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Contents
Preface to the first edition xviii Preface to the second edition xix Preface to the third edition xx Preface to the fourth edition xxi Preface to the seventh edition xxii How to use this book xxiii Useful background information xxv Part 1 Foundation topics 1 Programme F.1 Arithmetic 3
Learning outcomes 3 Quiz F.1 4 Types of number 6 The natural numbers – Numerals and place value – Points on a line and order – The integers – Brackets – Addition and subtraction Multiplication and division – Brackets and precedence rules – Basic laws of arithmetic – Estimating – Rounding – Review summary – Review exercise Factors and prime numbers 15 Factors – Prime numbers – Prime factorization – Fundamental theorem of arithmetic – Highest common factor (HCF) – Lowest common multiple (LCM) – Review summary – Review exercise Fractions, ratios and percentages 18 Division of integers – Multiplying fractions – Of – Equivalent fractions Dividing fractions – Adding and subtracting fractions – Fractions on a calculator – Ratios – Percentages – Review summary – Review exercise Decimal numbers 27 Division of integers – Rounding – Significant figures – Decimal places Trailing zeros – Fractions as decimals – Decimals as fractions – Unending decimals – Unending decimals as fractions – Rational, irrational and real numbers – Review summary – Review exercise Powers 33 Raising a number to a power – The laws of powers – Powers on a calculator – Fractional powers and roots – Surds – Multiplication and division by integer powers of 10 – Precedence rules – Standard form Working in standard form – Using a calculator – Preferred standard form – Checking calculations – Accuracy – Review summary – Review exercise Number systems 44 Denary (or decimal) system – Binary system – Octal system (base 8) Duodecimal system (base 12) – Hexadecimal system (base 16) An alternative method – Review summary – Review exercise Change of base from denary to a new base 52 Binary form – Octal form – Duodecimal form – A denary decimal in octal form – Use of octals as an intermediate step Reverse method 56 Review summary – Review exercise
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Can You? Checklist F.1 58 Test exercise F.1 59 Further problems F.1 60
Programme F.2 Introduction to algebra 63
Learning outcomes 63 Quiz F.2 64 Algebraic expressions 65 Symbols other than numerals – Constants – Variables – Rules of algebra Rules of precedence – Terms and coefficients – Collecting like terms Similar terms – Expanding brackets – Nested brackets – Review summary Review exercise Powers and logarithms 72 Powers – Rules of indices – Logarithms – Rules of logarithms – Base 10 and base e – Change of base – Logarithmic equations – Review summary Review exercise Algebraic multiplication and division 81 Multiplication – Division – Review summary – Review exercise Algebraic fractions 85 Addition and subtraction – Multiplication and division – Review summary – Review exercise Factorization of algebraic expressions 88 Common factors – Common factors by grouping – Useful products of two simple factors – Quadratic expressions as the product of two simple factors – Factorization of a quadratic expression – Test for simple factors – Review summary – Review exercise Can You? Checklist F.2 98 Test exercise F.2 98 Further problems F.2 99
Programme F.3 Expressions and equations 101
Learning outcomes 101 Quiz F.3 102 Expressions and equations 103 Evaluating expressions – Equations – Evaluating independent variables Transposition of formulas – The evaluation process – Review summary Review exercise Polynomial equations 113 Polynomial expressions – Evaluation of polynomials – Evaluation of a polynomial by nesting – Remainder theorem – Factor theorem Factorization of fourth-order polynomials – Review summary – Review exercise Can You? Checklist F.3 124 Test exercise F.3 124 Further problems F.3 125
Programme F.4 Graphs 127
Learning outcomes 127 Quiz F.4 128 Graphs of equations 129 Equations – Ordered pairs of numbers – Cartesian axes – Drawing a graph – Review summary – Review exercise
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Using a spreadsheet 137 Spreadsheets – Rows and columns – Text and number entry – Formulas Clearing entries – Construction of a Cartesian graph – Displays – Review summary – Review exercise Inequalities 147 Less than or greater than – Review summary – Review exercise Absolute values 148 Modulus – Graphs – Inequalities – Interaction – Review summary Review exercise Can You? Checklist F.4 157 Test exercise F.4 158 Further problems F.4 158
Programme F.5 Linear equations 161
Learning outcomes 161 Quiz F.5 162 Linear equations 163 Solution of simple equations – Simultaneous linear equations with two unknowns – Simultaneous equations with three unknowns Pre-simplification – Review summary – Review exercise Can You? Checklist F.5 173 Test exercise F.5 174 Further problems F.5 174
Programme F.6 Polynomial equations 177
Learning outcomes 177 Quiz F.6 178 Polynomial equations 179 Quadratic equations – Cubic equations having at least one simple linear factor – Fourth-order equations having at least two linear factors Review summary – Review exercise Can You? Checklist F.6 193 Test exercise F.6 193 Further problems F.6 193
Programme F.7 Binomials 195
Learning outcomes 195 Quiz F.7 196 Factorials and combinations 197 Factorials – Combinations – Three properties of combinatorial coefficients – Review summary – Review exercise Binomial expansions 205 Pascal’s triangle – Binomial expansions – The general term of the binomial expansion – Review summary – Review exercise The Æ (sigma) notation 211 General terms – The sum of the first n natural numbers – Rules for manipulating sums – The exponential number e – Review summary Review exercise Can You? Checklist F.7 220 Test exercise F.7 220 Further problems F.7 221
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Programme F.8 Partial fractions 223
Learning outcomes 223 Quiz F.8 224 Partial fractions 225 Review summary – Review exercise Denominators with repeated and quadratic factors 232 Review summary – Review exercise Can You? Checklist F.8 239 Test exercise F.8 240 Further problems F.8 240 Programme F.9 Trigonometry 243
Learning outcomes 243 Quiz F.9 244 Angles 245 Rotation – Radians – Triangles – Trigonometric ratios – Reciprocal ratios Pythagoras’ theorem – Special triangles – Half equilateral – Review summary – Review exercise Trigonometric identities 258 The fundamental identity – Two more identities – Identities for compound angles – Trigonometric formulas – Review summary Review exercise Can You? Checklist F.9 264 Test exercise F.9 264 Further problems F.9 265 Programme F.10 Functions 267
Learning outcomes 267 Quiz F.10 268 Processing numbers 269 Functions are rules but not all rules are functions – Functions and the arithmetic operations – Inverses of functions – Graphs of inverses The graph of y ¼ x3 – The graph of y ¼ x1=3 – The graphs of y ¼ x3 and y ¼ x1=3 plotted together – Review summary – Review exercise Composition 280 Function of a function – Inverses of compositions – Review summary Review exercise Can You? Checklist F.10 284 Test exercise F.10 285 Further problems F.10 285 Programme F.11 Trigonometric and exponential functions 287
Learning outcomes 287 Quiz F.11 288 Trigonometric functions 289 Rotation – The tangent – Period – Amplitude – Phase difference Inverse trigonometric functions – Trigonometric equations – Equations of the form a cos x þ b sin x = c – Review summary – Review exercise Exponential and logarithmic functions 303 Exponential functions – Indicial equations – Review summary – Review exercise
