tan of 30°, 45° and 60°......Page 159
4.A.(h) Special relations of sin, cos and tan......Page 160
4.B.(a) The Sine Rule for any triangle......Page 162
4.B.(b) Another area formula for triangles......Page 164
4.B.(c) The Cosine Rule for any triangle......Page 165
4.C.(a) The parts of a circle......Page 170
4.C.(b) Special properties of chords and tangents of circles......Page 171
4.C.(c) Special properties of angles in circles......Page 172
4.C.(d) Finding and working with the equations which give circles......Page 174
4.C.(e) Circles and straight lines – the different possibilities......Page 176
4.C.(f) Finding the equations of tangents to circles......Page 179
4.D.(a) Measuring angles in radians......Page 181
4.D.(b) Finding the perimeter and the area of a sector of a circle......Page 183
4.D.(d) What do we do if the angle is given in degrees?......Page 184
4.D.(e) Very small angles in radians – why we like them......Page 185
4.E.(a) The sum of interior and exterior angles of polygons......Page 188
4.E.(b) Can we draw circles round all triangles and quadrilaterals?......Page 189
5.A.(a) Extending sin and cos......Page 191
5.A.(b) The graph of y = tan x from 0° to 90°......Page 194
5.A.(c) Defining the sin, cos and tan of angles of any size......Page 195
5.A.(d) How does X move as P moves round its circle?......Page 198
5.A.(e) The graph of tan Theta for any value of Theta......Page 199
5.A.(f) Can we find the angle from its sine?......Page 200
5.A.(g) sin-1 x and cos-1 x: what are they?......Page 202
5.A.(h) What do the graphs of sin-1 x and cos-1 x look like?......Page 203
5.A.(i) Defining the function tan x......Page 205
5.B.(c) Some examples of proving other trig identities......Page 206
5.B.(d) What do the graphs of the trig reciprocal functions look like?......Page 209
5.B.(e) Drawing other reciprocal graphs......Page 210
5.C.(a) Stretching, shifting and shrinking trig functions......Page 212
5.C.(b) Relating trig functions to how P moves round its circle and SHM......Page 214
5.C.(c) New shapes from putting together trig functions......Page 218
5.C.(d) Putting together trig functions with different periods......Page 220
5.D.(a) How else can we write sin(A + B)?......Page 221
5.D.(b) A summary of results for similar combinations......Page 222
5.D.(d) The rules for sin 2A, cos 2A and tan 2A......Page 223
5.D.(f) Using sin(A + B) to find another way of writing 4 sin t + 3 cos t......Page 224
5.D.(g) More examples of the R sin (t ± Alpha) and R cos (t ± Alpha) forms......Page 227
5.D.(h) Going back the other way – the Factor Formulas......Page 230
5.E.(a) Laying some useful foundations......Page 231
5.E.(b) Finding solutions for equations in cos x......Page 233
5.E.(c) Finding solutions for equations in tan x......Page 235
5.E.(d) Finding solutions for equations in sin x......Page 237
5.E.(e) Solving equations using R sin (x + Alpha) etc......Page 240
6.A.(a) Finding patterns in sequences of numbers......Page 242
6.A.(b) How to describe number patterns mathematically......Page 243
6.B.(a) What are arithmetic progressions?......Page 246
6.B.(b) Finding a rule for summing APs......Page 247
6.B.(d) Solving a typical problem......Page 248
6.C.(a) What are geometric progressions?......Page 249
6.C.(b) Summing geometric progressions......Page 250
6.C.(c) The sum to infinity of a GP......Page 251
6.C.(d) What do ‘convergent’ and ‘divergent’ mean?......Page 252
6.C.(e) More examples using GPs; chain letters......Page 253
6.C.(f) A summary of the results for GPs......Page 254
6.C.(g) Recurring decimals, and writing them as fractions......Page 257
6.C.(h) Compound interest: a faster way of getting rich......Page 259
6.C.(k) What is the fate of the frog down the well?......Page 261
6.D.(a) What does Sigma stand for?......Page 262
6.D.(c) Summing by breaking down to simpler series......Page 263
6.E.(a) Introducing partial fractions for summing series......Page 265
6.E.(b) General rules for using partial fractions......Page 267
6.E.(d) Coping with possible complications......Page 268
