7th Edition 2012136 Link to Contents Introduction Formula Handbook including Engineering Formulae, Mathematics, Statistics and Computer Algebra Name__________________________ Course__________________________ http://is.gd/formulahandbook - pdf http://ubuntuone.com/p/dAn - print Introduction This handbook was designed to provide engineering students at Aberdeen College with the formulae required for their courses up to Higher National level (2nd year university equivalent). In order to use the interactive graphs you will need to have access to Geogebra (see 25 ). If you are using a MS Windows operating system and you already have Java Runtime Environment loaded then no changes will be required to the registry. This should mean that no security issues should be encountered. If you have problems see http://www.geogebra.org/cms/en/portable I have the handbook copied as an A5 booklet with a spiral binding. The covers are printed on thin card rather than paper. It is typed in LibreOffice Writer. Future developments will include more hyperlinks within the handbook and to other maths sites, with all the illustrations in it produced with Geogebra (see 25) or LibreOffice. Any contributions will be gratefully accepted and acknowledged in the handbook. If you prefer, you can make changes or add to the handbook within the terms of the Creative Commons licence . I will send you a copy the original LibreOffice file on request. Please send me a copy of your work and be prepared to have it incorporated or adapted for inclusion in my version. My overriding concern is for the handbook to live on and be continuously improved. I hope that you find the handbook useful and that you will enjoy using it and that that you will feel inspired to contribute material and suggest hyperlinks that could be added. Many thanks to my colleagues at Aberdeen College and elsewhere for their contributions and help in editing the handbook. Special thanks are due to Mark Perkins at Bedford College who adopted the handbook for his students, helped to format the contents and contributed to the contents. Without Mark's encouragement this project would have never taken off. If you find any errors or have suggestions for changes please contact the editor: Peter K Nicol. (
[email protected]) (
[email protected]) Contents Peter K Nicol Aberdeenshire, Scotland 7th Edition VI/MMXII 13/03/13 136 Contents 1 Recommended Books..................................3 9 Mathematical Notation – what the symbols mean...........................................................17 1.1 Maths.................................................3 9.1.1 Notation for Indices and Logarithms 18 1.2 Mechanical and Electrical Engineering . .3 9.1.2 Notation for Functions........................18 2 Useful Web Sites.........................................4 10 Laws of Mathematics...............................19 3 Evaluation..................................................6 10.1 Algebra...........................................20 3.1.1 Accuracy and Precision........................6 10.1.1 Sequence of operations...................20 3.1.2 Units........................................................6 10.1.2 Changing the subject of a Formula (Transposition)...............................................21 3.1.3 Rounding................................................6 4 Areas and Volumes......................................7 11 The Straight Line ...................................22 5 Electrical Formulae and Constants ...............8 12 Quadratic Equations ..............................23 5.1 Basic .................................................8 13 Simultaneous Equations with 2 variables....24 14 Matrices.................................................25 5.2 Electrostatics.......................................8 15 The Circle...............................................28 5.3 Electromagnetism ...............................8 15.1.1 Radian Measure................................28 5.4 AC Circuits .........................................9 16 Trigonometry...........................................29 