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Odd and even functions 307 Odd and even parts – Odd and even parts of the exponential function Limits of functions – The rules of limits – Review summary – Review exercise Can You? Checklist F.11 312 Test exercise F.11 313 Further problems F.11 314 Programme F.12 Differentiation 315
Learning outcomes 315 Quiz F.12 316 Gradients 317 The gradient of a straight line – The gradient of a curve at a given point Algebraic determination of the gradient of a curve – Derivatives of powers of x – Differentiation of polynomials – Second and higher derivatives – alternative notation – Review summary – Review exercise Standard derivatives and rules 330 Limiting value of sin = as ! 0 – Standard derivatives – Derivative of a product of functions – Derivative of a quotient of functions – Derivative of a function of a function – Derivative of ax – Review summary Review exercise Newton–Raphson iterative method 344 Notation – Tabular display of results – Review summary – Review exercise Can You? Checklist F.12 350 Test exercise F.12 351 Further problems F.12 352 Programme F.13 Integration 353
Learning outcomes 353 Quiz F.13 354 Integration 355 Constant of integration – Standard integrals – Review summary Review exercise Integration of polynomial expressions 358 Functions of a linear function of x – Review summary – Review exercise Integration by partial fractions 363 Review summary – Review exercise Areas under curves 366 Review summary – Review exercise Integration as a summation 370 The area between a curve and an intersecting line – Review summary Review exercise Can You? Checklist F.13 379 Test exercise F.13 380 Further problems F.13 381 Part II 383 Programme 1 Complex numbers 1 385
Learning outcomes 385 Introduction 386 Ideas and symbols
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The symbol j 386 Quadratic equations Powers of j 389 Positive integer powers – Negative integer powers Complex numbers 390 Addition and subtraction – Multiplication – Division – Equal complex numbers – Review exercise Graphical representation of a complex number 399 Argand diagram – Graphical addition of complex numbers Polar form of a complex number 401 Exponential form of a complex number 406 Review summary Can You? Checklist 1 409 Test exercise 1 409 Further problems 1 410 Programme 2 Complex numbers 2 412
Learning outcomes 412 Polar form calculations 413 Review exercise Roots of a complex number 421 Expansions 427 Expansions of sin n and cos n , where n is a positive integer Expansions of cosn and sinn Loci problems 430 Review summary Can You? Checklist 2 434 Test exercise 2 434 Further problems 2 435 Programme 3 Hyperbolic functions 437
Learning outcomes 437 Introduction 438 Graphs of hyperbolic functions 440 Review exercise Evaluation of hyperbolic functions 445 Inverse hyperbolic functions 446 Log form of the inverse hyperbolic functions 448 Hyperbolic identities 451 Relationship between trigonometric and hyperbolic functions 453 Can You? Checklist 3 456 Test exercise 3 457 Further problems 3 457 Programme 4 Determinants 459
Learning outcomes 459 Determinants 460 Determinants of the third order 466 Evaluation of a third-order determinant Simultaneous equations in three unknowns 470 Review exercise Consistency of a set of equations 478 Properties of determinants 481
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Can You? Checklist 4 485 Test exercise 4 485 Further problems 4 486 Programme 5 Matrices 489
Learning outcomes 489 Matrices – definitions 490 Matrix notation 491 Equal matrices 492 Addition and subtraction of matrices 492 Multiplication of matrices 493 Scalar multiplication – Multiplication of two matrices Transpose of a matrix 496 Special matrices 497 Determinant of a square matrix 499 Cofactors – Adjoint of a square matrix Inverse of a square matrix 501 Product of a square matrix and its inverse Solution of a set of linear equations 504 Gaussian elimination method for solving a set of linear equations Eigenvalues and eigenvectors 509 Eigenvalues – Eigenvectors – Review summary Can You? Checklist 5 515 Test exercise 5 515 Further problems 5 516 Programme 6 Vectors 519
Learning outcomes 519 Introduction: scalar and vector quantities 520 Vector representation 521 Two equal vectors – Types of vector – Addition of vectors – The sum of a number of vectors Components of a given vector 524 Components of a vector in terms of unit vectors Vectors in space 530 Direction cosines 532 Scalar product of two vectors 533 Vector product of two vectors 535 Angle between two vectors 537 Direction ratios 540 Review summary Can You? Checklist 6 541 Test exercise 6 541 Further problems 6 542 Programme 7 Differentiation 544
Learning outcomes 544 Standard derivatives 545 Functions of a function 546 Products – Quotients Logarithmic differentiation 552 Review exercise Implicit functions 555
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Parametric equations 557 Can You? Checklist 7 559 Test exercise 7 560 Further problems 7 560 Programme 8 Differentiation applications 562
Learning outcomes 562 Differentiation of inverse trigonometric functions 563 Review exercise Derivatives of inverse hyperbolic functions 565 Review exercise Maximum and minimum values 570 Points of inflexion 574 Can You? Checklist 8 580 Test exercise 8 580 Further problems 8 580 Programme 9 Tangents, normals and curvature 583
Learning outcomes 583 Equation of a straight line 584 Tangents and normals to a curve at a given point 587 Curvature 592 Centre of curvature Can You? Checklist 9 601 Test exercise 9 602 Further problems 9 602 Programme 10 Sequences 605
Learning outcomes 605 Functions with integer input 606 Sequences – Graphs of sequences – Arithmetic sequence – Geometric sequence – Harmonic sequence – Recursive prescriptions – Other sequences – Review summary – Review exercise Difference equations 618 Solving difference equations – Second-order homogeneous equations Equal roots of the characteristic equation – Review summary – Review exercise Limits of sequences 626 Infinity – Limits – Infinite limits – Rules of limits – Indeterminate limits Review summary – Review exercise Can You? Checklist 10 635 Test exercise 10 636 Further problems 10 637 Programme 11 Series 1 639
Learning outcomes 639 Series 640 Arithmetic series – Arithmetic mean – Geometric series – Geometric mean Series of powers of the natural numbers 645 Sum of natural numbers – Sum of squares – Sum of cubes Infinite series 648
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Contents xiii
Limiting values 650 Convergent and divergent series – Tests for convergence – Absolute convergence – Review summary Can You? Checklist 11 660 Test exercise 11 660 Further problems 11 661 Programme 12 Series 2 663
Learning outcomes 663 Power series 664 Introduction – Maclaurin’s series – Standard series – The binomial series Approximate values 674 Taylor’s series Limiting values – indeterminate forms 677 L’Hopital’s rule for finding limiting values Can You? Checklist 12 684 Test exercise 12 685 Further problems 12 685 Programme 13 Curves and curve fitting 688
Learning outcomes 688 Introduction 689 Standard curves 689 Straight line – Second-degree curves –Third-degree curves – Circle Ellipse – Hyperbola – Logarithmic curves – Exponential curves Hyperbolic curves – Trigonometrical curves Asymptotes 697 Determination of an asymptote – Asymptotes parallel to the x- and y- axes Systematic curve sketching, given the equation of the curve 702 Symmetry – Intersection with the axes – Change of origin – Asymptotes Large and small values of x and y – Stationary points – Limitations Curve fitting 707 Straight line law – Graphs of the form y ¼ axn, where a and n are constants – Graphs of the form y ¼ aenx Method of least squares 712 Fitting a straight line graph Correlation 718 Correlation – Measures of correlation – The Pearson product-moment correlation coefficient – Spearman’s rank correlation coefficient Review summary Can You? Checklist 13 726 Test exercise 13 727 Further problems 13 728 Programme 14 Partial differentiation 1 731
