Maths: A Student's Survival Guide - Amazon Web Services

643 Pages·2004·3.72 MB·English

Maths
A Student’s Survival Guide
Contents
I have split the chapters up in the following way so that you can easily find particular topics.
Also, it makes it easy for me to tell you where to go if you need help, and easy for you to
find this help.
Introduction 1
Introduction to the second edition 3
1
Basic algebra: some reminders of how it works 5
1.A Handling unknown quantities 5
(a) Where do you start? Self-test 1 5
(b) A mind-reading explained 6
(c) Some basic rules 7
(d) Working out in the right order 9
(e) Using negative numbers 10
(f) Putting into brackets, or factorising 11
1.B Multiplications and factorising: the next stage 11
(a) Self-test 2 11
(b) Multiplying out two brackets 12
(c) More factorisation: putting things back into brackets 14
1.C Using fractions 16
(a) Equivalent fractions and cancelling down 16
(b) Tidying up more complicated fractions 18
(c) Adding fractions in arithmetic and algebra 20
(d) Repeated factors in adding fractions 22
(e) Subtracting fractions 24
(f) Multiplying fractions 25
(g) Dividing fractions 26
1.D The three rules for working with powers 26
(a) Handling powers which are whole numbers 26
(b) Some special cases 28
1.E The different kinds of numbers 30
(a) The counting numbers and zero 30
(b) Including negative numbers: the set of integers 30
(c) Including fractions: the set of rational numbers 30
(d) Including everything on the number line: the set of real numbers 31
(e) Complex numbers: a very brief forwards look 33
1.F Working with different kinds of number: some examples 33
(a) Other number bases: the binary system 33
(b) Prime numbers and factors 35
(c) A useful application – simplifying square roots 36
(d) Simplifying fractions with (cid:1) signs underneath 36
Contents v
2
Graphs and equations 38
2.A Solving simple equations 38
(a) Do you need help with this? Self-test 3 38
(b) Rules for solving simple equations 39
(c) Solving equations involving fractions 40
(d) A practical application – rearranging formulas to fit different situations 43
2.B Introducing graphs 45
(a) Self-test 4 46
(b) A reminder on plotting graphs 46
(c) The midpoint of the straight line joining two points 47
(d) Steepness or gradient 49
(e) Sketching straight lines 50
(f) Finding equations of straight lines 52
(g) The distance between two points 53
(h) The relation between the gradients of two perpendicular lines 54
(i) Dividing a straight line in a given ratio 54
2.C Relating equations to graphs: simultaneous equations 56
(a) What do simultaneous equations mean? 56
(b) Methods of solving simultaneous equations 57
2.D Quadratic equations and the graphs which show them 60
(a) What do the graphs which show quadratic equations look like? 60
(b) The method of completing the square 63
(c) Sketching the curves which give quadratic equations 64
(d) The ‘formula’ for quadratic equations 65
(e) Special properties of the roots of quadratic equations 67
(f) Getting useful information from ‘b2 – 4ac’ 68
(g) A practical example of using quadratic equations 70
(h) All equations are equal – but are some more equal than others? 72
2.E Further equations – the Remainder and Factor Theorems 76
(a) Cubic expressions and equations 76
(b) Doing long division in algebra 79
(c) Avoiding long division – the Remainder and Factor Theorems 80
(d) Three examples of using these theorems, and a red herring 81
3
Relations and functions 84
3.A Two special kinds of relationship 84
(a) Direct proportion 84
(b) Some physical examples of direct proportion 85
(c) More exotic examples 87
(d) Partial direct proportion – lines not through the origin 89
(e) Inverse proportion 90
(f) Some examples of mixed variation 92
3.B An introduction to functions 92
(a) What are functions? Some relationships examined 92
(b) y = f(x) – a useful new shorthand 95
(c) When is a relationship a function? 96
(d) Stretching and shifting – new functions from old 96
vi Contents
(e) Two practical examples of shifting and stretching 102
(f) Finding functions of functions 104
(g) Can we go back the other way? Inverse functions 106
(h) Finding inverses of more complicated functions 109
(i) Sketching the particular case of f(x) = (x + 3)/(x – 2), and its inverse 111
(j) Odd and even functions 115
3.C Exponential and log functions 116
(a) Exponential functions – describing population growth 116
(b) The inverse of a growth function: log functions 118
(c) Finding the logs of some particular numbers 119
(d) The three laws or rules for logs 120
(e) What are ‘e’ and ‘exp’? A brief introduction 122
(f) Negative exponential functions – describing population decay 124