6 Mechanical Engineering.............................10 16.1.1 Notation for Trigonometry................29 6.1.1 Dynamics: Terms and Equations......10 16.2 Pythagoras’ Theorem.......................29 6.1.2 Conversions.........................................10 16.3 The Triangle....................................30 6.2 Equations of Motion...........................10 6.3 Newton's Second Law........................11 6.3.1 Centrifugal Force.................................11 6.4 Work done and Power........................11 6.5 Energy..............................................11 6.6 Momentum / Angular Impulse..............12 6.7 Specific force / torque values...............12 16.3.1 Sine and Cosine Rules and Area Formula...........................................................30 16.4 Trigonometric Graphs.......................31 16.4.1 Degrees - Radians Conversion.......33 16.4.2 Sinusoidal Wave................................34 16.5 Trigonometric Identities.....................35 16.6 Multiple / double angles....................35 16.7 Products to Sums.............................36 6.8 Stress and Strain...............................12 17 Complex Numbers...................................37 6.9 Fluid Mechanics.................................13 18 Vectors...................................................38 6.10 Heat Transfer..................................13 6.11 Thermodynamics..............................14 7 Maths for Computing..................................15 7.1.1 Notation for Set Theory and Boolean Laws ...............................................................15 8 Combinational Logic..................................16 8.1.1 Basic Flowchart Shapes and Symbols .........................................................................16 Contents p1 9 Notation 1 18.1 Co-ordinate Conversion using Scientific Calculators...............................................39 18.2 Graphical Vector Addition..................41 19 Functions................................................42 19.1 Indices and Logarithms.....................42 19.2 Infinite Series and Hyberbolic Functions ...............................................................43 24 Computer Input 19.3 Exponential and Logarithmic Graphs..44 21.1.1 Notation for Statistics........................60 19.4 Graphs of Common Functions...........45 21.2 Statistical Formulae..........................61 20 Calculus ................................................46 21.2.1 Regression Line ...............................62 20.1.1 Notation for Calculus........................46 20.2 Differential Calculus - Derivatives.......47 21.2.2 T Test .................................................62 21.2.3 Statistical Tables ..............................63 21.2.3.1 Normal Distribution...................63 20.2.1 Maxima and Minima..........................49 21.2.3.2 Far Right Tail Probabilities .....63 20.2.2 Differentiation Rules.........................49 21.2.3.3 Critical Values of the t Distribution................................................64 20.2.3 Formula for the Newton-Raphson Iterative Process............................................50 21.2.4 Normal Distribution Curve................65 20.2.4 Partial Differentiation .......................50 21.2.5 Binomial Theorem.............................65 20.2.5 Implicit Differentiation.......................50 21.2.6 Permutations and Combinations....65 20.2.6 Parametric Differentiation................50 22 Financial Mathematics.............................66 20.3 Integral Calculus - Integrals...............51 23 Recommended Computer Programs..........67 20.3.1 Integration by Substitution...............52 20.3.2 Integration by Parts...........................52 20.3.3 Indefinite Integration.........................53 20.3.4 Area under a Curve..........................53 20.3.5 Mean Value........................................53 24 Computer Input ......................................68 24.1 wxMaxima Input...............................69 24.1.1 Differential Equations.......................69 24.1.2 