Learning outcomes 731 Partial differentiation 732 Review exercise Small increments 744 Can You? Checklist 14 749 Test exercise 14 749 Further problems 14 750
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Programme 15 Partial differentiation 2 752
Learning outcomes 752 Partial differentiation 753 Rate-of-change problems 755 Change of variables 763 Can You? Checklist 15 765 Test exercise 15 765 Further problems 15 766
Programme 16 Integration 1 768
Learning outcomes 768 Introduction 769 Standard integrals Functions of a linear functionÐ of x Ð 772 Integrals of the forms f 0ðxÞ=f ðxÞ dx and f ðxÞf 0ðxÞ dx 774 Integration of products – integration by parts 778 Integration by partial fractions 782 Integration of trigonometric functions 787 Can You? Checklist 16 791 Test exercise 16 792 Further problems 16 792
Programme 17 Integration 2 794
Learning outcomes 794 Can You? Checklist 17 819 Test exercise 17 819 Further problems 17 820
Programme 18 Reduction formulas 821
Learning outcomes 821 Can You? Checklist 18 831 Test exercise 18 831 Further problems 18 832
Programme 19 Integration applications 1 834
Learning outcomes 834 Basic applications 835 Areas under curves – Definite integrals Parametric equations 843 Mean values 844 Root mean square (rms) value 846 Review summary Can You? Checklist 19 849 Test exercise 19 849 Further problems 19 849
Programme 20 Integration applications 2 851
Learning outcomes 851 Introduction 852 Volume of a solid of revolution 852 Centroid of a plane figure 856
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Centre of gravity of a solid of revolution 859 Length of a curve 860 Parametric equations Surface of revolution 864 Parametric equations Rules of Pappus 867 Review summary Can You? Checklist 20 869 Test exercise 20 870 Further problems 20 870 Programme 21 Integration applications 3 873
Learning outcomes 873 Moments of inertia 874 Radius of gyration – Parallel axes theorem – Perpendicular axes theorem (for thin plates) – Useful standard results Second moments of area 888 Composite figures Centre of pressure 892 Pressure at a point P, depth z below the surface – Total thrust on a vertical plate immersed in a liquid – Depth of the centre of pressure – Review summary Can You? Checklist 21 899 Test exercise 21 899 Further problems 21 900 Programme 22 Approximate integration 902
Learning outcomes 902 Introduction 903 Approximate integration 904 Series – Simpson’s rule Proof of Simpson’s rule 916 Can You? Checklist 22 918 Test exercise 22 918 Further problems 22 919 Programme 23 Polar coordinate systems 921
Learning outcomes 921 Introduction to polar coordinates 922 Polar curves 924 Standard polar curves 926 Applications 929 Review summary Can You? Checklist 23 941 Test exercise 23 941 Further problems 23 942 Programme 24 Multiple integrals 943
Learning outcomes 943 Summation in two directions 944 Double integrals 947 Triple integrals 949 Applications 951 Review exercise
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Alternative notation 956 Determination of areas by multiple integrals 960 Determination of volumes by multiple integrals 962 Can You? Checklist 24 965 Test exercise 24 965 Further problems 24 966
Programme 25 First-order differential equations 968
Learning outcomes 968 Introduction 969 Formation of differential equations 970 Solution of differential equations 972 By direct integration – By separating the variables – Review exercise Solution of differential equations 979 Homogeneous equations – by substituting y ¼ vx – Review exercise Solution of differential equations 985 Linear equations – use of integrating factor – Review exercise Bernoulli’s equation 993 Review summary Can You? Checklist 25 999 Test exercise 25 1000 Further problems 25 1000
Programme 26 Second-order differential equations 1004
Learning outcomes 1004 Homogeneous equations 1005 Review exercise Inhomogeneous equations 1013 Review summary Particular solution 1020 Review summary Can You? Checklist 26 1024 Test exercise 26 1025 Further problems 26 1025
Programme 27 Introduction to Laplace transforms 1027
Learning outcomes 1027 The Laplace transform 1028 The inverse Laplace transform, Table of Laplace transforms Review summary – Review exercise The Laplace transform 1032 Laplace transform of a derivative – Two properties of Laplace transforms Table of Laplace transforms Review summary – Review exercise The Laplace transform 1036 Generating new transforms – Laplace transforms of higher derivatives Table of Laplace transforms – Linear, constant coefficient, inhomogeneous differential equations – Review summary – Review exercise Can You? Checklist 27 1043 Test exercise 27 1043 Further problems 27 1044
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Programme 28 Data handling and statistics 1045
Learning outcomes 1045 Introduction 1046 Arrangement of data 1046 Tally diagram – Grouped data – Grouping with continuous data Relative frequency – Rounding off data – Class boundaries Histograms 1053 Frequency histogram – Relative frequency histogram Measures of central tendency 1055 Mean – Coding for calculating the mean – Decoding – Coding with a grouped frequency distribution – Mode – Mode with grouped data Median – Median with grouped data Measures of dispersion 1063 Mean deviation – Range – Standard deviation – Alternative formula for the standard deviation Distribution curves 1066 Frequency polygons – Frequency curves – Normal distribution curve Standardized normal curve 1069 Review summary Can You? Checklist 28 1072 Test exercise 28 1072 Further problems 28 1073
Programme 29 1076
Learning outcomes 1076 Probability 1077 Random experiments – Events – Sequences of random experiments Combining events Events and probabilities 1079 Probability – Assigning probabilities – Review summary Probabilities of combined events 1082 Or – Non-mutually exclusive events – And – Dependent events Independent events – Probability trees – Review summary Conditional probability 1087 Probability distributions 1090 Random variables – Expectation – Variance and standard deviation Bernoulli trials – Binomial probability distribution – Expectation and standard deviation – The Poisson probability distribution – Binomial and Poisson compared Continuous probability distributions 1106 Normal distribution curve Standard normal curve 1106 Review summary Can You? Checklist 29 1112 Test exercise 29 1113 Further problems 29 1113
Answers 1117 Index 1145
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Programme F.1 Frames 1 to 154
Arithmetic
Learning outcomes When you have completed this Programme you will be able to: Carry out the basic rules of arithmetic with integers Check the result of a calculation making use of rounding Write a whole as a product of prime numbers Find the highest common factor and lowest common multiple of two whole numbers Manipulate fractions, ratios and percentages Manipulate decimal numbers Manipulate powers Use standard or preferred standard form and complete a calculation to the required level of accuracy Understand the construction of various number systems and convert from one number system to another. If you already feel confident about these why not try the quiz over the page? You can check your answers at the end of the book.
3
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Questions marked with this icon can be found at www.palgrave.com/stroud. There you can go through the question step by step and follow Quiz F.1 online hints, with full working provided.