3.D Unveiling secrets – logs and linear forms 126
(a) Relationships of the form y = axn 126
(b) Relationships of the form y = anx 129
(c) What can we do if logs are no help? 130
4
Some trigonometry and geometry of triangles and circles 132
4.A Trigonometry in right-angled triangles 132
(a) Why use trig ratios? 132
(b) Pythagoras’ Theorem 137
(c) General properties of triangles 139
(d) Triangles with particular shapes 139
(e) Congruent triangles – what are they, and when? 140
(f) Matching ratios given by parallel lines 142
(g) Special cases – the sin, cos and tan of 30°, 45° and 60° 143
(h) Special relations of sin, cos and tan 144
4.B Widening the field in trigonometry 146
(a) The Sine Rule for any triangle 146
(b) Another area formula for triangles 148
(c) The Cosine Rule for any triangle 149
4.C Circles 154
(a) The parts of a circle 154
(b) Special properties of chords and tangents of circles 155
(c) Special properties of angles in circles 156
(d) Finding and working with the equations which give circles 158
(e) Circles and straight lines – the different possibilities 160
(f) Finding the equations of tangents to circles 163
4.D Using radians 165
(a) Measuring angles in radians 165
(b) Finding the perimeter and area of a sector of a circle 167
(c) Finding the area of a segment of a circle 168
(d) What do we do if the angle is given in degrees? 168
(e) Very small angles in radians – why we like them 169
4.E Tidying up – some thinking points returned to 172
(a) The sum of interior and exterior angles of polygons 172
(b) Can we draw circles round all triangles and quadrilaterals? 173
Contents vii
5
Extending trigonometry to angles of any size 175
5.A Giving meaning to trig functions of any size of angle 175
(a) Extending sin and cos 175
(b) The graph of y = tan x from 0° to 90° 178
(c) Defining the sin, cos and tan of angles of any size 179
(d) How does X move as P moves round its circle? 182
(e) The graph of tan θ for any value of θ 183
(f) Can we find the angle from its sine? 184
(g) sin–1 x and cos–1 x: what are they? 186
(h) What do the graphs of sin–1 x and cos–1 x look like? 187
(i) Defining the function tan–1 x 189
5.B The trig reciprocal functions 190
(a) What are trig reciprocal functions? 190
(b) The trig reciprocal identities: tan2 θ + 1 = sec2 θ and cot2 θ + 1 = cosec2 θ 190
(c) Some examples of proving other trig identities 190
(d) What do the graphs of the trig reciprocal functions look like? 193
(e) Drawing other reciprocal graphs 194
5.C Building more trig functions from the simplest ones 196
(a) Stretching, shifting and shrinking trig functions 196
(b) Relating trig functions to how P moves round its circle and SHM 198
(c) New shapes from putting together trig functions 202
(d) Putting together trig functions with different periods 204
5.D Finding rules for combining trig functions 205
(a) How else can we write sin (A + B)? 205
(b) A summary of results for similar combinations 206
(c) Finding tan (A + B) and tan (A – B) 207
(d) The rules for sin 2A, cos 2A and tan 2A 207
(e) How could we find a formula for sin 3A? 208
(f) Using sin (A + B) to find another way of writing 4 sin t + 3 cos t 208
(g) More examples of the R sin (t ± α) and R cos (t ± α) forms 211
(h) Going back the other way – the Factor Formulas 214
5.E Solving trig equations 215
(a) Laying some useful foundations 215
(b) Finding solutions for equations in cos x 217
(c) Finding solutions for equations in tan x 219
(d) Finding solutions for equations in sin x 221
(e) Solving equations using R sin (x + α) etc. 224
6
Sequences and series 226
6.A Patterns and formulas 226
(a) Finding patterns in sequences of numbers 226
(b) How to describe number patterns mathematically 227
6.B Arithmetic progressions (APs) 230
(a) What are arithmetic progressions? 230
(b) Finding a rule for summing APs 231
(c) The arithmetic mean or ‘average’ 232
(d) Solving a typical problem 232
(e) A summary of the results for APs 233
viii Contents
6.C Geometric progressions (GPs) 233
(a) What are geometric progressions? 233
(b) Summing geometric progressions 234
(c) The sum to infinity of a GP 235
(d) What do ‘convergent’ and ‘divergent’ mean? 236
(e) More examples using GPs; chain letters 237
(f) A summary of the results for GPs 238
(g) Recurring decimals, and writing them as fractions 241
(h) Compound interest: a faster way of getting rich 243
(i) The geometric mean 245
(j) Comparing arithmetic and geometric means 245
(k) Thinking point: what is the fate of the frog down the well? 245
6.D A compact way of writing sums: the ∑ notation 246
(a) What does ∑ stand for? 246
(b) Unpacking the ∑s 247
