engineering math reference

7/29/2019 Engineering Math Reference

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Engineering Mathematics

Dr Colin Turner

October 15, 2009


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i

Copyright Notice

The contents of this document are protected by a Creative CommonsLicense, that allows copying and modification with attribution, but not

commercial reuse. Please contact me for more details.

http://creativecommons.org/licenses/by-nc/2.0/uk/

Queries

Should you have any queries about these notes, you should approach yourlecturer or your tutor as soon as possible. Dont be afraid to ask questions, itis possible you may have found an error, and if you have not, your questionswill help your lecturer / tutor understand the problems you are experiencing.As mathematics is cumulative, it will be very hard to continue the modulewith outstanding problems from the start, a bit of work at this point willmake the rest much easier going.

Practice

Mathematics requires practice. No matter how simple a procedure may

look when demonstrated in a lecture or a tutorial you can have no idea howwell you can perform it until you try. As there is very little opportunityfor practice in the university environment it is vitally important that youattempt the questions provided in the tutorial, preferably before attendingthe relevant tutorial class. Your time with your tutor will be best spent whenyou arrive at the class with a list of problems you are unable to tackle, themore specific the better. If you find the questions too hard before the tutorial,do not become discouraged, the mere act of thinking about the problem willhave a positive affect on your understanding of the problem once explainedto you in the tutorial.

Contact Details

My contact details are as follows

Name Dr Colin TurnerRoom 5F10Phone 68084 (+44-28-9036-8084 externally)Email [email protected] http://newton.engj.ulst.ac.uk/crt/
http://creativecommons.org/licenses/by-nc/2.0/uk/http://creativecommons.org/licenses/by-nc/2.0/uk/


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Contents

1 Preliminaries 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 The law of signs . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Order of precedence . . . . . . . . . . . . . . . . . . . . 3

1.4 Decimal Places & Significant Figures . . . . . . . . . . . . . . 31.4.1 Decimal Places . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Significant Figures . . . . . . . . . . . . . . . . . . . . 4

1.5 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 Standard prefixes . . . . . . . . . . . . . . . . . . . . . 5

2 Number Systems 82.1 Natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Basic Algebra 113.1 Rearranging Equations . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Order of Rearranging . . . . . . . . . . . . . . . . . . . 123.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Function Notation . . . . . . . . . . . . . . . . . . . . . . . . 15


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3.3 Expansion of Brackets . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Brackets upon Brackets . . . . . . . . . . . . . . . . . . 183.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Laws of Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5.1 Example proofs . . . . . . . . . . . . . . . . . . . . . 223.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Laws of Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . 243.7.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 243.7.2 Graphical interpretation . . . . . . . . . . . . . . . . . 253.7.3 Factorization . . . . . . . . . . . . . . . . . . . . . . . 263.7.4 Quadratic solution formula . . . . . . . . . . . . . . . . 273.7.5 The discriminant . . . . . . . . . . . . . . . . . . . . . 283.7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 283.7.7 Special cases . . . . . . . . . . . . . . . . . . . . . . . . 29

3.8 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.8.1 Modulus or absolute value . . . . . . . . . . . . . . . . 30

3.8.2 Sigma notation . . . . . . . . . . . . . . . . . . . . . . 313.8.3 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . 323.9 Exponential and Logarithmic functions . . . . . . . . . . . . . 32

3.9.1 Exponential functions . . . . . . . . . . . . . . . . . . 323.9.2 Logarithmic functions . . . . . . . . . . . . . . . . . . 333.9.3 Logarithms to solve equations . . . . . . . . . . . . . . 343.9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 353.9.5 Anti-logging . . . . . . . . . . . . . . . . . . . . . . . . 363.9.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.10 Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . . 383.10.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.10.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 403.10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 403.10.4 High values ofn . . . . . . . . . . . . . . . . . . . . . . 41

3.11 Arithmetic Progressions . . . . . . . . . . . . . . . . . . . . . 423.11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 423.11.2 Sum of an arithmetic progression . . . . . . . . . . . . 423.11.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.12 Geometric Progressions . . . . . . . . . . . . . . . . . . . . . . 45


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3.12.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.12.2 Sum of a geometric progression . . . . . . . . . . . . . 463.12.3 Sum to infinity . . . . . . . . . . . . . . . . . . . . . . 463.12.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 473.12.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Trigonometry 494.1 Right-angled triangles . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Labelling . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Pythagoras Theorem . . . . . . . . . . . . . . . . . . . 504.1.3 Basic trigonometric functions . . . . . . . . . . . . . . 50

4.1.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 514.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Table of values . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . 534.5 Multiple Solutions . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5.1 CAST diagram . . . . . . . . . . . . . . . . . . . . . . 554.5.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 564.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Scalene triangles . . . . . . . . . . . . . . . . . . . . . . . . . 584.6.1 Labelling . . . . . . . . . . . . . . . . . . . . . . . . . 584.6.2 Scalene trigonmetry . . . . . . . . . . . . . . . . . . . . 584.6.3 Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . 584.6.4 Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . 594.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7 Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 614.7.1 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . 614.7.2 Length of Arc . . . . . . . . . . . . . . . . . . . . . . . 614.7.3 Area of Sector . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.8.1 Basic identities . . . . . . . . . . . . . . . . . . . . . . 644.8.2 Compound angle identities . . . . . . . . . . . . . . . . 644.8.3 Double angle identities . . . . . . . . . . . . . . . . . . 64

4.9 Trigonmetric equations . . . . . . . . . . . . . . . . . . . . . . 654.9.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 654.9.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 654.9.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 66


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5 Complex Numbers 68

5.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1.1 Imaginary and Complex Numbers . . . . . . . . . . . . 69

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Argand Diagram Representation . . . . . . . . . . . . . . . . . 705.4 Algebra of Complex Numbers . . . . . . . . . . . . . . . . . . 70

5.4.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4.2 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 725.4.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . 725.4.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5.1 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5.2 Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . 755.5.3 Real part . . . . . . . . . . . . . . . . . . . . . . . . . 765.5.4 Imaginary part . . . . . . . . . . . . . . . . . . . . . . 76

5.6 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 765.6.1 Cartesian form . . . . . . . . . . . . . . . . . . . . . . 765.6.2 Polar form . . . . . . . . . . . . . . . . . . . . . . . . . 775.6.3 Exponential form . . . . . . . . . . . . . . . . . . . . . 785.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 805.7 De Moivres Theorem . . . . . . . . . . . . . . . . . . . . . . . 805.7.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 825.7.2 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . 825.7.3 Roots of other numbers . . . . . . . . . . . . . . . . . . 83