Frames 1 Place the appropriate symbol < or > between each of the following pairs of numbers: (a) 3 2 (b) 8 13 (c) 25 0 1to4 2 Find the value of each of the following: (a) 13 þ 9 3 2 5 (b) ð13 þ 9Þ ð3 2Þ 5 5to12 3 Round each number to the nearest 10, 100 and 1000: (a) 1354 (b) 2501 (c) 2452 (d) 23 625 13 to 15 4 Write each of the following as a product of prime factors: (a) 170 (b) 455 (c) 9075 (d) 1140 19 to 22 5 Find the HCF and the LCM of each pair of numbers: (a) 84, 88 (b) 105, 66 23 to 24 6 Reduce each of the following fractions to their lowest terms: 12 144 49 64 (a) (b) (c) (d) 28 to 36 18 21 14 4 7 Evaluate the following: 3 2 11 5 3 4 5 4 (a) (b) (c) þ (d) 37 to 46 7 3 30 6 7 13 16 3 8 Write the following proportions as ratios: 1 1 3 (a) of A, of B and of C 2 5 10 1 1 1 (b) of P, of Q , of R and the remainder S 47 to 48 4 3 5 9 Complete the following: 4 (a) ¼ % (b) 48% of 50 ¼ 5 9 (c) ¼ % (d) 15% of 25 ¼ 49 to 52 14 10 Round each of the following decimal numbers, first to 3 significant figures and then to 2 decimal places: (a) 21.355 (b) 0.02456 (c) 0.3105 (d) 5134.555 56 to 65 11 Convert each of the following to decimal form to 3 decimal places: 4 7 9 28 (a) (b) (c) (d) 66 to 67 15 13 5 13 12 Convert each of the following to fractional form in lowest terms: (a) 0:8 (b) 2:8 (c) 3:3_ 2_ (d) 5:5 68 to 73 13 Write each of the following in abbreviated form: (a) 1.010101 . . . (b) 9.2456456456 . . . 70 to 71 14 Write each of the following as a number raised to a power: 3 8 (a) 36 33 (b) 43 25 (c) 92 (d) 70 78 to 89
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Frames 15 Find the value of each of the following to 3 dp: pffiffiffi 1 1 1 5 (a) 153 (b) 5 (c) ð 27Þ3 (d) ð 9Þ2 90 to 94 16 Write each of the following as a single decimal number: (a) 3:2044 103 (b) 16:1105 10 2 95 to 98 17 Write each of the following in standard form: (a) 134.65 (b) 0.002401 99 to 101 18 Write each of the following in preferred standard form: (a) 16:1105 10 2 (b) 9.3304 102 to 104 19 In each of the following the numbers have been obtained by measurement. Evaluate each calculation to the appropriate level of accuracy: (a) 11:4 0:0013 5:44 8:810 1:01 0:00335 (b) 105 to 108 9:12 6:342 20 Express the following numbers in denary form: : : (a) 1011 012 (b) 456 7218 : : (c) 123 2912 (d) CA1 B2216 112 to 126 : 21 Convert 15 60510 to the equivalent octal, binary, duodecimal and hexadecimal forms. 127 to 149
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6 Programme F.1
Types of number
1 The natural numbers The first numbers we ever meet are the whole numbers. These, together with zero, are called the natural numbers, and are written down using numerals. Numerals and place value The natural numbers are written using the ten numerals 0, 1, ..., 9 where the position of a numeral dictates the value that it represents. For example: 246 stands for 2 hundreds and 4 tens and 6 units. That is 200 þ 40 þ 6 Here the numerals 2, 4 and 6 are called the hundreds, tens and unit coefficients respectively. This is the place value principle. Points on a line and order The natural numbers can be represented by equally spaced points on a straight line where the first natural number is zero 0.
01234567 8
The natural numbers are ordered – they progress from small to large. As we move along the line from left to right the numbers increase as indicated by the arrow at the end of the line. On the line, numbers to the left of a given number are less than (<) the given number and numbers to the right are greater than (>) the given number. For example, 8 > 5 because 8 is represented by a point on the line to the right of 5. Similarly, 3 < 6 because 3 is to the left of 6.
Now move on to the next frame
2 The integers If the straight line displaying the natural numbers is extended to the left we can plot equally spaced points to the left of zero.
–5 –4 –3 –2 3210–1
These points represent negative numbers which are written as the natural number preceded by a minus sign, for example 4. These positive and negative whole numbers and zero are collectively called the integers. The notion of order still applies. For example, 5 < 3 and 2 > 4 because the point on the line representing 5isto the left of the point representing 3. Similarly, 2 is to the right of 4. The numbers 10, 4, 0, 13 are of a type called ......
You can check your answer in the next frame
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Arithmetic 7
Integers 3
They are integers. The natural numbers are all positive. Now try this: Place the appropriate symbol < or > between each of the following pairs of numbers: (a) 3 6 (b) 2 4 (c) 712 Complete these and check your results in the next frame
(a) 3 > 6 4 (b) 2 > 4 (c) 7 < 12
The reasons being: (a) 3 > 6 because 3 is represented on the line to the right of 6 (b) 2 > 4 because 2 is represented on the line to the right of 4 (c) 7 < 12 because 7 is represented on the line to the left of 12
Now move on to the next frame
Brackets 5 Brackets should be used around negative numbers to separate the minus sign attached to the number from the arithmetic operation symbol. For example, 5 3 should be written 5 ð 3Þ and 7 2 should be written 7 ð 2Þ. Never write two arithmetic operation symbols together without using brackets. Addition and subtraction Adding two numbers gives their sum and subtracting two numbers gives their difference. For example, 6 þ 2 ¼ 8. Adding moves to the right of the first number and subtracting moves to the left of the first number, so that 6 2 ¼ 4 and 4 6 ¼ 2:
6 + 2 6 – 2 4 – 6
–2 0 2 4 6 8 10
Adding a negative number is the same as subtracting its positive counterpart. For example 7 þð 2Þ¼7 2. The result is 5. Subtracting a negative number is the same as adding its positive counterpart. For example 7 ð 2Þ¼7 þ 2 ¼ 9. So what is the value of: (a) 8 þð 3Þ (b) 9 ð 6Þ (c) ð 4Þþð 8Þ (d) ð 14Þ ð 7Þ?
When you have finished these check your results with the next frame
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8 Programme F.1
6 (a) 5 (b) 15 (c) 12 (d) 7
Move now to Frame 7
7 Multiplication and division Multiplying two numbers gives their product and dividing two numbers gives their quotient. Multiplying and dividing two positive or two negative numbers gives a positive number. For example: 12 2 ¼ 24 and ð 12Þ ð 2Þ¼24 12 2 ¼ 6 and ð 12Þ ð 2Þ¼6 Multiplying or dividing a positive number by a negative number gives a negative number. For example: 12 ð 2Þ¼ 24, ð 12Þ 2 ¼ 6 and 8 ð 4Þ¼ 2 So what is the value of: (a) ð 5Þ 3 (b) 12 ð 6Þ (c) ð 2Þ ð 8Þ (d) ð 14Þ ð 7Þ?
When you have finished these check your results with the next frame
8 (a) 15 (b) 2 (c) 16 (d) 2
Move on to Frame 9
9 Brackets and precedence rules Brackets and the precedence rules are used to remove ambiguity in a calculation. For example, 14 3 4 could be either: 11 4 ¼ 44 or 14 12 ¼ 2 depending on which operation is performed first. To remove the ambiguity we rely on the precedence rules: In any calculation involving all four arithmetic operations we proceed as follows:
(a) Working from the left evaluate divisions and multiplications as they are encountered; this leaves a calculation involving just addition and subtraction. (b) Working from the left evaluate additions and subtractions as they are encountered.
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Arithmetic 9
For example, to evaluate: 4 þ 5 6 2 12 4 2 1 a first sweep from left to right produces: 4 þ 30 2 3 2 1 a second sweep from left to right produces: 4 þ 15 6 1 and a final sweep produces: 19 7 ¼ 12 If the calculation contains brackets then these are evaluated first, so that: ð4 þ 5 6Þ 2 12 4 2 1 ¼ 34 2 6 1 ¼ 17 7 ¼ 10 This means that: 14 3 4 ¼ 14 12 ¼ 2 because, reading from the left we multiply before we subtract. Brackets must be used to produce the alternative result: ð14 3Þ 4 ¼ 11 4 ¼ 44 because the precedence rules state that brackets are evaluated first. So that 34 þ 10 ð2 3Þ 5 ¼ ......
Result in the next frame
16 10
Because 34 þ 10 ð2 3Þ 5 ¼ 34 þ 10 ð 1Þ 5 we evaluate the bracket first ¼ 34 þð 10Þ 5 by dividing ¼ 34 þð 50Þ by multiplying ¼ 34 50 finally we subtract ¼ 16 Notice that when brackets are used we can omit the multiplication signs and replace the division sign by a line, so that: 5 ð6 4Þ becomes 5ð6 4Þ and 25 10 ð25 10Þ 5 becomes ð25 10Þ=5or 5
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10 Programme F.1
When evaluating expressions containing nested brackets the innermost brackets are evaluated first. For example: 3ð4 2½5 1 Þ ¼ 3ð4 2 4Þ evaluating the innermost bracket ½... first ¼ 3ð4 8Þ multiplication before subtraction inside the ð...Þ bracket ¼ 3ð 4Þ subtraction completes the evaluation of the ð...Þ bracket ¼ 12 multiplication completes the calculation so that 5 f8 þ 7½4 1 9=3g¼......