Runge-Kutta.......................................69 20.3.6 Root Mean Square (RMS)...............53 24.2 Mathcad Input .................................70 20.3.7 Volume of Revolution ......................54 24.3 SMath.............................................70 20.3.8 Centroid..............................................54 20.3.9 Partial Fractions................................54 24.4 Spreadsheet procedures ..................71 20.3.10 Approximation of Definite Integrals .........................................................................55 24.4.1 Find the ‘best fit’ formula for a set of data..................................................................71 20.3.10.1 Simpson's Rule.......................55 24.4.2 Euler's Method...................................71 20.3.10.2 Trapezium Method.................55 25 Calibration Error......................................74 20.4 Laplace Transforms .........................56 26 Mechanical Tables...................................75 20.5 Approximate numerical solution of differential equations.................................57 20.6 Fourier Series. ...............................58 20.6.1 Fourier Series - wxMaxima method. .........................................................................59 26.1.1 Properties of Materials.....................75 26.1.2 Young's Modulus- approximate.......75 27 Periodic Table of The Elements.................76 SI Units - Commonly used prefixes..................77 28 Electrical Tables......................................77 21 Statistics.................................................60 29 THE GREEK ALPHABET.........................78 Contents p1 9 Notation 2 24 Computer Input ISBN 0-582-42429-1 Contents p1 9 Notation 3 24 Computer Input . ISBN 0-333-92224-7.2 Mechanical and Electrical Engineering NC Advanced Physics for You. J Bird ISBN 0750652284 Electrical Engineering NC and HN Basic Electrical Engineering Science Ian McKenzie Smith.4039-4246-3 Degree ----------------------------------------------------------------------------------------------1. K A Stroud ISBN 978-1.com/science/engineering/singh Engineering Mathematics. Kuldeep Singh.1 Maths General pre-NC and NC : Countdown to Mathematics. Croft and Davison ISBN 0-131-97921-3 NC and HN and Degree: Engineering Mathematics through Applications. 2 ISBN 0-201-13731-3 NC Foundation Maths. (1st Edition) (1) 978-0-230-27479-2 (2nd Edition) (2) K Singh www. 1 ISBN 0-201-13730-5. C Ross. 6th Edition.1 Recommended Books referred to by author name in this handbook 1. ISBN 0 7487 5296 X Mechanical Engineering NC and HN Mechanical Engineering Principles. S Hewett et al. Vol. J Bird ISBN 1-8561-7767-X HN and degree: J Bird Higher Engineering Mathematics. K Johnson. J Bird. 4th Edition.palgrave. Graham and Sargent Vol. ISBN 0-7506-6266-2 Engineering Mathematics 6th Edition . org Freestudy Mechanical engineering notes and exercises and Maths notes and exercises.the most powerful mathematical software. http://www. Excellent calculators (like quickmath). http://BetterExplained.coventry.freestudy.hu An online calculator which also does calculus and produces graphs.ac.khanacademy.com Just the Maths A complete text book – all in pdf format http://nestor.com/ how to learn maths how to learn maths The "free classroom of the World" Many video lectures using a blackboard http://www. Efunda A US service providing a wealth of engineering information on materials. Maths.com Mathcentre MC Try the Video Tutorials.mathtutor. http://www. See section 24 of this book.htm Contents p1 9 Notation 4 24 Computer Input .efunda.2 Useful Web Sites If you use any of the sites below please read the instructions first.matek. http://www. If you find anything really useful in the sites below or any other site please tell us so that we can pass on the information to other students. It is well worth trying the examples first. Also see http://www. See 26 for input syntax.mathway.ac. trig and calculus and it draws graphs too.com Mathway Try the problem solver for algebra. http://www.uk The other stuff is excellent too.hu