6.F The fate of the frog down the well......Page 274
7.A.(a) Looking for the patterns......Page 277
7.A.(b) Permutations or arrangements......Page 279
7.A.(c) Combinations or selections......Page 281
7.A.(d) How selections give binomial expansions......Page 282
7.A.(e) Writing down rules for binomial expansions......Page 283
7.A.(f) Linking Pascal’s Triangle to selections......Page 285
7.A.(g) Some more binomial examples......Page 287
7.B.(a) Tossing coins and throwing dice......Page 288
7.B.(b) What do the probabilities we have found mean?......Page 289
7.B.(d) Lotteries: winning the jackpot…or not......Page 290
7.C.(a) Can we expand (1 + x)n if n is negative or a fraction? If so, when?......Page 291
7.C.(b) Working out some expansions......Page 292
7.C.(c) Dealing with slightly different situations......Page 293
7.D.(a) Truth from patterns – or false mirages?......Page 295
7.D.(b) Proving the Binomial Theorem by induction......Page 299
7.D.(c) Two non-series applications of induction......Page 300
8 Differentiation......Page 302
8.A.(a) How can we find a speed from knowing the distance travelled?......Page 303
8.A.(b) How does y = xn change as x changes?......Page 308
8.A.(c) Different ways of writing differentiation…......Page 309
8.A.(d) Some special cases of y = axn......Page 310
8.A.(e) Differentiating x = cos t answers another thinking point......Page 311
8.A.(f) Can we always differentiate? If not, why not?......Page 315
8.B Natural growth and decay – the number et......Page 316
8.B.(a) Even more money – compound interest and exponential growth......Page 317
8.B.(b) What is the equation of this smooth growth curve?......Page 320
8.B.(c) Getting numerical results from the natural growth law of x = et......Page 321
8.B.(d) Relating ln x to the log of x using other bases......Page 323
8.B.(e) What do we get if we differentiate ln t?......Page 324
8.C.(a) The Chain Rule......Page 325
8.C.(c) Differentiating functions with angles in degrees or logs to base 10......Page 328
8.C.(d) The Product Rule, or ‘uv’ Rule......Page 329
8.C.(e) The Quotient Rule or ‘u/v’ Rule......Page 331
8.D.(a) Getting symmetries from ex and e-x......Page 334
8.D.(c) Using sinh x and cosh x to get other hyperbolic functions......Page 337
8.D.(d) Comparing other hyperbolic and trig formulas – Osborn’s Rule......Page 338
8.D.(e) Finding the inverse function for sinh x......Page 339
8.D.(f) Can we find an inverse function for cosh x?......Page 341
8.D.(g) tanh x and its inverse function tanh-1 x......Page 343
8.D.(h) What’s in a name? Why ‘hyperbolic’ functions?......Page 346
8.D.(i) Differentiating inverse trig and hyperbolic functions......Page 347
8.E.(a) Finding the equations of tangents to particular curves......Page 350
8.E.(b) Finding turning points and points of inflection......Page 352
8.E.(c) General rules for sketching curves......Page 356
8.E.(d) Some practical uses of turning points......Page 359
8.E.(e) A clever use for tangents – the Newton–Raphson Rule......Page 364
8.F.(a) How implicit differentiation works, using circles as examples......Page 369
8.F.(b) Using implicit differentiation with more complicated relationships......Page 372
8.F.(c) Differentiating inverse functions implicitly......Page 374
8.F.(d) Differentiating exponential functions like x = 2t......Page 377
8.F.(e) A practical application of implicit differentiation......Page 378
8.G Writing functions in an alternative form using series......Page 379
9.A.(a) What could this tell us?......Page 386
9.A.(b) A physical interpretation of this process......Page 387
9.A.(c) Finding the area under a curve......Page 389
9.A.(d) What happens if the area we are finding is below the horizontal axis?......Page 394
9.A.(e) What happens if we change the order of the limits?......Page 395
9.A. (f) What is…......Page 396
9.B Techniques of integration......Page 398
9.B.(a) Making use of what we already know......Page 399
9.B.(b) Integration by substitution......Page 400
9.B.(c) A selection of trig integrals with some hyperbolic cousins......Page 405