5.8 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . 84

6 Vectors & Matrices 856.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1.1 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1.2 Unit Vector . . . . . . . . . . . . . . . . . . . . . . . . 866.1.3 Cartesian unit vectors . . . . . . . . . . . . . . . . . . 866.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 876.1.5 Signs of vectors . . . . . . . . . . . . . . . . . . . . . . 876.1.6 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 886.1.7 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 886.1.8 Zero vector . . . . . . . . . . . . . . . . . . . . . . . . 886.1.9 Scalar Product . . . . . . . . . . . . . . . . . . . . . . 896.1.10 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 90


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6.1.11 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.12 Vector Product . . . . . . . . . . . . . . . . . . . . . . 916.1.13 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.1 Square matrices . . . . . . . . . . . . . . . . . . . . . . 936.2.2 Row and Column vectors . . . . . . . . . . . . . . . . . 946.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.4 Zero and Identity . . . . . . . . . . . . . . . . . . . . . 94

6.3 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 956.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.3 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 956.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 956.3.5 Multiplication by a scalar . . . . . . . . . . . . . . . . 966.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.7 Domino Rule . . . . . . . . . . . . . . . . . . . . . . . 966.3.8 Multiplication . . . . . . . . . . . . . . . . . . . . . . . 976.3.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 Determinant of a matrix . . . . . . . . . . . . . . . . . . . . . 996.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4.2 Sign rule for matrices . . . . . . . . . . . . . . . . . . . 996.4.3 Order 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1006.4.5 Order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5 Inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . 1016.5.1 Order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1026.5.3 Other orders . . . . . . . . . . . . . . . . . . . . . . . . 1036.5.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.6 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . 1046.6.3 Mixed . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.7 Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . 1056.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.7.3 Row reduction . . . . . . . . . . . . . . . . . . . . . . . 108

6.8 Row Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.8.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . 1096.8.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 110


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6.9 Solving systems of equations . . . . . . . . . . . . . . . . . . . 110

6.9.1 Gaussian Elimination . . . . . . . . . . . . . . . . . . . 1116.9.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.9.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.9.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.9.5 Number of equations vs. unknowns . . . . . . . . . . . 116

6.10 Inversion by Row operations . . . . . . . . . . . . . . . . . . . 1176.11 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.11.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.11.2 Systems of equations . . . . . . . . . . . . . . . . . . . 1196.11.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.11.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.11.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1216.11.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.12 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 1226.12.1 Finding Eigenvalues . . . . . . . . . . . . . . . . . . . 1226.12.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.12.3 Finding eigenvectors . . . . . . . . . . . . . . . . . . . 1236.12.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.12.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.12.6 Other orders . . . . . . . . . . . . . . . . . . . . . . . . 1256.13 Diagonalisation . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.13.1 Powers of diagonal matrices . . . . . . . . . . . . . . . 1266.13.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.13.3 Powers of other matrices . . . . . . . . . . . . . . . . . 1276.13.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7 Graphs of Functions 1287.1 Simple graph plotting . . . . . . . . . . . . . . . . . . . . . . . 128

7.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.2 Important functions . . . . . . . . . . . . . . . . . . . . . . . . 130

7.2.1 Direct Proportion . . . . . . . . . . . . . . . . . . . . . 1307.2.2 Inverse Proportion . . . . . . . . . . . . . . . . . . . . 1317.2.3 Inverse Square Proportion . . . . . . . . . . . . . . . . 1 3 27.2.4 Exponential Functions . . . . . . . . . . . . . . . . . . 1347.2.5 Logarithmic Functions . . . . . . . . . . . . . . . . . . 134

7.3 Transformations on graphs . . . . . . . . . . . . . . . . . . . . 135


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7.3.1 Addition or Subtraction . . . . . . . . . . . . . . . . . 1 3 6

7.3.2 Multiplication or Division . . . . . . . . . . . . . . . . 1 3 67.3.3 Adding to or Subtracting from x . . . . . . . . . . . . 1367.3.4 Multiplying or Dividing x . . . . . . . . . . . . . . . . 1 3 7

7.4 Even and Odd functions . . . . . . . . . . . . . . . . . . . . . 1387.4.1 Even functions . . . . . . . . . . . . . . . . . . . . . . 1387.4.2 Odd functions . . . . . . . . . . . . . . . . . . . . . . . 1387.4.3 Combinations of functions . . . . . . . . . . . . . . . . 1 3 97.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 Coordinate geometry 142

8.1 Elementary concepts . . . . . . . . . . . . . . . . . . . . . . . 1428.1.1 Distance between two points . . . . . . . . . . . . . . . 1438.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1.4 Midpoint of two points . . . . . . . . . . . . . . . . . . 1448.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.1.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.1.7 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.1.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.1.9 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2 Equation of a straight line . . . . . . . . . . . . . . . . . . . . 1458.2.1 Meaning of equation of line . . . . . . . . . . . . . . . 1468.2.2 Finding the equation of a line . . . . . . . . . . . . . . 1468.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9 Differential Calculus 1499.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.3 Rules & Techniques . . . . . . . . . . . . . . . . . . . . . . . . 150

9.3.1 Power Rule . . . . . . . . . . . . . . . . . . . . . . . . 1509.3.2 Addition and Subtraction . . . . . . . . . . . . . . . . 1 5 19.3.3 Constants upon functions . . . . . . . . . . . . . . . . 1 5 19.3.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . 1519.3.5 Product Rule . . . . . . . . . . . . . . . . . . . . . . . 1519.3.6 Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . 1529.3.7 Trigonometric Rules . . . . . . . . . . . . . . . . . . . 1529.3.8 Exponential Rules . . . . . . . . . . . . . . . . . . . . 1529.3.9 Logarithmic Rules . . . . . . . . . . . . . . . . . . . . 152


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9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9.4.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.5 Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.6 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.6.1 Types of turning point . . . . . . . . . . . . . . . . . . 1589.6.2 Finding turning points . . . . . . . . . . . . . . . . . . 1599.6.3 Classification of turning points . . . . . . . . . . . . . . 1609.6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 162

9.6.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1639.7 Newton Rhapson . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1649.7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 165

9.8 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . 1669.8.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9.9 Small Changes . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.9.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.9.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 169

10 Integral Calculus 17210.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

10.1.1 Constant of Integration . . . . . . . . . . . . . . . . . . 17210.2 Rules & Techniques . . . . . . . . . . . . . . . . . . . . . . . . 173

10.2.1 Power Rule . . . . . . . . . . . . . . . . . . . . . . . . 17310.2.2 Addition & Subtraction . . . . . . . . . . . . . . . . . 1 7 310.2.3 Multiplication by a constant . . . . . . . . . . . . . . . 17410.2.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . 17410.2.5 Limited Chain Rule . . . . . . . . . . . . . . . . . . . . 17410.2.6 Logarithm rule . . . . . . . . . . . . . . . . . . . . . . 175

10.2.7 Partial Fractions . . . . . . . . . . . . . . . . . . . . . 17610.2.8 Integration by Parts . . . . . . . . . . . . . . . . . . . 17710.2.9 Other rules . . . . . . . . . . . . . . . . . . . . . . . . 178