QuickMath Links you to a computer running MATHEMATICA .uk/jtm/contents.uk matek. Most sites have examples as well as instructions. http://www. unit conversion and more.co. processes. http://www.ac.wolframalpha.com/ BetterExplained BE Khan Academy It is true – maths and some other topics explained better.mathcentre. When entering mathematical expressions the syntax MUST be correct. (Based on Maxima).quickmath.uk WolframAlpha Almost any maths problem solved! http://www. ac.waldomaths. http://www.uk/ Read the text very carefully on all the pages and then go to http://mathschoices.php?id=8&perpage=15&page=1 The one below is really useful Using a Scientific Calculator Do you have a Casio fx-83 ES scientific calculator (or a compatible model) and want to learn how to use it? This unit will help you to understand how to use the different facilities and functions and discover what a powerful tool this calculator can be! http://openlearn.ac.maths. Facebook.uk/ Android Scientific Calculator Try HEXFLASHOR The Narrow Road Maths Explained by Leland McInnes http://zenandmath.p.wordpress. Banff and Buchan College http://tutorial. browse our blog.ac. medicine.html and try the quizzes.uk Contents p1 9 Notation 5 24 Computer Input .ac. with all its beauty and applications.ac.edu/ Waldomaths Some excellent interactive tools . by providing articles from the top mathematicians and science writers on topics as diverse as art.uk/course/view.open. You can read the latest mathematical news on the site every week. listen to our podcasts and keep up-to-date by subscribing to Plus (on email. RSS.co.try them http://openlearn.nicol@abcol. cosmology and sport.open.hndengineer.uk/routes/p6/index.lamar.Equations 1 and 2 in particular for transposition practice.com/ HND Engineer http://www.uk/course/category.php? name=MU123_1 Plus Magazine Plus magazine opens a door to the world of maths. http://plus.The Open University There are a lot of excellent courses to study and if you want to improve your maths I suggest that you start here http://mathschoices.open. OU Learning Space Many courses for many levels .math.com/ If you come across any Engineering or Mathematics sites that might be useful to students on your course please tell me (Peter Nicol) .org/content/ Paul's Online Math Notes Recommended by June Cardno.open. iTunes or Twitter). 1.234 ------------------------------------------------------------------------------------------------------3. not Precise 1.276 Precise and Accurate 1.425 Accurate but not Precise 1. 1. Round to significant figures preferably in engineering form 2 d Example: where d =40 A= 4 3 A=1256.3 Evaluation 3.276.835. 1.1. Contents p1 9 Notation 6 24 Computer Input .1 Accuracy and Precision Example: Target = 1. 1.231. 1.2 Units MC Treat units as algebra m 1 2 for example KE = m v where m=5 kg and v=12 .637061 A=1. 1. 1. 1.234. 1. 0. 1.234.130.276.234.234 - 4 possible answers Not Accurate.276. s 2 2 1 m KE = ×5×kg×12× Standard workshop 2 s 2 1 m 2 KE = ×5×kg×12 × 2 tolerance ±0. 1.232.2 mm 2 s 2 1 2 kg ×m KE = ×5×12 × 2 2 s 2 kg⋅m KE =360 KE =360 J 2 s ------------------------------------------------------------------------------------------------------3.257×10 rounded to 4 sig fig ( A=1257 ) There should be at least 2 more significant figures in the calculation than in the answer.235. 2.270.256637061×10 3 A=1.3 Rounding ( ) Do not round calculations until the last line.1.236 Precise but not Accurate 1. 4 Areas and Volumes Volume of a Prism b Rectangle A=l b Area length Volume = Area x length l (Uniform cross sectional area) h Triangle 1 A= b h 2 Circle πd2 A= 4 A=π r 2 C=π d C=2 π r b r d Cylinder V = Area of circular base times height πd2 V cyl = ×h 4 h V cyl =π r 2×h πd2 Total surface area = A=π d h+ 2 4 side + 2 ends d A=2 π r h+ 2π r 2 Cone l h V cyl πd2h V cone = or V = 3 12 π r2 h V= 3 πd l A=π r l 2 A=π r l+ π r 2 Curved surface area A= d Total surface area d Sphere V sphere= 2V cyl 3 V= πd3 6 Total surface area A=π d 2 d Contents p1 9 Notation 7 24 Computer Input V= 4 π r3 3 A=4 π r 2 . 