Work this out, the result is in the following frame
11 21
Because 5 f8 þ 7½4 1 9=3g¼5 f8 þ 7 3 9 3g ¼ 5 f8 þ 21 3g ¼ 5 f29 3g ¼ 5 26 ¼ 21
Now move to Frame 12
12 Basic laws of arithmetic All the work that you have done so far has been done under the assumption that you know the rules that govern the use of arithmetic operations as, indeed, you no doubt do. However, there is a difference between knowing the rules innately and being consciously aware of them, so here they are. The four basic arithmetic operations are: addition and subtraction multiplication and division where each pair may be regarded as consisting of ‘opposites’ – in each pair one operation is the reverse operation of the other.
1 Commutativity Two integers can be added or multiplied in either order without affecting the result. For example: 5 þ 8 ¼ 8 þ 5 ¼ 13 and 5 8 ¼ 8 5 ¼ 40 We say that addition and multiplication are commutative operations
The order in which two integers are subtracted or divided does affect the result. For example: 4 2 6¼ 2 4 because 4 2 ¼ 2 and 2 4 ¼ 2 Notice that 6¼ means is not equal to. Also 4 2 6¼ 2 4 We say that subtraction and division are not commutative operations
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Arithmetic 11
2 Associativity The way in which three or more integers are associated under addition or multiplication does not affect the result. For example: 3 þð4 þ 5Þ¼ð3 þ 4Þþ5 ¼ 3 þ 4 þ 5 ¼ 12 and 3 ð4 5Þ¼ð3 4Þ 5 ¼ 3 4 5 ¼ 60 We say that addition and multiplication are associative operations The way in which three or more integers are associated under subtraction or division does affect the result. For example: 3 ð4 5Þ 6¼ð3 4Þ 5 because 3 ð4 5Þ¼3 ð 1Þ¼3 þ 1 ¼ 4 and ð3 4Þ 5 ¼ð 1Þ 5 ¼ 6 Also 24 ð4 2Þ 6¼ð24 4Þ 2 because 24 ð4 2Þ¼24 2 ¼ 12 and ð24 4Þ 2 ¼ 6 2 ¼ 3 We say that subtraction and division are not associative operations
3 Distributivity Consider the equations: 3 ð4 þ 5Þ¼3 9 ¼ 27 and ð3 4Þþð3 5Þ¼12 þ 15 ¼ 27 From this we can deduce that: 3 ð4 þ 5Þ¼ð3 4Þþð3 5Þ We say that multiplication distributes itself over addition from the left. Multiplication is also distributive over addition from the right. For example: ð3 þ 4Þ 5 ¼ð3 5Þþð4 5Þ¼35 The same can be said of multiplication and subtraction: multiplication is distributive over subtraction from both the left and the right. For example: 3 ð4 5Þ¼ð3 4Þ ð3 5Þ¼ 3 and ð3 4Þ 5 ¼ð3 5Þ ð4 5Þ¼ 5 Division is distributed over addition and subtraction from the right but not from the left. For example: ð60 þ 15Þ 5 ¼ð60 5Þþð15 5Þ because ð60 þ 15Þ 5 ¼ 75 5 ¼ 15 and ð60 5Þþð15 5Þ¼12 þ 3 ¼ 15 However, 60 ð15 þ 5Þ 6¼ð60 15Þþð60 5Þ because 60 ð15 þ 5Þ¼60 20 ¼ 3 and ð60 15Þþð60 5Þ¼4 þ 12 ¼ 16 Also: ð20 10Þ 5 ¼ð20 5Þ ð10 5Þ because ð20 10Þ 5 ¼ 10 5 ¼ 2 and ð20 5Þ ð10 5Þ¼4 2 ¼ 2 but 20 ð10 5Þ 6¼ð20 10Þ ð20 5Þ because 20 ð10 5Þ¼20 5 ¼ 4 and ð20 10Þ ð20 5Þ¼2 4 ¼ 2
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
12 Programme F.1
13 Estimating Arithmetic calculations are easily performed using a calculator. However, by pressing a wrong key, wrong answers can just as easily be produced. Every calculation made using a calculator should at least be checked for the reasonableness of the final result and this can be done by estimating the result using rounding. For example, using a calculator the sum 39 þ 53 is incorrectly found to be 62 if 39 þ 23 is entered by mistake. If, now, 39 is rounded up to 40, and 53 is rounded down to 50 the reasonableness of the calculator result can be simply checked by adding 40 to 50 to give 90. This indicates that the answer 62 is wrong and that the calculation should be done again. The correct answer 92 is then seen to be close to the approximation of 90. Rounding An integer can be rounded to the nearest 10 as follows: If the number is less than halfway to the next multiple of 10 then the number is rounded down to the previous multiple of 10. For example, 53 is rounded down to 50. If the number is more than halfway to the next multiple of 10 then the number is rounded up to the next multiple of 10. For example, 39 is rounded up to 40. If the number is exactly halfway to the next multiple of 10 then the number is rounded up. For example, 35 is rounded up to 40. This principle also applies when rounding to the nearest 100, 1000, 10 000 or more. For example, 349 rounds up to 350 to the nearest 10 but rounds down to 300 to the nearest 100, and 2501 rounds up to 3000 to the nearest 1000. Try rounding each of the following to the nearest 10, 100 and 1000 respectively: (a) 1846 (b) 638 (c) 445 Finish all three and check your results with the next frame
14 (a) 1850, 1800, 2000 (b) 640, 600, 1000 (c) 450, 400, 0
Because (a) 1846 is nearer to 1850 than to 1840, nearer to 1800 than to 1900 and nearer to 2000 than to 1000. (b) 638 is nearer to 640 than to 630, nearer to 600 than to 700 and nearer to 1000 than to 0. The negative sign does not introduce any complications. (c) 445 rounds to 450 because it is halfway to the next multiple of 10, 445 is nearer to 400 than to 500 and nearer to 0 than 1000. How about estimating each of the following using rounding to the nearest 10: (a) 18 21 19 11 (b) 99 101 49 8 Check your results in Frame 15
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index
A priori 1080 Decimal numbers 27 Absolute values 148 Division 8 Accuracy 42 Estimating 12 Acute angle 245 Factors 15 Addition 7 Fractions 18 Algebraic fractions 85 Fundamental theorem 16 Complex numbers 391 Highest common factor (HCF) 16 Fractions 22 Laws 10 Graphical 400 Lowest common multiple (LCM) 16 Matrices 492 Multiplication 8 Powers 33 Negative numbers 6 Vectors 522 Number systems 44 Adjoint matrix 500 Percentages 18 Algebra Precedence rules 8, 13 Brackets 69 Prime factorization 15 Constants 66 Prime numbers 15 Factorization 68, 88 Quotient 8 Fractions 85 Rational number 18 Introduction 63 Ratios 18 Symbols 65 Rounding 12 Variables 66 Subtraction 7 Algebraic equations 65, 101 Types of number 6 Algebraic expressions Arithmetic operations 272 Factorization 88 Arithmetic sequence 608 Algebraic fractions 85 Common difference 608 Algebraic symbols 65 Arrangement of data 1046 Amplitude 294 Arrangements 200 And 1084 Assigning equations 105 Angle between two vectors 537 Assigning probabilities 1080 Angles 245 Associativity 11, 67 Rotation 245 Asymptote 134, 697, 703 Applications of multiple integrals 951 Determination 698 Applications of polar curves 929 Parallel to axes 699 Area of surface of revolution 937 Augmented matrix 507 Length of arc 934 Auxiliary equation 1006 Volume of rotation 932 Complex roots 1009 Approximate integration 902 Real and different roots 1007 By series 904 Real and equal roots 1008 By Simpson’s rule 908 Area between a curve and a line 374 Bernoulli trials 1094 Areas under curves 366, 835 Bernoulli’s equation 993 Argand diagram 399 Binary system 45 Arithmetic 3 Binomial expansions 205, 207 Addition 7 General term 209 Brackets 7, 8 Binomial expectation 1099 Calculations 13 Binomial probability distribution 1095, 1102
1145