9.B.(d) Integrals which use inverse trig and hyperbolic functions......Page 407
9.B.(e) Using partial fractions in integration......Page 411
9.B.(f) Integration by parts......Page 413
9.B.(g) Finding rules for doing integrals like…......Page 418
9.B.(h) Using the t = tan (x/2) substitution......Page 422
9.C.(a) Solving equations where we can split up the variables......Page 425
9.C.(b) Putting flesh on the bones – some practical uses for differential equations......Page 427
9.C.(c) A forwards look at some other kinds of differential equation, including ones which describe SHM......Page 435
10.A.(a) Finding the missing roots......Page 438
10.A.(b) Finding roots for all quadratic equations......Page 441
10.A.(c) Modulus and argument (or mod and arg for short)......Page 442
10.B.(a) Addition and subtraction......Page 446
10.B.(b) Multiplication of complex numbers......Page 447
10.B.(c) Dividing complex numbers in mod/arg form......Page 451
10.B.(d) What are complex conjugates?......Page 452
10.B.(e) Using complex conjugates to simplify fractions......Page 453
10.C.(a) Two for the price of one – equating real and imaginary parts......Page 454
10.C.(b) How does e get involved?......Page 456
10.C.(c) What is the geometrical meaning of z = e?......Page 457
10.C.(d) What is e and what does it do geometrically?......Page 458
10.C.(e) A summary of the sin/cos and sinh/cosh links......Page 459
10.C.(g) Another example: writing cos 5Theta in terms of cos Theta......Page 460
10.C.(h) More examples of writing trig functions in different forms......Page 462
10.C.(i) Solving a differential equation which describes SHM......Page 463
10.C.(j) A first look at how we can use complex numbers to describe electric circuits......Page 464
10.D.(a) Finding the n roots of zn = a + bj......Page 466
10.D.(b) Solving quadratic equations with complex coefficients......Page 470
10.D.(c) Solving cubic and quartic equations with complex roots......Page 471
10.E.(a) Some simple examples of paths or regions where z must lie......Page 474
10.E.(b) What do we do if z has been shifted?......Page 476
10.E.(c) Using algebra to find where z can be......Page 478
10.E.(d) Another example involving a relationship between w and z......Page 482
11.A.(a) What are vectors?......Page 486
11.A.(b) Adding vectors and what this can mean physically......Page 487
11.A.(c) Using components to describe vectors......Page 492
11.A.(d) Vector components in three-dimensional space......Page 494
11.A.(e) Finding the magnitude of a three-dimensional vector......Page 495
11.A.(f) Finding unit vectors......Page 496
11.B.(a) Defining the scalar or dot product of two vectors......Page 497
11.B.(b) Working out the dot product of two vectors......Page 498
11.B.(c) Defining the vector or cross product of two vectors......Page 502
11.B.(d) Working out the cross product of two vectors......Page 505
11.B.(f) The vector triple product......Page 507
11.B.(g) The scalar triple product and what it means geometrically......Page 508
11.C.(a) Finding a vector equation for a line......Page 509
11.C.(b) Dealing with lines in two dimensions......Page 510
11.C.(c) Dealing with lines in three dimensions......Page 513
11.C.(d) Finding the Cartesian equation of a line in three dimensions......Page 514
11.C.(f) Finding vector equations for planes......Page 517
11.C.(g) Finding equations of planes using normal vectors......Page 519
11.C.(h) Finding the perpendicular distance from the origin to a plane......Page 520
11.C.(i) The Cartesian form of the equation of a plane......Page 521
11.C.(k) Finding the line of intersection of two planes......Page 523
11.D.(a) Finding the angle between two lines......Page 524
11.D.(b) Finding the angle between two planes......Page 526
11.D.(c) Finding the acute angle between a line and a plane......Page 527
11.D.(d) Finding the shortest distance from a point to a line......Page 528
11.D.(e) Finding the shortest distance from a point to a plane......Page 529
11.D.(f) Finding the shortest distance between two skew lines......Page 532
Answers to the exercises......Page 535
Index......Page 647