10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17910.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 17910.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 18010.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 184


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10.4 Definite Integration . . . . . . . . . . . . . . . . . . . . . . . . 185

10.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4.3 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.4.6 Volumes of Revolution . . . . . . . . . . . . . . . . . . 18710.4.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18710.4.8 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . 18810.4.9 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18810.4.10Example . . . . . . . . . . . . . . . . . . . . . . . . . . 188

10.4.11RMS Values . . . . . . . . . . . . . . . . . . . . . . . . 18910.4.12Example . . . . . . . . . . . . . . . . . . . . . . . . . . 18910.4.13Example . . . . . . . . . . . . . . . . . . . . . . . . . . 190

10.5 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 19110.5.1 Simpsons rule . . . . . . . . . . . . . . . . . . . . . . . 19210.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 192

11 Power Series 19411.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

11.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 194

11.2 Maclaurins Expansion . . . . . . . . . . . . . . . . . . . . . . 19511.2.1 Odd and Even . . . . . . . . . . . . . . . . . . . . . . . 19511.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 19611.2.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 19611.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 19611.2.5 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 19711.2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 197

11.3 Taylors Expansion . . . . . . . . . . . . . . . . . . . . . . . . 19711.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 19811.3.2 Identification of Turning Points . . . . . . . . . . . . . 198

12 Differential Equations 20012.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20012.2 Exact D.E.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 20112.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 20112.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 202

12.3 Variables separable D.E.s . . . . . . . . . . . . . . . . . . . . . 20312.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 203


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12.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 204

12.4 First order linear D.E.s . . . . . . . . . . . . . . . . . . . . . . 20612.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 20712.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 208

12.5 Second order D.E.s . . . . . . . . . . . . . . . . . . . . . . . . 20912.5.1 Homogenous D.E. with constant coefficients . . . . . . 20912.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 21112.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 21112.5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 21212.5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 21212.5.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 213

13 Differentiation in several variables 21413.1 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . 214

13.1.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 21413.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 21513.1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 21613.1.4 Higher Derivatives . . . . . . . . . . . . . . . . . . . . 21613.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 217

13.2 Taylors Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21813.3 Stationary Points . . . . . . . . . . . . . . . . . . . . . . . . . 218

13.3.1 Types of points . . . . . . . . . . . . . . . . . . . . . . 21813.3.2 Finding points . . . . . . . . . . . . . . . . . . . . . . . 21913.3.3 Classifying points . . . . . . . . . . . . . . . . . . . . . 21913.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 22013.3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 221

13.4 Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . 22213.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 223

13.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 22413.6 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

13.6.1 Differential . . . . . . . . . . . . . . . . . . . . . . . . 226

13.7 Parametric functions . . . . . . . . . . . . . . . . . . . . . . . 22613.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 22713.8 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

14 Integration in several variables 22914.1 Double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 229

14.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 22914.2 Change of order . . . . . . . . . . . . . . . . . . . . . . . . . . 23014.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


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14.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 232

14.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 23214.4 Triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 233

14.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 23314.5 Change of variable . . . . . . . . . . . . . . . . . . . . . . . . 234

14.5.1 Polar coordinates . . . . . . . . . . . . . . . . . . . . . 23514.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 23514.5.3 Cylindrical Polar Coordinates . . . . . . . . . . . . . . 23614.5.4 Spherical Polar Coordinates . . . . . . . . . . . . . . . 236

15 Fourier Series 237

15.1 Periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . 23715.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 23715.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 238

15.2 Sets of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 23915.2.1 Orthogonal functions . . . . . . . . . . . . . . . . . . . 23915.2.2 Orthonormal functions . . . . . . . . . . . . . . . . . . 23915.2.3 Norm of a function . . . . . . . . . . . . . . . . . . . . 239

15.3 Fourier concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 23915.3.1 Fourier coefficents . . . . . . . . . . . . . . . . . . . . . 23915.3.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . 240

15.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 24015.4 Important functions . . . . . . . . . . . . . . . . . . . . . . . . 240

15.4.1 Trigonometric system . . . . . . . . . . . . . . . . . . . 24115.4.2 Exponential system . . . . . . . . . . . . . . . . . . . . 242

15.5 Trigonometric expansions . . . . . . . . . . . . . . . . . . . . . 24215.5.1 Even functions . . . . . . . . . . . . . . . . . . . . . . 24215.5.2 Odd functions . . . . . . . . . . . . . . . . . . . . . . . 24315.5.3 Other Ranges . . . . . . . . . . . . . . . . . . . . . . . 243

15.6 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24315.6.1 Odd and Even Harmonics . . . . . . . . . . . . . . . . 2 4 4

15.6.2 Trigonometric system . . . . . . . . . . . . . . . . . . . 24415.6.3 Exponential system . . . . . . . . . . . . . . . . . . . . 24515.6.4 Percentage harmonic . . . . . . . . . . . . . . . . . . . 245

15.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24515.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 245

15.8 Exponential Series . . . . . . . . . . . . . . . . . . . . . . . . 246


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16 Laplace transforms 248

16.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24816.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 24816.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 24916.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 24916.1.4 Inverse Transform . . . . . . . . . . . . . . . . . . . . . 25016.1.5 Elementary properties . . . . . . . . . . . . . . . . . . 25016.1.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 250

16.2 Important Transforms . . . . . . . . . . . . . . . . . . . . . . 25116.2.1 First shifting property . . . . . . . . . . . . . . . . . . 25116.2.2 Further Laplace transforms . . . . . . . . . . . . . . . 253

16.3 Transforming derivatives . . . . . . . . . . . . . . . . . . . . . 25416.3.1 First derivative . . . . . . . . . . . . . . . . . . . . . . 25416.3.2 Second derivative . . . . . . . . . . . . . . . . . . . . . 25416.3.3 Higher derivatives . . . . . . . . . . . . . . . . . . . . . 254

16.4 Transforming integrals . . . . . . . . . . . . . . . . . . . . . . 25416.5 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 255

16.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 25516.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 25616.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 25816.5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 260

16.5.5 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 26116.5.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 26116.5.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 262

16.6 Other theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 26316.6.1 Change of Scale . . . . . . . . . . . . . . . . . . . . . . 26316.6.2 Derivative of the transform . . . . . . . . . . . . . . . . 2 6 316.6.3 Convolution Theorem . . . . . . . . . . . . . . . . . . . 26416.6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 26416.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 26416.6.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 265

16.7 Heaviside unit step function . . . . . . . . . . . . . . . . . . . 26516.7.1 Laplace transform ofu(t c) . . . . . . . . . . . . . . 26816.7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 26816.7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 27016.7.4 Delayed functions . . . . . . . . . . . . . . . . . . . . . 27016.7.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 271