(c) Summing by breaking down to simpler series 247
6.E Partial fractions 249
(a) Introducing partial fractions for summing series 249
((bb)) General rules for using partial fractions 251
(c) The cover-up rule 252
(d) Coping with possible complications 252
6.F The fate of the frog down the well 258
7
Binomial series and proof by induction 261
7.A Binomial series for positive whole numbers 261
(a) Looking for the patterns 261
(b) Permutations or arrangements 263
(c) Combinations or selections 265
(d) How selections give binomial expansions 266
(e) Writing down rules for binomial expansions 267
(f) Linking Pascal’s Triangle to selections 269
(g) Some more binomial examples 271
7.B Some applications of binomial series and selections 272
(a) Tossing coins and throwing dice 272
(b) What do the probabilities we have found mean? 273
(c) When is a game fair? (Or are you fair game?) 274
(d) Lotteries: winning the jackpot ... or not 274
7.C Binomial expansions when n is not a positive whole number 275
(a) Can we expand (1 + x)n if n is negative or a fraction? If so, when? 275
(b) Working out some expansions 276
(c) Dealing with slightly different situations 277
7.D Mathematical induction 279
(a) Truth from patterns – or false mirages? 279
(b) Proving the Binomial Theorem by induction 283
(c) Two non-series applications of induction 284
Contents ix
8
Differentiation 286
8.A Some problems answered and difficulties solved 287
(a) How can we find a speed from knowing the distance travelled? 287
(b) How does y = xn change as x changes? 292
(c) Different ways of writing differentiation: dx/dt, f(cid:1)(t), x˙, etc. 293
(d) Some special cases of y = axn 294
(e) Differentiating x = cos t answers another thinking point 295
(f) Can we always differentiate? If not, why not? 299
8.B Natural growth and decay – the number e 300
(a) Even more money – compound interest and exponential growth 301
(b) What is the equation of this smooth growth curve? 304
(c) Getting numerical results from the natural growth law of x = et 305
(d) Relating ln x to the log of x using other bases 307
(e) What do we get if we differentiate ln t? 308
8.C Differentiating more complicated functions 309
(a) The Chain Rule 309
(b) Writing the Chain Rule as F(cid:1)(x) = f(cid:1)(g(x))g(cid:1)(x) 312
(c) Differentiating functions with angles in degrees or logs to base 10 312
(d) The Product Rule, or ‘uv’ Rule 313
(e) The Quotient Rule, or ‘u/v’ Rule 315
8.D The hyperbolic functions of sinh x and cosh x 318
(a) Getting symmetries from ex and e–x 318
(b) Differentiating sinh x and cosh x 321
(c) Using sinh x and cosh x to get other hyperbolic functions 321
(d) Comparing other hyperbolic and trig formulas – Osborn’s Rule 322
(e) Finding the inverse function for sinh x 323
(f) Can we find an inverse function for cosh x? 325
(g) tanh x and its inverse function tanh–1 x 327
(h) What’s in a name? Why ‘hyperbolic’ functions? 330
(i) Differentiating inverse trig and hyperbolic functions 331
8.E Some uses for differentiation 334
(a) Finding the equations of tangents to particular curves 334
(b) Finding turning points and points of inflection 336
(c) General rules for sketching curves 340
(d) Some practical uses of turning points 343
(e) A clever use for tangents – the Newton–Raphson Rule 348
8.F Implicit differentiation 353
(a) How implicit differentiation works, using circles as examples 353
(b) Using implicit differentiation with more complicated relationships 356
(c) Differentiating inverse functions implicitly 358
(d) Differentiating exponential functions like x = 2t 361
(e) A practical application of implicit differentiation 362
8.G Writing functions in an alternative form using series 363
x Contents
9
Integration 370
9.A Doing the opposite of differentiating 370
(a) What could this tell us? 370
(b) A physical interpretation of this process 371
(c) Finding the area under a curve 373
(d) What happens if the area we are finding is below the horizontal axis? 378
(e) What happens if we change the order of the limits? 379
(f) What is (cid:2)(1/x)dx? 380
9.B Techniques of integration 382
(a) Making use of what we already know 383
(b) Integration by substitution 384
(c) A selection of trig integrals with some hyperbolic cousins 389
(d) Integrals which use inverse trig and hyperbolic functions 391
(e) Using partial fractions in integration 395
(f) Integration by parts 397
(g) Finding rules for doing integrals like I = (cid:2) sinn x dx 402
n
(h) Using the t = tan (x/2) substitution 406
9.C Solving some more differential equations 409
(a) Solving equations where we can split up the variables 409
(b) Putting flesh on the bones – some practical uses for differential equations 411