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
1146 Index
Binomial standard deviation 1099 Checking calculations 42 Binomials 195 Circle 692 Arrangements 200 Class boundaries 1051 Binomial expansions 205, 207, 209 Class interval 1051 Combinations 200 Co-domain 271 Combinatorial coefficients 203 Coding for a grouped distribution 1058 Factorials 197 Coding for calculating the mean 1057 Pascal’s triangle 205 Coefficients 6, 68, 72 Permutations 200 Cofactor 500 Brackets 7, 8 Collecting like terms 68 Expanding 69 Combinations 200 Nested 69 Combinatorial coefficients 203 Properties 203 Calculations 13 Combining events 1079 Accuracy 42 Common denominator 22 Checking 42 Common difference 608 Decimal places 27 Common factors 68 Significant figures 27 Algebra 88 Standard form 39 Common ratio 609 Calculator 23 Commutativity 10, 67 Powers 36 Complementary function 1013 Standard form 40 Completing the square 181 Cancelling 20 Complex conjugate 393 Cartesian axes 130 Complex numbers 385, 412 Cartesian graph 139 Addition and subtraction 391 Catenary 438 Argand diagram 399 Cause-and-effect 718 Complex conjugate 393 Centre of curvature 594, 598 Complex plane 399 Centre of gravity of a solid of revolution 859 De Moivre’s theorem 422 Centre of pressure 892 Division 394 Depth of centre of pressure 895 Equal 396 Pressure at a point 892 Exponential form 406 Total thrust 892 Graphical representation 399 Centroid of a plane figure 856 Imaginary part 390 Certainty 1079 Loci problems 430 Chain rule 338 Multiplication 392 Change of base of logarithm 77 Polar form 401, 413 Change of number base Principal root 425 Denary decimal to duodecimal 54 Real part 390 Denary decimal to octal 53 Roots 421 Denary to binary 52 Trigonometric expansions 427 Denary to duodecimal 52 Complex plane 399 Octals as an intermediate step 55 Components of a vector 524 Reverse method 56 Composition 280 Change of variables 763 Conditional equations 104 Characteristic determinant 510 Conditional probability 1087 Characteristic equation 510, 620 Consistency of a set of equations 478 Equal roots 623 Constant of integration 355 Characteristic values 509 Constants 66 Characteristic vectors 509 Continuous variable 1046
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index 1147
Convergence 628 Trigonometric expansions 427 Correlation 718 Decimal numbers 27 Measures 719 As fractions 29 Pearson product-moment coefficient 719 Standard form 38 Spearman’s rank coefficient 723 Trailing zeros 29 Cosecant 251 Unending 30 Cosine 249 Decimal places (dp) 27, 28 Cosine function 290 Decimal point 27 Cotangent 251 Decoding 1058 Cube root 36 Defining equations 104 Cubic equations 184 Definite integrals 838 Curvature 592 Definition of a Laplace transform 1028 Centre of curvature 594. 598 Degrees 245 Parametric equations 596 Denary system 44 Radius of curvature 594 Denominator 18 Curve fitting 707 Common 22 Straight-line law 707 Dependent events 1084 Curve sketching 702 Dependent variables 104 Asymptotes 703 Depth of centre of pressure 895 Change of origin 703 Derivative Intersection with axes 702 Implicit functions 759 Large and small values of the variables 704 Partial 732 Limitations 705 Derivative of a function of a function 337 Stationary points 704 Derivative of a product 333 Symmetry 702 Derivative of a quotient 334 Curves 688 Derivative of powers of x 323 Asymptotes 697 Derived curve 572 Circle 692 Determinants 459 Ellipse 693 Properties 481 Exponential curves 695 Second-order 461 Hyperbola 693 Simultaneous equations in three Hyperbolic curves 696 unknowns 470 Logarithmic curves 695 Square matrix 499 Second-degree 690 Third-order 466 Standard curves 689 Diagonal matrix 497 Third-degree 691 Difference equations 618 Trigonometrical curves 696 Characteristic equation 620 Homogeneous equation 619 Data handling 1045 Second-order, homogeneous 621 Arrangement of data 1046 Solving 619 Class boundaries 1051 Difference of two numbers 7 Class interval 1051 Differential equations Frequency distribution 1047 First-order 968 Grouped data 1048 Formation 970 Grouping with continuous data 1049 Solution 972 Histograms 1053 Differentials 322 Relative frequency 1050 Differentiation 315, 544 Rounding off data 1050 Function of a function 546 Tally diagram 1047 Implicit functions 555 De Moivre’s theorem 422 Inverse hyperbolic functions 565
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1148 Index
Differentiation (cont.) Defining 104 Inverse trigonometric functions 563 Formula 105 Logarithmic differentiation 552 Fourth-order polynomials 187 Maximum and minimum values 570 Graphs of 129 Partial differentiation 731 Homogeneous equation 619 Point of inflexion 571 Identity 104 Products 549 Linear equations 163 Quotients 550 Logarithmic 78 Standard derivatives 545 Newton–Raphson method 344 Stationary points 571 Polynomial equations 113, 177 Parametric equations 557 Simultaneous equations 165 Differentiation of polynomials 326 Solution of simple equations 163 Direction cosines 532 Equivalent fractions 20 Direction ratios 540 Estimating 12 Discontinuity 135 Evaluating Discrete variable 1046 Expressions 103 Distribution curves 1066 Hyperbolic functions 445 Frequency curves 1067 Independent variables 106 Frequency polygons 1066 Third-order determinant 467 Normal distribution curve 1067 Evaluation by nesting 114 Distributivity 11, 67 Evaluation of areas 960 Divergence 629 Evaluation of polynomials 114 Division 8 Evaluation of volumes 962 Algebraic expressions 82 Evaluation process 111 Algebraic fractions 86 Even functions 307 Complex numbers 394 Events 1077 Fractions 21 Events and probabilities 1079 By integer powers of 10 37 A priori 1080 Subtraction of powers 34 Assigning probabilities 1080 Division of integers 27 Certainty 1079 Domain 271 Impossibility 1079 Double integrals 947 Probability 1079 Double suffix notation 491 Probability distribution 1080 Drawing a graph 131 Statistical regularity 1080 Duodecimal system 46 Excel 137 Expanding brackets 69 e 217 Expectation 1092 Eigenvalues 509 Explicit function 555 Eigenvectors 509, 511 Exponent 38 Ellipse 693 Exponential curves 695 Equal complex numbers 396 Exponential form of a complex number 406 Equal matrices 492 Exponential functions 287, 303 Equal roots of characteristic equation 623 Exponential number e 217 Equal vectors 521 Expressions Equation of a straight line 584 Algebraic 65 Equations 104 Evaluating 103 Assigning 105 Polynomial 113 Conditional 104 Consistency of a set 478 Factor theorem 116 Cubic equations 184 Factorials 197
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index 1149