16.8 The Dirac Delta . . . . . . . . . . . . . . . . . . . . . . . . . . 27216.8.1 Delayed impulse . . . . . . . . . . . . . . . . . . . . . . 27316.8.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 274

16.9 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 274


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16.9.1 Impulse Response . . . . . . . . . . . . . . . . . . . . . 276

16.9.2 Initial value theorem . . . . . . . . . . . . . . . . . . . 27716.9.3 Final value theorem . . . . . . . . . . . . . . . . . . . . 277

17 Z-transform 27917.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27917.2 Important Z-transforms . . . . . . . . . . . . . . . . . . . . . 281

17.2.1 Unit step function . . . . . . . . . . . . . . . . . . . . 28117.2.2 Linear function . . . . . . . . . . . . . . . . . . . . . . 28217.2.3 Exponential function . . . . . . . . . . . . . . . . . . . 28317.2.4 Elementary properties . . . . . . . . . . . . . . . . . . 283

17.2.5 Real translation theorem . . . . . . . . . . . . . . . . . 283

18 Statistics 28518.1 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 285

18.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 28518.2 Populations and Samples . . . . . . . . . . . . . . . . . . . . . 287

18.2.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 28718.3 Parameters and Statistics . . . . . . . . . . . . . . . . . . . . 28818.4 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28818.5 Measures of Location . . . . . . . . . . . . . . . . . . . . . . . 288

18.5.1 Arithmetic Mean . . . . . . . . . . . . . . . . . . . . . 28918.5.2 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28918.5.3 Median . . . . . . . . . . . . . . . . . . . . . . . . . . 28918.5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 28918.5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 290

18.6 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . 29018.6.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . 29118.6.2 Standard deviation . . . . . . . . . . . . . . . . . . . . 29118.6.3 Inter-quartile range . . . . . . . . . . . . . . . . . . . . 292

18.7 Frequency Distributions . . . . . . . . . . . . . . . . . . . . . 292

18.7.1 Class intervals . . . . . . . . . . . . . . . . . . . . . . . 29218.8 Cumulative frequency . . . . . . . . . . . . . . . . . . . . . . . 293

18.8.1 Calculating the median . . . . . . . . . . . . . . . . . . 29318.8.2 Calculating quartiles . . . . . . . . . . . . . . . . . . . 29418.8.3 Calculating other ranges . . . . . . . . . . . . . . . . . 2 9 4

18.9 Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29418.10Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

18.10.1Linear regression . . . . . . . . . . . . . . . . . . . . . 29518.10.2Correlation coefficient . . . . . . . . . . . . . . . . . . 295


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19 Probability 297

19.1 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29719.1.1 Probability of an Event . . . . . . . . . . . . . . . . . . 29719.1.2 Exhaustive lists . . . . . . . . . . . . . . . . . . . . . . 297

19.2 Multiple Events . . . . . . . . . . . . . . . . . . . . . . . . . . 29819.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 29819.2.2 Relations between events . . . . . . . . . . . . . . . . . 2 9 8

19.3 Probability Laws . . . . . . . . . . . . . . . . . . . . . . . . . 29919.3.1 A or B (mutually exclusive events) . . . . . . . . . . . 29919.3.2 not A . . . . . . . . . . . . . . . . . . . . . . . . . . . 29919.3.3 1 event ofN . . . . . . . . . . . . . . . . . . . . . . . . 300

19.3.4 n events ofN . . . . . . . . . . . . . . . . . . . . . . . 30019.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 30019.3.6 A and B (independent events) . . . . . . . . . . . . . . 30019.3.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 30119.3.8 A or B or C or ... . . . . . . . . . . . . . . . . . . . . . 30119.3.9 A and B and C and ... . . . . . . . . . . . . . . . . . . 30119.3.10Example . . . . . . . . . . . . . . . . . . . . . . . . . . 30219.3.11 A or B revisited . . . . . . . . . . . . . . . . . . . . . . 30219.3.12Example . . . . . . . . . . . . . . . . . . . . . . . . . . 30219.3.13 A and B revisited . . . . . . . . . . . . . . . . . . . . . 303

19.3.14Conditional probability . . . . . . . . . . . . . . . . . . 30319.3.15Example . . . . . . . . . . . . . . . . . . . . . . . . . . 30419.3.16Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . 304

19.4 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . 30419.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 30419.4.2 Expected Value . . . . . . . . . . . . . . . . . . . . . . 30519.4.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 30619.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 307

19.5 Continuous Random Variables . . . . . . . . . . . . . . . . . . 30819.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 30919.5.2 Probability Density Function . . . . . . . . . . . . . . 309

20 The Normal Distribution 31020.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31020.2 Standard normal distribution . . . . . . . . . . . . . . . . . . 311

20.2.1 Transforming variables . . . . . . . . . . . . . . . . . . 31120.2.2 Calculation of areas . . . . . . . . . . . . . . . . . . . . 31220.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 31220.2.4 Confidence limits . . . . . . . . . . . . . . . . . . . . . 313


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CONTENTS xvi

20.2.5 Sampling distribution . . . . . . . . . . . . . . . . . . . 313

20.3 The central limit theorem . . . . . . . . . . . . . . . . . . . . 31420.4 Finding the Population mean . . . . . . . . . . . . . . . . . . 31520.5 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . 315

20.5.1 Two tailed tests . . . . . . . . . . . . . . . . . . . . . . 31620.6 Difference of two normal distributions . . . . . . . . . . . . . . 317

A Statistical Tables 318

B Greek Alphabet 321


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List of Tables

1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The law of signs . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . 31.4 SI prefixes for large numbers . . . . . . . . . . . . . . . . . . . 61.5 SI prefixes for small numbers . . . . . . . . . . . . . . . . . . . 7

3.1 The laws of indices . . . . . . . . . . . . . . . . . . . . . . . . 213.2 The laws of surds . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Examples of quadratic equations . . . . . . . . . . . . . . . . . 253.4 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Pascals Triangle . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Table of trigonometric values . . . . . . . . . . . . . . . . . . . 544.2 Conversion between degrees and radians . . . . . . . . . . . . 62

5.1 Powers ofj . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.1 Matrix algebra - Addition . . . . . . . . . . . . . . . . . . . . 1046.2 Matrix algebra - Multiplication . . . . . . . . . . . . . . . . . 1056.3 Matrix algebra - Mixed operations . . . . . . . . . . . . . . . . 1 0 5

7.1 Adding and Subtracting even and odd functions . . . . . . . . 1 3 9

7.2 Multiplying even and odd functions . . . . . . . . . . . . . . . 140

9.1 Second derivative test . . . . . . . . . . . . . . . . . . . . . . . 1609.2 First derivative turning point classification . . . . . . . . . . . 161