3 Electromagnetism Magnetomotive Force F=I N Magnetisation H= Reluctance S= At IN ℓ At/m l l = A o r A At/Wb −7 0=4 ×10 Absolute Permeability H/m -------------------------------------------------------Contents p1 9 Notation 8 24 Computer Input .5 Electrical Formulae and Constants Circuit Construction Kit 5.2 Electrostatics Series Capacitors 1 1 1 1 = …. F Charge Q=I t or Q=C V C Capacitance C= A A 0r = d d F −12 0≈8.854×10 Absolute Permittivity F/m ------------------------------------------------------- 5. R T R1 R2 R 3 Potential Difference V =I R V Power P= I V or P= I 2 R or P= Energy (work done) W =P t Frequency f= V2 R 8 W J or kWh 1 T Hz ------------------------------------------------------- 5.1 Basic Unit symbol Series Resistors R T =R 1 R 2R 3 … . Parallel Resistors 1 1 1 1 = …. CT C1 C2 C3 F Parallel Capacitors C T =C 1C 2C 3 …. 5.4 AC Circuits Unit Symbol Force on a conductor F =B I ℓ N Electromotive Force E= B ℓ v V Instantaneous emf e= E sin V Induced emf e= N RMS Voltage V rms= Average Voltage 2 V AV = ×V peak Angular Velocity =2 f d dt e= L di dt 1 ×V 2 peak V rms≈0.2 Transformation Ratios V s Ns I p = = V p N p Is Potential Difference V =I Z Power Factor pf =cos(ϕ) Capacitive Reactance X C= 1 2 f C Inductive Reactance X L=2 f L Admittance Y= True Power P=V I cos(ϕ) W Reactive Power Q=V I sin (ϕ) VAr Apparent Power S=V I * VA V 1 Z S = P j Q Note: I * is the complex conjugate of the phasor current.4. Aberdeen College Contents p1 9 Notation 9 24 Computer Input . See 17 _________________________________________________________ Thanks to Iain Smith.707 V peak V V V AV ≈0.637 V peak V rad/s 16. e.1 Dynamics: Terms and Equations Linear Angular (m) s= displacement = angular displacement (rad) 1= initial velocity (m/s) (rad/s) u= initial velocity 2= final velocity (m/s) (rad/s) v= final velocity 2 (m/s ) (rad/s2) a= acceleration = acceleration (s) (s) t = time t= time -------------------------------------------------------6. i.73 (2) 2-99] 6. 1 rad = ≈57.1.6 Mechanical Engineering [K Singh(1) 2–98 especially 32 – 40 and 69 .4.1.2 Conversions Displacement s=r Velocity v=r Acceleration a=r v= s t = t o 360 o 2 radians = 1 revolution = 360 .3 see 16.1 2 2 N If N = rotational speed in revolutions per minute (rpm).2 Equations of Motion Linear Angular MC v=ua t 2= 1 t 1 s= uvt 2 1 = 12 t 2 1 s=ut a t 2 2 1 =1 t t 2 2 v 2=u 22 a s 22= 212 a= v–u t = 2− 1 t ------------------------------------------------------------- Contents p1 9 Notation 10 24 Computer Input . then = rad/s 60 o -------------------------------------------------------- 6. I =m k 2 and k = radius of gyration --------------------------------------------------------6.6.4 Work done and Power Work Done Linear Angular WD= F s WD=T Work done Time taken Fs = t =F v P= Power P=T -------------------------------------------------------- 6.3 Newton's Second Law Linear Angular ∑ F =ma ∑T=I where T = F r .5 Energy Linear Angular Kinetic Energy 1 KE= m v 2 2 Potential Energy PE=m g h 1 KE= I 2 2 1 KE= m k 2 2 2 KE of a rolling wheel = KE (linear) + KE (angular) -------------------------------------------------------- Contents p1 9 Notation 11 24 Computer Input .3.1 Centrifugal Force m v2 CF= r CF=m 2 r -------------------------------------------------------- 6. 7 Specific force / torque values Force to move a load: F = m g cos m g sin m a Force to hoist a load vertically =90 o F =m gm a=m ga Force to move a load along a horizontal surface =0 o F = m gm a T app =T F F r I Winch drum torque -------------------------------------------------------- 6.6.6 Momentum / Angular Impulse Impulse = Change in momentum MC Linear Angular Ft=m2 v – m 1 u Tt= I 2 2− I 1 1 If the mass does not change: Ft=m v−mu -------------------------------------------------------- 6. Aberdeen College Contents p1 9 Notation 12 24 Computer Input .8 Stress and Strain Stress = load / area = Strain = change in length / original length = E= Stress / Strain F A l x or = l l E= M E = = I y R Bending of Beams b d3 I= 12 b d3 I= + A h2 12 2nd Moment of Area (rectangle) Including the Parallel axis Theorem T G = = J r L Torsion Equation D4 d 4 