Maths by Olive, Jenny -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Contents Acknowledgements Dedication Page Introduction Introduction to the second edition 1 Basic algebra: some reminders of how it works

1.A Handling unknown quantities

(a) Where do you start? Self-test (b) A mind-reading explained (c) Some basic rules (d) Working out in the right order (e) Using negative numbers (f) Putting into brackets, or factorising

l.B Multiplications and factorising: the next stage

(a) Self-test 2 (b) Multiplying out two brackets (c) More factorisation: putting things back into brackets

l.C Using fractions

(a) Equivalent fractions and cancelling down (b) Tidying up more complicated fractions (c) Adding fractions in arithmetic and algebra (d) Repeated factors in adding fractions (e) Subtracting fractions (f) Multiplying fractions (g) Dividing fractions

l.D The three rules for working with powers

(a) Handling powers which are whole numbers (b) Some special cases

l.E The different kinds of numbers

(a) The counting numbers and zero (b) Including negative numbers: the set of integers (c) Including fractions: the set of rational numbers (d) Including everything on the number line: the set of real numbers (e) Complex numbers: a very brief forwards look

l.F Working with different kinds of number: some examples

(a) Other number bases: the binary system (b) Prime numbers and factors (c) A useful application - simplifying square roots (d) Simplifying fractions with √ signs underneath

2 Graphs and equations

2.A Solving simple equations

(a) Do you need help with this? Self-test 3 (b) Rules for solving simple equations (c) Solving equations involving fractions (d) A practical application – rearranging formulas to fit different situations

2.B Introducing graphs

(a) Self-test 4 (b) A reminder on plotting graphs (c) The midpoint of the straight line joining two points (d) Steepness or gradient (e) Sketching straight lines (f) Finding equations of straight lines (g) The distance between two points (h) The relation between the gradients of two perpendicular lines (i) Dividing a straight line in a given ratio

2.C Relating equations to graphs: simultaneous equations

(a) What do simultaneous equations mean? (b) Methods of solving simultaneous equations

2.D Quadratic equations and the graphs which show them

(a) What do the graphs which show quadratic equations look like? (b) The method of completing the square (c) Sketching the curves which give quadratic equations (d) The ‘formula’ for quadratic equations (e) Special properties of the roots of quadratic equations (f) Getting useful information from ‘b2 – 4ac’ (g) A practical example of using quadratic equations (h) All equations are equal – but are some more equal than others?

2.E Further equations – the Remainder and Factor Theorems

(a) Cubic expressions and equations (b) Doing long division in algebra (c) Avoiding long division – the Remainder and Factor Theorems (d) Three examples of using these theorems, and a red herring

3 Relations and functions

3.A Two special kinds of relationship

(a) Direct proportion (b) Some physical examples of direct proportion (c) More exotic examples (d) Partial direct proportion – lines not through the origin (e) Inverse proportion (f) Some examples of mixed variation

3.B An introduction to functions

(a) What are functions? Some relationships examined (b) y = f(x) – a useful new shorthand (c) When is a relationship a function? (d) Stretching and shifting – new functions from old (e) Two practical examples of shifting and stretching (f) Finding functions of functions (g) Can we go back the other way? Inverse functions (h) Finding inverses of more complicated functions (i) Sketching the particular case of f(x) = (x + 3)/(x- 2), and its inverse (j) Odd and even functions

3.C Exponential and log functions

(a) Exponential functions - describing population growth (b) The inverse of a growth function: log functions (c) Finding the logs of some particular numbers (d) The three laws or rules for logs (e) What are ‘e’ and ‘exp’? A brief introduction (f) Negative exponential functions – describing population decay

3.D Unveiling secrets - logs and linear forms

(a) Relationships of the form y = axn (b) Relationships of the form y = anx (c) What can we do if logs are no help?

4 Some trigonometry and geometry of triangles and circles

4.A Trigonometry in right-angled triangles

(a) Why use trig ratios? (b) Pythagoras’ Theorem (c) General properties of triangles (d) Triangles with particular shapes (e) Congruent triangles – what are they, and when? (f) Matching ratios given by parallel lines (g) Special cases – the sin, cos and tan of 30°, 45° and 60° (h) Special relations of sin, cos and tan

4.B Widening the field in trigonometry

(a) The Sine Rule for any triangle (b) Another area formula for triangles (c) The Cosine Rule for any triangle

4.C Circles

(a) The parts of a circle (b) Special properties of chords and tangents of circles (c) Special properties of angles in circles (d) Finding and working with the equations which give circles (e) Circles and straight lines – the different possibilities (f) Finding the equations of tangents to circles

4.D Using radians

(a) Measuring angles in radians (b) Finding the perimeter and area of a sector of a circle (c) Finding the area of a segment of a circle (d) What do we do if the angle is given in degrees? (e) Very small angles in radians – why we like them

4.E Tidying up - some thinking points returned to

(a) The sum of interior and exterior angles of polygons (b) Can we draw circles round all triangles and quadrilaterals?