15.1 Symmetry in Fourier Series . . . . . . . . . . . . . . . . . . . . 244

16.1 Common Laplace transforms . . . . . . . . . . . . . . . . . . . 25216.2 Further Laplace transforms . . . . . . . . . . . . . . . . . . . . 25316.3 Examples of transfer function denominators . . . . . . . . . . 276


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LIST OF TABLES xviii

18.1 An example of class intervals . . . . . . . . . . . . . . . . . . . 293

19.1 Probabilities for total of two rolled dice . . . . . . . . . . . . . 30519.2 Calculating E(X) and var (X) for two rolled dice. . . . . . . . 308

A.1 Table of (x) (Normal Distribution) . . . . . . . . . . . . . . 319A.2 Table of2 distribution (Part I) . . . . . . . . . . . . . . . . . 320A.3 Table of2 distribution (Part II) . . . . . . . . . . . . . . . . 3 2 0


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List of Figures

2.1 The real number line . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 The quadratic equation . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Labelling right-angled triangles . . . . . . . . . . . . . . . . . 504.2 Generating the trigonometric graphs . . . . . . . . . . . . . . 554.3 The graphs of sin and cos . . . . . . . . . . . . . . . . . . . 564.4 The CAST diagram . . . . . . . . . . . . . . . . . . . . . . 574.5 Labelling a scalene triangle . . . . . . . . . . . . . . . . . . . . 594.6 Length of Arc, Area of Sector . . . . . . . . . . . . . . . . . . 63

5.1 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . 715.2 Polar representation of a complex number . . . . . . . . . . . 77

6.1 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 Vector Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1 The graph ofx2 + 2x 3 . . . . . . . . . . . . . . . . . . . . . 1297.2 The graph of 2x + 3 . . . . . . . . . . . . . . . . . . . . . . . 1307.3 The graph of 1

x. . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.4 The graph of 1x2

. . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5 The graph ofex . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.6 The graph of ln x . . . . . . . . . . . . . . . . . . . . . . . . . 1357.7 Closeup of graph of ln x . . . . . . . . . . . . . . . . . . . . . 1367.8 Graph of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.9 Graph of sin x + 1 . . . . . . . . . . . . . . . . . . . . . . . . . 1387.10 Graph of 2 sin x . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.11 Graph of sin(x + 90) . . . . . . . . . . . . . . . . . . . . . . . 1407.12 Closeup of graph of sin(2x) . . . . . . . . . . . . . . . . . . . . 141

8.1 Relationships between two points . . . . . . . . . . . . . . . . 1 4 2


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LIST OF FIGURES xx

9.1 Types of turning point . . . . . . . . . . . . . . . . . . . . . . 159

9.2 The Newton-Rhapson method . . . . . . . . . . . . . . . . . . 171

13.1 The graph off(x, y) = xy2 + 1 . . . . . . . . . . . . . . . . . . 21513.2 The graph off(x, y) = x3(sin xy + 3x + y + 3) . . . . . . . . . 2 1 613.3 A graph with four turning points . . . . . . . . . . . . . . . . 2 2 2

14.1 Double integration over the simple region R. . . . . . . . . . . 23014.2 Double integration over x then y. . . . . . . . . . . . . . . . . 2 3 114.3 Double integration over y then x. . . . . . . . . . . . . . . . . 2 3 2

16.1 An L and R curcuit. . . . . . . . . . . . . . . . . . . . . . . . 261

16.2 An L, C, and R curcuit. . . . . . . . . . . . . . . . . . . . . . 26216.3 The unit step function u(t). . . . . . . . . . . . . . . . . . . . 26616.4 The displaced unit step function u(t c). . . . . . . . . . . . . 26616.5 Building functions that are on and off when we please. . . . . 26716.6 A positive waveform built from steps. . . . . . . . . . . . . . . 26916.7 A waveform built from steps. . . . . . . . . . . . . . . . . . . 27016.8 A waveform built from delayed linear functions. . . . . . . . . 2 7 216.9 An impulse train built from Dirac deltas. . . . . . . . . . . . . 274

17.1 A continuous (analog) function . . . . . . . . . . . . . . . . . 2 7 917.2 Sampling the function . . . . . . . . . . . . . . . . . . . . . . 28017.3 The digital view . . . . . . . . . . . . . . . . . . . . . . . . . . 280

20.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . 31020.2 Close up of the Normal Distribution . . . . . . . . . . . . . . . 311


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Chapter 1

Preliminaries

1.1 Introduction

We shall start the course, by recapping many definitions and results thatmay already be well known. As mathematics is a cumulative subject, it isnecessary however, to ensure that all the basics are in place before we cango on.

We assume the reader is familiar with the elementary arithmetic of num-bers positive, negative and zero. We also assume the reader is familiar withthe decimal representation of numbers, and that they can evaluate simpleexpressions, including fractional arithmetic.

1.2 Notation

We now list some mathematical notation that we may use in the course, or

may be encountered elsewhere, this is shown in table 1.1.Another important bit of notation is . . . , which is used as a sort ofmathematicians etcetera. For example

1, 2, 3, . . . , 10 is short hand for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1, 2, 3, . . . is short hand for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc.

Its probably worth noting that in algebra when we use letters to representnumbers, then


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1.3 Arithmetic 2

= equal to= not equal to

< less than less than or equal to> greater than greater than or equal to equivalent to approximately equal to implies infinity sum of what follows product of what follows

Table 1.1: Basic notation

3a is a shorthand for 3 a a is a shorthand for 1 a a is a shorthand for 1 a

1.3 Arithmetic

Two often forgotten pieces of arithmetic are:

1.3.1 The law of signs

When we combine signs, either by multiplying two numbers, or by subtractinga negative number for example, we use table 1.2 to determine the sign ofthe outcome. Put simply, a sign reverses our direction, and so two ofthem take us back to the + direction and so on.

+ +

+

+ + +

Table 1.2: The law of signs


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1.4 Decimal Places & Significant Figures 3

1.3.2 Order of precedence

We are familiar with the fact that expressions inside brackets must be evalu-ated first, that is what the bracket signifies. However, without brackets thereis still an inherent order in which operations must be done. Consider thissimple calculation

2 + 3 4opinion is usually split as to whether the answer is 20, or 14. The rea-

son is that multiplication should be performed before addition, and so the3 4 segment should be calculated first. Be aware that not all calculatorsunderstand this, test yours with this calculuation.

Calculations should be performed in the order shown in table 1.3.

B Brackets first - they override all priority

O Order (Powers, roots)

D Division

M Multiplication

A Addition

S Subtraction

Table 1.3: Order of precedence

We note that the table provides us with a handy reminder, BODMAS.

1.4 Decimal Places & Significant Figures

Often we are required to produce answers to a specific degree of accuracy.The most well known way to do this is with the number of decimal places.