J= − 32 32 nd 2 Moment of Area (cylinder) -------------------------------------------------------Thanks to Frank McClean Scott Smith and William Livie. h= 2gd Re= Reynold’s number Darcy formula for head loss 2gh m –1 f A1 –1 A2 Efunda Calculator 2gh m –1 f D 1– 0 D1 4 Reynold's Number video VD Efunda calculator 4 f l v2 energy loss h= 2d Re= Efunda Calculator ----------------------------------------------------------------------------------------------------- 6.6.9 Fluid Mechanics Mass continuity m= A V .10 Heat Transfer k AT 1 – T 2 x x x ˙ T where R= 1 2 1 1 … Q= k 1 k 2 h1 h2 R ˙ T Q= R R R ln 2 ln 3 R1 R2 1 1 R= 2 R 1 h 1 2 k 1 2 k 2 2 R 3 h 3 ˙ Q= Through a slab Through a composite Through a cylindrical pipe where -----------------------------------------------------------------------------------------------------Contents p1 9 Notation 13 24 Computer Input . or ˙ Bernoulli’s Equation p C2 z = constant g 2g or m= AC ˙ p 1 C 21 p 2 C 22 z 1= z 2z F g 2g g 2g Q=A v Volumetric flow rate Actual flow for a venturi-meter Qactual = A1 c d Actual flow for an orifice plate Q= A0 c d ρV D v 4 f l v2 . Contents p1 9 Notation 14 24 Computer Input .11 Thermodynamics Boyle’s Law p 1 V 1= p 2 V 2 Charles’s Law V1 V2 = T1 T2 Combined Gas Law p1 V 1 p 2 V 2 = T1 T2 Perfect Gas pV =m R T Mass flow rate m= AC ˙ Polytropic Process pV n = constant Isentropic Process γ cP cV (reversible adiabatic) pV Gas constant R=c p −c v Enthalpy (specific) h=u p v Steady flow energy equation C2 C1 ˙ m ˙ Q= ˙ h 2 – h 1 – g z 2 – z 1 W 2 2 Vapours v x= x v g where γ= = constant 2 2 u x =u f x u g −u f h x =h f xh g – h f or h x =h f x h f g ___________________________________________________________________ Thanks to Richard Kaczkowski.6. Aberdeen College. Canada and Scott Smith. Calgary. 7 Maths for Computing an a to the base n a 10 decimal. denary ( a d) a2 binary ( a b) a 16 hexadecimal ( a h) a8 octal ( a o) ----------------------------------------------------------------------------------------------- 10 3 (1000) 106 kilo 2 10 1024 kilobyte Mega 2 20 1024 2 megabyte Giga 2 30 1024 3 gigabyte Tera 2 40 1024 4 terabyte but 10 9 1012 5 Peta petabyte 2 50 1024 _____________________________________________________________ 7.b .1. b .c B⊂ A E A B is a subset of A B A∪ B A B Set theory Boolean ∪ union ∨ OR ∩ intersection ∧ ⋅ AND E A B A∩ B A⋅B E A' A complement of A A NOT A' Contents p1 9 Notation 15 24 Computer Input A .396] E universal set A={ a . c …} a set A with elements a .a B .1 Notation for Set Theory and Boolean Laws 1015 [J Bird pp 377 . c etc a∈ A a is a member of A { } the empty set ( Ø is also used) B⊂ A E A . b . A BC. -----------------------------------------------------------------------------------------------------8...8 Combinational Logic A0= A A⋅0=0 A1=1 A⋅1= A A⋅A= A A A= A A A=0 A A=1 A= A A⋅B=B⋅A A B= B A A⋅ BC = A⋅B A⋅C A B⋅C = A B⋅ AC A⋅ B⋅C =C⋅ A⋅B A BC=C AB A⋅ AB= A A A⋅B= A De Morgan's Laws A⋅B⋅C⋅..= ABC..1 Basic Flowchart Shapes and Symbols Start / End Input / Output Action or Process Connector Decision Flow Line ______________________________________________________________ Contents p1 9 Notation 16 24 Computer Input ....1..= A⋅B⋅C⋅. . ≤ less than or equal to... 1.... Numbers represented by drawing a continuous number line. 0. ( x∈ℝ means x is a member of ℝ ) ℕ the set of natural numbers 1.9 Mathematical Notation – what the symbols mean MC ∈ is a member of.. . Numbers represented by drawing vectors. 3.˙. with respect to ∗ used as a multiplication sign ( × ) (in computer algebra) ^ used as “power of” ( x y ) in computer algebra ≠ not equal to ≈ approximately equal to greater than. p . ( 3×10 3=3000 ) use EXP or ×10x key on a calculator n! “ n factorial” Contents p1 n×n – 1×n – 2×n−3×. therefore w. ℤ the set of all integers . -1. ℚ the set of rational numbers including ℤ and fractions p . . a≤ x≤b x is greater than or equal to a and less than or equal to b ab abbreviation for a×b or a∗b or a⋅b a×10n a number in scientific (or standard) form......