5 Extending trigonometry to angles of any size

5.A Giving meaning to trig functions of any size of angle

(a) Extending sin and cos (b) The graph of y = tan x from 0° to 900 (c) Defining the sin, cos and tan of angles of any size (d) How does X move as P moves round its circle? (e) The graph of tan θ for any value of θ (f) Can we find the angle from its sine? (g) sin-1 x and cos-1 x: what are they? (h) What do the graphs of sin-1 x and cos-1 x look like? (i) Defining the function tan-1 x

5.B The trig reciprocal functions

(a) What are trig reciprocal functions? (b) The trig reciprocal identities: tan2 θ + 1 = sec2 θ and cot2 θ + 1 = cosec2 θ (c) Some examples of proving other trig identities (d) What do the graphs of the trig reciprocal functons look like? (e) Drawing other reciprocal graphs

5.C Building more trig functions from the simplest ones

(a) Stretching, shifting and shrinking trig functions (b) Relating trig functions to how P moves round its circle and SHM (c) New shapes from putting together trig functions (d) Putting together trig functions with different periods

5.D Finding rules for combining trig functions

(a) How else can we write sin (A + B)? (b) A summary of results for similar combinations (c) Finding tan (A + B) and tan (A - B) (d) The rules for sin 2A, cos 2A and tan 2A (e) How could we find a formula for sin 3A? (f) Using sin (A + B) to find another way of writing 4 sin t + 3 cos t (g) More examples of the R sin (t ± a) and R cos (t ± a) forms (h) Going back the other way – the Factor Formulas

5.E Solving trig equations

(a) Laying some useful foundations (b) Finding solutions for equations in cos x (c) Finding solutions for equations in tan x (d) Finding solutions for equations in sin x (e) Solving equations using R sin (x + α) etc.

6 Sequences and series

6.A Patterns and formulas

(a) Finding patterns in sequences of numbers (b) How to describe number patterns mathematically

6.B Arithmetic progressions (APs)

(a) What are arithmetic progressions? (b) Finding a rule for summing APs (c) The arithmetic mean or ‘average’ (d) Solving a typical problem (e) A summary of the results for APs

6.C Geometric progressions (GPs)

(a) What are geometric progressions? (b) Summing geometric progressions (c) The sum to infinity of a GP (d) What do ‘convergent’ and ‘divergent’ mean? (e) More examples using GPs; chain letters (f) A summary of the results for GPs (g) Recurring decimals, and writing them as fractions (h) Compound interest: a faster way of getting rich (i)The geometric mean (j) Comparing arithmetic and geometric means (k) Thinking point: what is the fate of the frog down the well?

6.D A compact way of writing sums: the ? notation

(a) What does? stand for? (b) Unpacking the ?s (c) Summing by breaking down to simpler series

6.E Partial fractions

(a) Introducing partial fractions for summing series (b) General rules for using partial fractions (c) The cover-up rule (d) Coping with possible complications

6.F The fate of the frog down the well

7 Binomial series and proof by induction

7.A Binomial series for positive whole numbers

(a) Looking for the patterns (b) Permutations or arrangements (c) Combinations or selections (d) How selections give binomial expansions (e) Writing down rules for binomial expansions (f) Linking Pascal’s Triangle to selections (g) Some more binomial examples

7.B Some applications of binomial series and selections 272

(a) Tossing coins and throwing dice (b) What do the probabilities we have found mean? (c) When is a game fair? (Or are you fair game?) (d) Lotteries: winning the jackpot ... or not

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? (b) Working out some expansions (c) Dealing with slightly different situations

7.D Mathematical induction 279

(a) Truth from patterns - or false mirages? (b) Proving the Binomial Theorem by induction (c) Two non-series applications of induction

8 Differentiation

8.A Some problems answered and difficulties solved

(a) How can we find a speed from knowing the distance travelled? (b) How does y = xn change as x changes? (c) Different ways of writing differentiation: dx/dt, f’(t), x, etc. (d) Some special cases of y = axn (e) Differentiating x = cos t answers another thinking point (f) Can we always differentiate? If not, why not?

8.B Natural growth and decay – the number e

(a) Even more money – compound interest and exponential growth (b) What is the equation of this smooth growth curve? (c) Getting numerical results from the natural growth law of x = ef (d) Relating In x to the log of x using other bases (e) What do we get if we differentiate In t?