1.4.1 Decimal Places

The number of decimal places is a measure of how to truncate answers to agiven accuracy. If four decimal places are required then we look at the fifthdecimal place and beyond, if it is 5 or greater, we round up the last decimalplaces that is written, otherwise we simply leave it alone. Let us take an


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1.4 Decimal Places & Significant Figures 4

example. is a mathematical constant that continues infinitely through all

its decimal places, never repeating its pattern.

= 3.141592653589793 . . .

rounded to five decimal places we obtain

3.14159

since the next digit is 2, wheras rounding to four decimal places give

3.1416

since the next digit is 9 which is clearly bigger than 5.It is good to use enough decimal places to obtain an accurate answer, but

one must always remember the context of the answer. There is little pointin calculating that the length of a piece of metal should be 2.328745 cm if allwe will have to measure it with is a ruler accurate to 1 mm.

1.4.2 Significant Figures

Sometimes decimal places are not the most appropriate way to define accu-

racy. There is no specific number of decimal places that suit all situations.For example, if we quote the radius of the Earth in metres, then probably nonumber of decimal places are appropriate for most purposes, as the answerwill not be that accurate, and there will be so many other figures before it,they are unlikely to be significant.

An alternative often used is to specify a number of significant figures.This is essential the number of non-zero numbers that should be displayed.Suppose that we specify four significant figures. Then the speed of light inm/s is written as:

c = 2, 997, 992, 458 2, 998, 000, 000 m/swhich can be written more simply again in standard form (see below). Theissue here is that the other figures are less likely to have any real impact onthe answer of a problem. Similarly the standard atomic mass of Uranium is

238.02891 g/mol 238.0 g/mol

since we only have four significant figures, we round after the zero. Note thatwriting the zero helps indicate the precision of the answer.


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1.5 Standard Form 5

1.5 Standard Form

In science, large and small numbers are often represented by standard form.This takes the form

a.bcd 10nif we are using four significant figures. For example, we saw above that tofour significant figures

c = 2, 998, 000, 000 m/s = 2, 998 1, 000, 000 m/sor, working a bit more, we move the decimal place each time to the left,

(which divides the left hand number by ten), and multiply by another tenon the right to compensate.

= 2.998 1, 000, 000, 000 m/snow all that remains to do, is to write the number on the right as a powerof ten. We count the zeros, there are nine, and so

c 2.998 109 m/s.The same applies for small numbers. The light emitted by a Helium-Neon

Laser has a wavelength of

= 0.000, 000, 632, 8 m

but this is clearly rather unwieldy to write down. This time we move thedecimal place to the right until we get to after the first non-zero digit. Eachtime we do this we essentially multiply by 10, and so to compensate wehave to divide by ten. This can be represented by increasingly large negativevalues of the power.1

So here, we need to move the decimal place seven times to the right, andso we will multiply by 107.

= 6.328 107 m.

1.5.1 Standard prefixes

There are a number of prefixes applied to large and small numbers to allowus to write them more meaningfully. You will have met many of them before.The prefixes for large numbers are shown in table 1.5.1.

1We will have to wait a while, until 3.5 to see exactly why this is.


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1.5 Standard Form 6

Name Prefix In English Power of Ten

deca da tens 101

hecto h hundreds 102

kilo k thousands 103

Mega M millions 106

Giga G billions 109

Tera T trillions 1012

Peta P quadrillions 1015

Exa E quintillions 1018

Table 1.4: SI prefixes for large numbers

Note that in the past there was a difference between billions as used inBritish English and American English. The English billion was one millionmillion, wheras the American billion is one thousand million. The latter haswon out now, and most references to a billion are to the American one. Also,

there is a slight disparity between normal quantities, and the bytes usedin computing. Since computing is based on binary, and therefore powers of2, a kilobyte (kB) is not 1000 bytes, but 1024 bytes.2So in computing, 1024is used rather than thousands to build up such quantities.

The prefixes for large numbers are shown in table 1.5.1.Because of these prefixes, it is normal within engineering to adapt stan-

dard scientific form to get the power of ten to be a multiple of three. Let usrevisit our wavelength example:

= 6.328 107 m.

So we would prefer to tweak the power of ten here. We could do this

= .6328 106 m = 0.6238 m,

but this is pretty ugly to have a fractional number. It would be more normalto write

632.8 109 m = 623.8 nm,

21024 = 210 and so is the closest power of two


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1.5 Standard Form 7

Name Prefix In English Power of Ten

deci d tenths 101

centi c hundredths 102

milli m thousandths 103

micro millionths 106

nano n billionths 109

pico p trillionths 1012

femto f quadrillionths 1015

atto a quintillionths 1018

Table 1.5: SI prefixes for small numbers


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Chapter 2

Number Systems

We remind ourselves of different sets of numbers that we will refer to later.

2.1 Natural numbers

The set of all numbers

1, 2, 3, . . .

is known as the set of positive integers, (or natural numbers, or wholenumbers, or counting numbers and is denoted by N.

2.2 Prime numbers

A prime number is a positive integer which has exactly two factors 1, namelyitself and one.

Thus 2 is the first prime number, and the only even prime number. Sothe prime numbers are given by

2, 3, 5, 7, 11, 13, 17, 19, . . .

1Recall that a factor of a number x is one that divides into x with no remainder.


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2.3 Integers 9

2.3 Integers

The set of all numbers given by

. . . , 3, 2, 1, 0, 1, 2, 3, . . .is known as the set of integers and is denoted by the symbol Z.

2.4 Real numbers

The collection of all numbers in ordinary arithmetic, i.e. including fractions,integers, zero, positive and negative numbers etc. is called the set of realnumbers and is denoted R. The set of real numbers can be visualised as aline, called the real number line or simply the real line. Each point on thelines represents a unique real number, and every number, including exoticexamples such as is represented by a unique point on the line.

Figure 2.1: The real number line

2.5 Rational numbers

The set of all numbers that can be written mn

where m and n are integers,

is known as the set of rational numbers and is noted by Q. (Note that ncannot by zero, as division by zero is not permitted).For example, 2

3, 165

2096, 2 = 2

1, 0 = 0

1are all rational numbers.

(Note that although division by zero is not peemitted, dividing zero byanother number is, and as no other number can fit into zero at all, the resultis zero).

It turns out that if you add, subtract, divide or multiply any two rationalnumbers together, you still get a rational number.


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2.6 Irrational Numbers 10

Theres an easy way of working out whether a given number is rational

or not. Simply expand it in its decimal form. Rational numbers always havea decimal expansion that ends, or repeats itself every so many digits.

For example

23

= 0.6666666 . . .3436.234523452345 . . .

34.68

are all rational.

2.6 Irrational Numbers

Of course, not all real numbers are rational, and in fact many numbers youwill already have met are not. These numbers are called irrational numbers.