(c) A forwards look at some other kinds of differential equation, including ones which
describe SHM 419
10
Complex numbers 422
10.A A new sort of number 422
(a) Finding the missing roots 422
(b) Finding roots for all quadratic equations 425
(c) Modulus and argument (or mod and arg for short) 426
10.B Doing arithmetic with complex numbers 430
(a) Addition and subtraction 430
(b) Multiplication of complex numbers 431
(c) Dividing complex numbers in mod/arg form 435
(d) What are complex conjugates? 436
(e) Using complex conjugates to simplify fractions 437
10.C How e connects with complex numbers 438
(a) Two for the price of one – equating real and imaginary parts 438
(b) How does e get involved? 440
(c) What is the geometrical meaning of z = ejθ? 441
(d) What is e–jθ and what does it do geometrically? 442
(e) A summary of the sin/cos and sinh/cosh links 443
(f) De Moivre’s Theorem 444
(g) Another example: writing cos 5θ in terms of cos θ 444
(h) More examples of writing trig functions in different forms 446
(i) Solving a differential equation which describes SHM 447
(j) A first look at how we can use complex numbers to describe electric circuits 448
Contents xi
10.D Using complex numbers to solve more equations 450
(a) Finding the n roots of zn = a + bj 450
(b) Solving quadratic equations with complex coefficients 454
(c) Solving cubic and quartic equations with complex roots 455
10.E Finding where z can be if it must fit particular rules 458
(a) Some simple examples of paths or regions where z must lie 458
(b) What do we do if z has been shifted? 460
(c) Using algebra to find where z can be 462
(d) Another example involving a relationship between w and z 466
11
Working with vectors 470
11.A Basic rules for handling vectors 470
(a) What are vectors? 470
(b) Adding vectors and what this can mean physically 471
(c) Using components to describe vectors 476
(d) Vector components in three-dimensional space 478
(e) Finding the magnitude of a three-dimensional vector 479
(f) Finding unit vectors 480
11.B Multiplying vectors 481
(a) Defining the scalar or dot product of two vectors 481
(b) Working out the dot product of two vectors 482
(c) Defining the vector or cross product of two vectors 486
(d) Working out the cross product of two vectors 489
(e) Can we multiply three vectors together by using dot or cross products? 491
(f) The vector triple product 491
(g) The scalar triple product and what it means geometrically 492
11.C Finding equations for lines and planes 493
(a) Finding a vector equation for a line 493
(b) Dealing with lines in two dimensions 494
(c) Dealing with lines in three dimensions 497
(d) Finding the Cartesian equation of a line in three dimensions 498
(e) Another form for the vector equation of a line 501
(f) Finding vector equations for planes 501
(g) Finding equations of planes using normal vectors 503
(h) Finding the perpendicular distance from the origin to a plane 504
(i) The Cartesian form of the equation of a plane 505
(j) Finding where a line intersects a plane 507
(k) Finding the line of intersection of two planes 507
11.D Finding angles and distances involving lines and planes 508
(a) Finding the angle between two lines 508
(b) Finding the angle between two planes 510
(c) Finding the acute angle between a line and a plane 511
(d) Finding the shortest distance from a point to a line 512
(e) Finding the shortest distance from a point to a plane 513
(f) Finding the shortest distance between two skew lines 516
Answers to the exercises 519
Index 631
xii Contents
Acknowledgements
I would particularly like to thank Rodie and
Tony Sudbery for their very helpful ideas
and comments on large parts of the text. I
am also very grateful to Neil Turok, Eleni
Haritou-Monioudis, John Szymanski, Jeremy
Jones and David Olive for detailed comments
on particular sections, and my father, William
Tutton, for his helpful advice on my
drawings. I would also like to thank the
mathematics department of the University of
Wales, Swansea, for helpful discussions
concerning the needs of incoming students.
The referees also all provided detailed and
useful input which was very helpful in
structuring the book and I thank them for
this.
I would also like to thank Rufus Neal, Harriet
Millward and Mairi Sutherland for their
patient and friendly editorial help and advice,
Phil Treble for his great design, and everyone
else at Cambridge University Press who has
worked on this book.
Finally, I am particularly grateful to my
daughter, Rosalind Olive, both for her helpful
comments and also for her excellent
guinea-pig drawings.
Acknowledgements xiii