8.C Differentiating more complicated functions

(a) The Chain Rule (b) Writing the Chain Rule as F‘(x) = f’(g(x))g’(x) (c) Differentiating functions with angles in degrees or logs to base 10 (d) The Product Rule, or ‘uv’ Rule (e) The Quotient Rule, or ‘u/v’ Rule

8.D The hyperbolic functions of sinh x and cosh x

(a) Getting symmetries from ex and e-x (b) Differentiating sinh x and cosh x (c) Using sinh x and cosh x to get other hyperbolic functions (d) Comparing other hyperbolic and trig formulas - Osborn’s Rule (e) Finding the inverse function for sinh x (f) Can we find an inverse function for cosh x? (g) tanh x and its inverse function tanh-1x (h) What’s in a name? Why ‘hyperbolic’ functions? (i) Differentiating inverse trig and hyperbolic functions

8.E Some uses for differentiation

(a) Finding the equations of tangents to particular curves (b) Finding turning points and points of inflection (c) General rules for sketching curves (d) Some practical uses of turning points (e) A clever use for tangents – the Newton–Raphson Rule

8.F Implicit differentiation

(a) How implicit differentiation works, using circles as examples (b) Using implicit differentiation with more complicated relationships (c) Differentiating inverse functions implicitly (d) Differentiating exponential functions like x = 2t (e) A practical application of implicit differentiation

8.G Writing functions in an alternative form using series

9 Integration

9.A Doing the opposite of differentiating

(a) What could this tell us? (b) A physical interpretation of this process (c) Finding the area under a curve (d) What happens if the area we are finding is below the horizontal axis? (e) What happens if we change the order of the limits? (f) What is ?(1/x)dx?

9.B Techniques of integration

(a) Making use of what we already know (b) Integration by substitution (c) A selection of trig integrals with some hyperbolic cousins (d) Integrals which use inverse trig and hyperbolic functions (e) Using partial fractions in integration (f) Integration by parts (g) Finding rules for doing integrals like ln = ? sinn x dx (h) Using the t = tan (x/2) substitution

9.C Solving some more differential equations

(a) Solving equations where we can split up the variables (b) Putting flesh on the bones – some practical uses for differential equations (c) A forwards look at some other kinds of differential equation, including ones which describe SHM

10 Complex numbers

10.A A new sort of number

(a) Finding the missing roots (b) Finding roots for all quadratic equations (c) Modulus and argument (or mod and arg for short)

10.B Doing arithmetic with complex numbers

(a) Addition and subtraction (b) Multiplication of complex numbers (c) Dividing complex numbers in mod/arg form (d) What are complex conjugates? (e) Using complex conjugates to simplify fractions

10.C How e connects with complex numbers

(a) Two for the price of one – equating real and imaginary parts (b) How does e get involved? (c) What is the geometrical meaning of z = ejθ? (d) What is e-jθand what does it do geometrically? (e) A summary of the sin/cos and sinh/cosh links (f) De Moivre’s Theorem (g) Another example: writing cos 5θ in terms of cos θ (h) More examples of writing trig functions in different forms (i) Solving a differential equation which describes SHM (j) A first look at how we can use complex numbers to describe electric circuits

10.D Using complex numbers to solve more equations

(a) Finding the n roots of zn = a + bj (b) Solving quadratic equations with complex coefficients (c) Solving cubic and quartic equations with complex roots

10.E Finding where z can be if it must fit particular rules

(a) Some simple examples of paths or regions where z must lie (b) What do we do if z has been shifted? (c) Using algebra to find where z can be (d) Another example involving a relationship between w and z

11 Working with vectors

ll.A Basic rules for handling vectors

(a) What are vectors? (b) Adding vectors and what this can mean physically (c) Using components to describe vectors (d) Vector components in three-dimensional space (e) Finding the magnitude of a three-dimensional vector (f) Finding unit vectors

11.B Multiplying vectors

(a) Defining the scalar or dot product of two vectors (b) Working out the dot product of two vectors (c) Defining the vector or cross product of two vectors (d) Working out the cross product of two vectors (e) Can we multiply three vectors together by using dot or cross products? (f) The vector triple product (g) The scalar triple product and what it means geometrically

11.C Finding equations for lines and planes

(a) Finding a vector equation for a line (b) Dealing with lines in two dimensions (c) Dealing with lines in three dimensions (d) Finding the Cartesian equation of a line in three dimensions (e) Another form for the vector equation of a line (f) Finding vector equations for planes (g) Finding equations of planes using normal vectors (h) Finding the perpendicular distance from the origin to a plane (i) The Cartesian form of the equation of a plane (j) Finding where a line intersects a plane (k) Finding the line of intersection of two planes

11.D Finding angles and distances involving lines and planes

(a) Finding the angle between two lines (b) Finding the angle between two planes (c) Finding the acute angle between a line and a plane (d) Finding the shortest distance from a point to a line (e) Finding the shortest distance from a point to a plane (f) Finding the shortest distance between two skew lines

Answers to the exercises Index

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