Examples are

2,

3, and in fact

p where p is prime.When written in their decimal forms, irrationals are never ending and

non-repeating. This means that irrational numbers can never be writtendown exactly.

Although in secondary schools we often write =22

7 , suggesting that is rational, this is only a simple (and not very accurate) approximation. Infact is also irrational, as is Eulers constant e.


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Chapter 3

Basic Algebra

Algebra is perhaps the most important part of mathematics to master. Ifyou do not, you will find problems in all the areas you study, all caused bythe underlying weakness in your algebra skills. In many ways, algebra isrepresentative of mathematics in that it deals with forming an easy problemout of a difficult one.

3.1 Rearranging Equations

Imagine an equation as a pair of balanced weighing scales. What will happenif we add 2kg on both sides? The scales will remain in balance. If we multiplythe weights by three on both sides? The scales will remain in balance. Infact, even if we take the sine of both weights, the scales remain in balance.The leads to the fundamental result you must remember.

You can do anything to both sides of an equation and you willobtain an equivalent equation.

3.1.1 Example

We shall look at an extremely trivial example of this concept in use. In ourlearning of algebraic manipulation we are often told that we can take thingsacross the equals sign and change the sign. Rarely are we told why thisworks. Lets examine it.


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3.1 Rearranging Equations 12

x + 4 = 9

Well, it is simple to see what value x has in this case. However, we wishto show how rearranging works in these very simple cases. We wish to find x,and this is really saying we want to manipulate the equation into the form:

x =??

Where ?? represents the answer. Therefore, we wish to have an x on itsown, on one side of the equation, with everything else on the other side ofthe equals sign. To that end, we start to look at what is attached to x, how

it is attached, and how we should remove it. In our example, 4 is attachedto the x by the process of addition. Now, how do you get rid of a 4 that hasbeen added? Of course, the answer is to subtract it, but we must not simplydo this on one side, rather in accordance with 3.1 we must do it on bothsides of the equation to maintain its validity.

So we obtain

x + 4 4 = 9 4Now the +44 on the L.H.S. cancel, leaving zero, and this step wouldnt

be written normally. So we finally obtain

x = 9 4 = 5If you observe that it appears that the +4 crossed the equals sign to

become a 4 on the R.H.S.. However, now we know what has actuallyhappened.

3.1.2 Order of Rearranging

Of course, in most examples, more than one thing is attached to the x,

and usually by a combination of operations. It may be equally correct torearrange by removing these in any order, but some ways will almost certainlybe easier than others.

We have already noted in 1.3.2 that some operations are naturally donebefore others, and so when we see an expression such as:

3x2 4 = 8it really means


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((3(x2)) 4) = 8where the brackets serve simply to underline the order in which things aredone. The effect is somewhat similar to an onion with the x in the very centre.We could peel the onion from the outside in for the most tidy approach. Thatis, we remove things in the reverse order to the way they were attached inthe first place.

So in our simple example, the 4 is subtracted last, so remove it first,(adding 4 on both sides).

3x2

4 + 4 = 8 + 4

3x2 = 12

Now the x still has two things attached, the three, which is multipliedon, and the 2 which is a power. Powers are done before multiplication, so weremove in the reverse order again. Therefore we divide by 3 on both sides.

3

3x2 =

12

3 x2 = 4

Now we only have one thing stuck to the x, and that is the power of 2.To remove this we simply take the square root on both sides:

x =

2.

Recall that -2 squared is also 4.

3.1.3 Example

Rearrange the following expression for x.

4x + 6 = 2x 3

SolutionWe still wish to rearrange to get x =, but we must notice here that x occursin two places. We could remove the 3 and 6 on the LHS to obtain

x =2x 9

4

(try it as an exercise), but this is not very helpful, as x is now defined interms of itself, so we still dont know its value.


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Instead we first gather all the x terms together, and we do this by per-

forming the same operation on both sides. For example, we dont want the2x on the RHS, it is positive and so it present by addition. We subtract iton both sides.

4x + 6 2x = 2x 3 2x 2x + 6 = 3So we now have a simpler equation, with x only on one side. We can

proceed as before now to remove things from the x in the LHS. Subtract 6on both sides.

2x + 6

6 =

3

6

2x =

9

Finally divide by 2 on both sides.

x = 92

3.1.4 Example

Rearrange the following expression for x:

3(2x + 3)

6 = 0

Solution

In this case we find that x is encased in brackets. To get at x so we canrearrange for it we could multiply out the bracket and rearrange from there.This is left as an exercise for the reader.

Another way to deal with it is to think of the bracket as an onion withinan onion to continue the analogy we began above. Begin by taking thingsoff this bracket, rather than x directly. We add 6 both sides

3(2x + 3) 6 + 6 = 0 + 6 3(2x + 3) = 6.Now divide by 3 on both sides to obtain

3(2x + 3)

3=

6

3 2x + 3 = 2.

Note the brackets can fall off at this point naturally. Now we startwith out new onion, subtracting 3 both sides.

2x + 3 3 = 2 3 2x = 1


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3.2 Function Notation 15

and finally divide by 2 both sides

2x

2= 1

2 x = 1

2

3.1.5 Example

Rearrange the following expression for x:

3 = 10x

5.

Solution

We have a more serious problem here, namely that x is on the bottom line.We begin by removing the 5 to clarify the equation, by adding 5 on bothsides of course.

3 + 5 = 10x

5 + 5 2 = 10x

Now, theres not much attached to x, but the x is still on the bottom line.That means the x has been divided into something (the 10 in this case). Tocancel the division by x, we multiply x on both sides.

2x =10x

x 2x = 10

which simplifies our equation quite a lot. We can now divide by two onboth sides to finish.

2x

2=

10

2 x = 5

3.2 Function NotationVery often when we wish to analyse the behaviour of an expression, we makeit a function. A function can be thought of as a box, into which goes a valueand out of which comes a, usually different, value.

You will have seen functions written as formulae before, for example

A = r2


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3.2 Function Notation 16

is a function for calculating the area of a circle. A value goes in (the

radius of the circle) and a value comes out (the area of the circle).At times we may use notation such as f(x) to represent a function. For

example

f(x) = 2x 3is a very simple function. For different values we put in (x), we will get

different values out f(x). The notation f(x) simply means the value of theoutput of the function.

This notation is very useful when we want to consider specific values thatwe insert. For example, we write f(2) to mean find the output value of thefunction f, when the input value for x is 2. You can see that we have simplyreplaced the x by a 2.

In our example stated above we get

f(2) = 2(2) 3 = 1f(3) = 2(3) 3 = 9

f(0) = 2(0) 3 = 3f(w) = 2(w) 3

In the last example, we had to replace x by w, but we cant work outanything further, so we stop there.

Very often we want to be able to undo the result of our function. Forexample, we need to be able to reverse multiplication with division in orderto rearrange equations, or reverse a square with a square root.