Factorization 68 Phase difference 295 Algebraic expressions 88 Processing numbers 269 Fourth-order polynomials 119 Range 271 Perfect square 94 Sine function 290 Quadratic expressions 91, 93 Tangent function 291 Test for simple factors 94 Trigonometric 287 Factors 15 Functions as rules 270 Fibonacci sequence 615 Functions of a linear function of x 772 Fitting a straight-line graph 712 Integration 360 Formula 105 Functions with integer input 606 Fourth-order polynomials 187 Fundamental theorem of arithmetic 16 Fractional powers 36 Fundamental trigonometric identity 258 Fractions 18 Addition 22 Gaussian elimination 507 Algebraic 85 Generating new Laplace transforms 1036 Calculator 23 Geometric sequence 609 Cancelling 20 Common ratio 609 As decimals 29 Gradient determined algebraically 322 Division 21 Gradient of a curve 320 Equivalent 20 Gradient of a straight line 317 Lowest terms 20 Gradient of a tangent 320 Reciprocal 21 Gradients 317 Subtraction 22 Gradients of perpendicular lines 585 Free vector 522 Graphical addition of complex numbers 400 Frequency curves 1067 Graphs 127 Frequency distribution 1047 Absolute values 149 Frequency polygons 1066 Drawing 131 Full angle 245 Hyperbolic functions 440 Function of a function 280, 546 Inverse functions 274 Functions 267 Sequences 606 Amplitude 294 Grouped data 1048 Arithmetic operations 272 Grouping with continuous data 1049 Co-domain 271 Composition 280 Half equilateral triangle 255 Cosine function 290 Harmonic sequence 610 Domain 271 Hexadecimal system 47 Even functions 307 Higher derivatives 327 Exponential 287, 303 Highest common factor (HCF) 16 Function of a function 280 Histograms 1053 Functions as rules 270 Frequency histogram 1054 Graphs of inverse functions 274 Relative frequency histogram 1055 Indicial equations 304 Homogeneous equation 619, 979 Inverse functions 272 Second-order 621 Inverse of a composition 282 Hyperbola 693 Inverse trigonometric functions 296 Hyperbolic cosine 438 Limits of functions 309 Hyperbolic curves 696 Logarithmic functions 303 Hyperbolic functions 437 Odd and even parts 307 Cosine 438 Odd functions 307 Evaluation 445 Period 292 Graphs 440
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
1150 Index
Hyperbolic functions (cont.) As a sum 370, 847 Identities 451 Sum of squares in the denominator 802 Inverse functions 446 Surface of revolution 864 Sine 438 Trigonometric functions 787 Tangent 439 Volume of revolution 852 And trigonometric functions 453 Integration as a sum 370, 847 Hyperbolic identities 451 Integration by partial fractions 782 Hyperbolic sine 438 Integration by parts 778 Hyperbolic tangent 439 Integration of polynomials 358 Hypotenuse 249 Integration by series 904 Integration of trigonometric functions 787 Identities 104 Inverse functions 272 Identities for compound angles 260 Inverse hyperbolic functions 446, 565 Imaginary number 386 Log form 448 Imaginary part of a complex number 390 Inverse of a composition 282 Implicit functions 555 Inverse of a square matrix 501 Impossibility 1079 Inverse transform 1029 Independent events 1085 Inverse trigonometric functions 296, 563 Independent variables 104 Irrational numbers 31 Indeterminate limits 633 Isosceles triangle 253 Index 33 Indicial equations 304 j 386 Inequalities 147 Powers of 389 Infinite limits 629 Infinity 627 Laplace transform of a derivative 1032 Input – process – output 111 Laplace transforms 1027 Integers 6 Definition 1028 Integrating factor 985 Generating new transforms 1036 Integration 353, 768, 794 Inverse transform 1029 Area between a curve and a line 374 Properties 1032 Areas under curves 366, 835 Table of transforms 1030, 1034, 1039 Centre of gravity of a solid of Transform of a derivative 1032 revolution 859 Transforms of higher derivatives 1037 Centroid of a plane figure 856 Laplace transforms of higher Change of variable 776 derivatives 1037 Definite integrals 838 Latent roots 509 Difference of squares in the Laws of arithmetic 10 denominator 795 Laws of powers 33 Functions of a linear function of x 360, 772 Length of a curve 860 Length of a curve 860 Like terms 68 Mean value 845 Limits 310 Numerator derivative of denominator 774 Limits of functions 309 Parametric equations 843, 862, 865 Limits of sequences 626, 628 Partial fractions 363, 782 Indeterminate 633 By parts 778 Infinity 627 Root mean square (rms) value 846 Linear equations 161 Rules of Pappus 867 Pre-simplification 169 Square roots in the denominator 804 Solution 504 Square roots in the numerator 810 Linear, constant coefficient, inhomogeneous Standard integrals 769 differential equations 1039
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index 1151
Loci problems 430 Measures of dispersion 1063 Log form of inverse hyperbolic Mean deviation 1063 functions 448 Range 1063 Logarithmic curves 695 Standard deviation 1063, 1064 Logarithmic differentiation 552 Variance 1063 Logarithmic equations 78 Median 1061 Logarithmic functions 303 With grouped data 1061 Logarithms 72 Method of least squares 712 Bases 10 and e 76 Using a spreadsheet 714 Change of base 77 Minimum values 570 Natural 76 Mode 1059 Lowest common multiple (LCM) 16 With grouped data 1059 Lowest terms 20 Modulus 148 Moments of inertia 874 Mantissa 38 Circular disc 883 Matrix 489 Parallel axes theorem 881 Addition and subtraction 492 Perpendicular axes theorem 884 Augmented 507 Radius of gyration 877 Cofactor 500 Rectangular plate 878 Diagonal 497 Rod 877 Double suffix notation 491 Standard results 885 Eigenvalues 509 Multiple integrals 943 Eigenvectors 509, 511 Applications 951 Equal 492 Areas 960 Gaussian elimination 507 Double integrals 947 Inverse 501 Triple integrals 949 Multiplication 493 Volumes 962 Notation 491 Multiplication 8 Null 498 Addition of powers 33 Order 490 Algebraic expressions 81 Scalar multiplication 493 Algebraic fractions 86 Single element 491 Complex numbers 392 Singular 499 Fractions 19 Skew-symmetric 497 By integer powers of 10 37 Square 497 Matrices 493 Symmetric 497 Powers 35 Transpose 496 Scalar 493 Unit 497 Mutually exclusive 1077 Maximum values 570 Mean 1055 Natural logarithms 76 Mean deviation 1063 Natural numbers 6 Mean values 844 Negative numbers 6 Measures of central tendency 1055 Negative powers 35 Coding for a grouped distribution 1058 Nested brackets 69 Coding for calculating the mean 1057 Nesting 114 Decoding 1058 Newton–Raphson iteration 344 Mean 1055 Non-mutually exclusive events 1083 Median 1061; with grouped data 1061 Normal distribution curve 1067, 1106 Mode 1059; with grouped data 1059 Normal to a curve 587 Measures of correlation 719 Parametric equations 590
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
1152 Index