To do this we use an inverse function. An inverse function can be thoughtof a complementary box to our original one, so that when we plug the outputof the first function into its input we get the original value. For example,with our simple f(x) above, we inserted 2 and got 1. Our inverse will haveto take 1 and give us 2.

To find the inverse function, it is usually easier to give f(x) a letter, likey. In our example we obtain

y = 2x 3Now we rearrange the equation for x, using the rules described above.

We obtain

x =y + 3

2


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3.3 Expansion of Brackets 17

this is left as an exercise for the reader.

Were pretty much done, but it is usual to label our inverse of f(x) withthe notation f1(x) and have our function in terms ofx, not y. So, we swapx and y to obtain

y =x + 3

2

and now use our inverse function formula

f1(x) =x + 3

2

Recall that with our simple example for f(x), that

f(2) = 2(2) 3 = 1.If we now feed this output into the input of the inverse, we should get

back to our starting position (2).

f1(1) =1 + 3

2= 2.

Just as before we insert the value in the brackets into x throughout theexpression for the inverse function, and you can see that indeed the inverse

function here has taken us back to the start.We will meet other examples of inverse functions throughout this module.Its not always possible to do this, and not all functions have inverses

unfortunately.

3.3 Expansion of Brackets

When we have to multiply something by a bracketed expression, we use theso called distributive law.

a(b + c) = ab + ac(a + b)c = ac + bc

We can easily show that this can be extended.

a(b + c + d + ) = ab + ac + ad + There are some simple things worth remembering

The abscence of a number before an expression is the same as multi-plying by 1.


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The sign preceding a number belongs to that number and must be

included in the multiplication.

In particular, a minus sign before a bracket means 1 multiplied bythat bracket.

3.3.1 Examples

Here are some examples.1.

2(3x

5y + z) =

6x + 10y

2z

2.3(2x y) (x + 2y) = 6x 3y x 2y = 5x 5y

3.

4x(y z+ 2(xy)) = 4x(y z+ 2x2y) = 4x(2xy z) = 8x2 4xy 4xz

4.2y(3x 4z(x + z)) = 2y(3x 4xz 4z2) = 6xy 4xyz 8yz2

3.3.2 Brackets upon Brackets

When we encounter a bracketed expression multiplied by another bracketedexpression we can apply the same technique, although it appears more com-plicated.

Consider

(a + b)(c + d)

For the moment, we shall call z = (a + b). Then our expression appearssimpler.

z(c + d) = zc + zd

Now we reinsert the true value of z.

= (a + b)c + (a + b)d = ac + bc + ad + bd

So we reduce the whole problem to two separate expansions of the typewe have already met. There are a number of rules of thumb to make thistechnique rather simpler, but many depend on multiplying only two brackets


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together, each of which with exactly two terms. We shall examine a general

technique without this shortcoming.Consider once more

(a + b)(c + d)

Pick any bracket, for the sake of demonstration, we shall pick the first.Now take the first term in it (which is a). We now multiply this term oneach term of the other bracket in turn, adding all the results.

= ac + ad +

When we reach the end of the other bracket, we return to the first bracketand move onto the next term, which is now b and do the same again, addingto our existing terms.

= ac + ad + bc + bd + Now we return to the first bracket, and move to the next term. We find

we have actually exhausted our supply of terms, and so our expansion isreally complete.

(a + b)(c + d) = ac + ad + bc + bd

To multiply several brackets together at once we should multiply two onlyat a time. For example

(a + b)(c + d)(e + f).

We begin my multiplying one pair together, let us say the first two, toobtain:

= (ac + ad + bc + bd)(e + f).

We may then complete the expansion, it is left to the reader as an exercise

to confirm that the full expansion will be:

= ace + ade + bce + bde + acf + adf + bcf + bdf.

3.3.3 Examples

Here are some examples.1.

(x + 1)(x + 2) = x2 + 2x + x + 2 = x2 + 3x + 2


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3.4 Factorization 20

2.

(x + y)2 = (x + y)(x + y) = x2 + xy + xy + y2 = x2 + 2xy + y2

(See binomial expansions later).3.

(x + y)(x y) = x2 xy + xy y2 = x2 y2

(This is called the difference of two squares).4.

(3x 2y)(x + 3) = 3x2 + 9x 2xy 6y5.

(2x

y)(x + 2y) = 2x2 + 4xy

xy

2y2 = 2x2 + 3xy

2y2

3.4 Factorization

Factorization is the opposite of expansion, we often prefer to condense andsimplify expressions rather than expand them. Indeed, even in the expansionexamples above we simplified the expressions along the way to make life easierfor ourselves.

Factorization is recognizing that an expression like this

4x + 2y

could be written as4x + 2y = 2(2x + y)

simply because we can clearly see that expanding the result gives us theoriginal. A factor common to all terms is observed - in this case the number2 clearly divides into all the terms. The factor is divided into the expressionand written outside the result which is bracketed.

The factor may often be some algebra, and not just a number. In theexpression

x2 3xwe see that x divides into both terms. We can thus write

x2 3x = x(x 3).The ability to spot factors does not come easily, but with a great deal of

practice.


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3.5 Laws of Indices 21

3.4.1 Examples

Let us look at some examples. Remember that in each case, expanding theend result should give us our original expression, and this allows you to checkand follow the logic.1.

3x + 12y2 6z = 3(x + 4y2 2z)2.

x3 + 3x2 + 4x = x(x2 + 3x + 4)

3.x3 + 3x2 = x2(x + 3)

4.2x2 + 4xy + 8x2z = 2x(x + 2y + 4xz)

We can also factorise expressions into two or more brackets multipliedtogether, but this is more difficult and we shall examine it later.

3.5 Laws of Indices

The term index is a formal term for a power, such as squaring, cubing etc,

and the plural of index is indices.There are some simple laws of indices, which are shown in table 3.1.

1 xa xb = xa+b

2 xa xb = xab

3 (xa)b = xab

4 x0 = 1

5 xb = 1xb

6 x1

b = bx7 x

a

b = b

xa

Table 3.1: The laws of indices


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3.5 Laws of Indices 22

3.5.1 Example proofs

We shall attempt to show how a selection of these results work, but suchdemonstrations are for understanding and are not examinable.

Let us consider the first law, with a concrete example:

x3 x2We dont know what the number x is, but all that is important is that

the base values of each number are the same.We recall that powers mean a string of the same thing multiplied together,

so that:

x3 x2 = x x x x3

x x x2

.

Clearly there is no difference between the inside the braced sectionand between them. In otherwords, this is just

x3 x2 = x x x x3

x x x2

= x x x x x x5

= x5

a string of five xs multiplied together, exactly the definition of x5, andthe 3 and 2 add to make 5.

We shall show how one other result works, using the more general a andb.

Consider

(xa)b.

By definition, this is ju

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