Null matrix 498 Points on a line 6 Number systems 44 Poisson mean 1101 Binary 45 Poisson probability distribution 1101, 1102 Change of base 52 Poisson standard deviation 1101 Denary 44 Poisson variance 1101 Duodecimal 46 Polar coordinate systems 921 Hexadecimal 47 Applications 929 Octal 45 Polar curves 924 Numerals 6 Standard polar curves 926 Numerator 18 Polar curves 924 Polar form calculations 413 Polar form of a complex number 401 Obtuse angle 245 Polynomial equations 113, 177 Octal system 45 Quadratic equations 179 Odd and even parts 307 Polynomials 113 Odd functions 307 Differentiation 326 Of 19 Factor theorem 116 Or 1082 Integration 358 Order 6 Remainder theorem 115 Ordered pairs of numbers 130 Position vector 522 Outcome tree 1078 Power unity 33 Power zero 34 Parabola 690 Powers 33, 72 Parallel axes theorem 881 On a calculator 36 Parameter 557 Fractional 36 Parametric equations 557, 843, 862, 865 Of j 389 Partial derivative 732 Negative 35 Second derivative 738 Roots 36 Partial differentiation 731, 753 Surds 37 Change of variables 763 Precedence rules 8, 13, 38, 68 Rate-of-change problems 755 Preferred standard form 40 Small increments 744 Prescription 606 Partial fractions 223 Pressure at a point in a fluid 892 Denominators with quadratic Prime factorization 15 factors 232 Prime numbers 15 Denominators with repeated factors 234 Principal root of a complex number 425 Integration 363 Probabilities of combined events 1082 Particular integral 1013 And 1084 Particular solution 1020 Dependent events 1084 Pascal’s triangle 205 Independent events 1085 Pearson product-moment correlation Non-mutually exclusive events 1083 coefficient 719 Or 1082 Percentages 18, 24 Probability trees 1085 Perfect square 94 Probability 1076, 1079 Period 292 Combining events 1079 Permutations 200 Events 1077 Perpendicular axes theorem 884 Mutually exclusive 1077 Phase difference 295 Outcome tree 1078 Place value 6 Random experiments 1077 Points of inflexion 571, 574 Sequences of random experiments 1078
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index 1153
Probability distributions 1080, 1090 Remainder theorem 115 Bernoulli trials 1094 Right angle 245 Binomial expectation 1099 Right-hand rule 530 Binomial probability distribution 1095, 1102 Root mean square (rms) value 846 Binomial standard deviation 1099 Roots 36 Expectation 1092 Roots of a complex number 421 Normal distribution curve 1106 Rotation 289 Poisson mean 1101 Rounding 12, 27 Poisson probability distribution 1101, 1102 Rounding off data 1050 Poisson standard deviation 1101 Rules for manipulating sums 215 Poisson variance 1101 Rules of algebra 67 Random variables 1091 Rules of derivatives 330 Standard deviation 1093 Rules of indices 72 Standard normal curve 1106 Rules of limits 310, 629 Variance 1093 Rules of logarithms 75 Probability trees 1085 Rules of Pappus 867 Processing numbers 269 Product of a square matrix and its inverse 504 Scalar multiplication 493 Product of simple factors 89 Scalar product of vectors 533 Properties of determinants 481 Scalar quantity 520 Properties of Laplace transforms 1032 Secant 251 Pythagoras’ theorem 252 Second-degree curves 690 Second derivatives 327 Quadratic equations 179 Second moment of area 886 Solution 386 Composite figures 892 Solution by completing the square 181 Rectangle 889 Solution by factors 179 Triangle 891 Solution by formula 183 Second-order differential equations 1004 Quadratic expressions 91 Auxiliary equation 1006 Factorization 91 Complementary function 1013 Quotient 8 Homogeneous equations 1005 Radians 245, 246 Inhomogeneous equations 1013 Radius of curvature 594 Particular integral 1013 Radius of gyration 877 Particular solution 1020 Random experiments 1077 Second partial derivative 738 Random variables 1091 Separating variables 973 Range 271, 1063 Sequences 605 Rate-of-change problems 755 Arithmetic sequence 608 Rational numbers 18, 31 Convergence 628 Ratios 18, 24 Divergence 629 Real numbers 31, 386 Fibonacci 615 Real part of a complex number 390 Geometric sequence 609 Reciprocal 21 Graphs 606 Reciprocal trigonometric ratios 251 Harmonic sequence 610 Recurrence relation 618 Indeterminate limits 633 Recursive prescriptions 611 Infinite limits 629 Recursive process 611 Limits 626 Reduction formulas 821 Prescription 606 Wallis’s formula 830 Recursive prescriptions 611 Relative frequency 1050 Rules of limits 629
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
1154 Index
Sequences of random experiments 1078 Square root 36 Sigma notation 211 Negative number 387 Significant figures (sig fig) 27, 28 Standard curves 689 Similar terms 68 Standard derivatives 330, 545 Similar triangles 247 Standard deviation 1063, 1064, 1093 Simpson’s rule 908 Standard form 38 Proof 916 Preferred standard form 40 Simultaneous equations 460 Standard integrals 355, 769 Three unknowns 470 Standard normal curve 1069, 1106 Simultaneous linear equations 165, 167 Table of values 1108 Solution by equating coefficients 166 Standard polar curves 926 Solution by substitution 165 Stationary points 571, 704 Sine 249 Statistical regularity 1080 Sine function 290 Statistics 1045 Single element matrix 491 Distribution curves 1066 Singular matrix 499 Measures of central tendency 1055 Skew-symmetric matrix 497 Measures of dispersion 1063 Small increments 744 Standardized normal curve 1069 Solution of a set of linear equations 504 Straight angle 245 Gaussian elimination 507 Straight line 689 Solution of first-order differential Straight-line law 707 equations 972 Subtraction 7 Bernoulli’s equation 993 Algebraic fractions 85 Direct integration 973 Complex numbers 391 Homogeneous equations 979 Fractions 22 Integrating factor 985 Matrices 492 Separating the variables 973 Powers 34 Spearman’s rank correlation Sum of first n natural numbers 215 coefficient 723 Sum of two numbers 7 Special matrices 497 Sums 215 Special triangles 253 Surds 37 Spreadsheet 137, 714 Surface of revolution 864 Absolute values 149 Symmetric matrix 497 Active cell 138 Symmetry 702 Cartesian graph 139 System 111 Clearing entries 139 Columns 138 Table of Laplace transforms 1039 Cursor 138 Tally diagram 1047 Displays 141 Tangent 249 Formulas 139 Tangent to a curve 587 Inequalities 147, 150 Parametric equations 590 Interaction 154 Tangent function 291 Mouse pointer 138 Tangents, normals and curvature 583 Number entry 138 Terms 68 Rows 138 Test for simple factors 94 Text entry 138 Third-degree curves 691 Square matrix 497 Total thrust 892 Adjoint 500 Trailing zeros 29 Determinant 499 Transpose of a matrix 496 Inverse 501 Transposing variables 106, 107
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Index 1155
Transposition of formulas 107 Real number 31 Triangles 247 Whole numbers 6 Half equilateral 255 Isosceles 253 Unending decimals 30 Special triangles 253 Unit matrix 497 Trigonometric equations 297 Unit vectors 528 Trigonometric formulas 262 Trigonometric functions 287 Variables 66 And hyperbolic functions 453 Variance 1063, 1093 Rotation 289 Vector product of vectors 535 Trigonometric identities 258 Vector quantity 520 Trigonometric ratios 249 Vector representation 521 Double angles 262 Vectors 519 Products of ratios 262 Addition 522 Sums and differences of angles 262 Angle between 537 Sums and differences of ratios 262 Components 524 Trigonometrical curves 696 Direction cosines 532 Trigonometry 243 Direction ratios 540 Identities for compound angles 260 Equal 521 Triple integrals 949 Representation 521 Types of number 6 Scalar product 533 Complex 385 In space 530 Decimal 27 Types 522 Fractions 18 In terms of unit vectors 528 Imaginary 386 Vector product 535 Integers 6 Volumes of revolution 852 Irrational numbers 31 Natural numbers 6 Negative numbers 6 Wallis’s formula 830 Rational numbers 18, 31 Whole numbers 6
Copyrighted material - ISBN 9781137031204 Copyrighted material - ISBN 9781137031204
Copyrighted material - ISBN 9781137031204