×1 9 Notation 17 24 Computer Input .r... 3. less than. 2.. .t.. q∈ℤ q ℝ the set of all real numbers. ℂ the set of complex numbers.. a2 means a is less than and not equal to 2. x2 means x is greater than and not equal to 2 ≥ greater than or equal to. -2. 2... 1. ------------------------------------------------------------------------------------------------------ Contents p1 9 Notation 18 24 Computer Input . The logarithm of x to the base 10 x= x = x 0. irrespective of the sign.. x ▄ or ^ or x y or y x or a b on a calculator. Also seen as g x .5 1 3 8=2 k a=a k . see 19. ∣−3∣=3=∣3∣ ∞ infinity ⇒ implies -------------------------------------------------------9.×a (n terms).71828. -------------------------------------------------------9. See 19. log e ( x) ln x on a calculator.. y x f −1 x the inverse of the function labelled f x g° f the composite function . h x . to the power of x ).2 Notation for Functions f x a function of x .first f then g .1..1 1 2 a the positive square root of the number a .1. The magnitude of the number x . k a k th root of a number a . The logarithm of x to the base e log 10 ( x) log x on a calculator. ex exp x (2. or g f x .A∝B implies A=k B where k is a constant (direct variation) ∣x∣ the modulus of x ...1 Notation for Indices and Logarithms MC a n abbreviation for a×a×a×a . for multiplication and division bc b c = a a a a bc=a ba c -------------------------------------------------------Arithmetical Identities x 0= x x×0=0 x ×1=x -------------------------------------------------------Algebraic Identities K Singh pp 73 – 75 ab2=abab=a 22 a bb 2 a 2 – b 2= aba−b ab3=ab a 22 a bb 2 =a 33 a 2 b3 ab2b3 see 19.......for addition and multiplication abc=abc a b c=a b c -------------------------------------------------------Commutative laws ..........10 Laws of Mathematics Associative laws ..4 -------------------------------------------------------Other useful facts a a 1 =a÷b= × b 1 b a – b=a−b a−−b=a−−b=ab ...9..3.....for addition and multiplication ab=ba but a – b≠b−a a b=b a but a b ≠ b a -------------------------------------------------------Distributive laws ..........------ a c a d b c = b d bd a c ac × = b d bd see 20... 5 a c a d ÷ = × b d b c MC FOIL -------------------------------------------------------- abcd =acad bcbd MC Contents p1 9 Notation 19 24 Computer Input . 1. Of comes before “square of x . ----------------------------------------------------------------- Brackets come before ------------------------------------------------------------------x 2.10.1 Algebra 10. x . sine of x ------------------------------------------------------------------× Multiplication comes before Division ÷ comes before ------------------------------------------------------------------ Addition comes before Subtraction − ------------------------------------------------------------------- 3sin a x 2b−5 would be read in this order left bracket x squared times a plus b right bracket sine of the result ( sin a x 2b ) times 3 minus 5 Contents p1 9 Notation 20 24 Computer Input .the same sequence as used by scientific calculators.1 Sequence of operations [K Singh (1) 40-43 (2) 40-43] Sequence of operations . e x . sin x . com/Equation3NLW.1.waldomaths. (to all the terms) + 5a . You must UNDO it by using the correct MATHEMATICAL operation. UNDO with − and − with UNDO × with ÷ and ÷ with × UNDO with x 2 and x2 with UNDO xn with and n with xn UNDO sin x with sin−1 x and sin−1 x with sin x UNDO ex with ln x ln x with UNDO 10 x dy dx with log 10 x and UNDO UNDO with n ∫ dx and and L[ y ] with L−1 [ y ] and ex log 10 x with 10 x dy ∫ dx with dx −1 L [ y ] with L[ y ] etc Generally (but not always) start with the terms FURTHEST AWAY from the new subject FIRST.take the layers off one by one.10. a x 2 b Try this first http://www.jsp There is a link to Equations 1 if you need a bit more help.7 + 3b = 7 7 You can’t move a term (or number) from one side of the equals sign to the other. MC Contents p1 9 Notation 21 24 Computer Input . Think of the terms in the formula as layers of an onion .2 Changing the subject of a Formula (Transposition) [K Singh (1) 53-66 (2) 53-56] An equation or formula must always be BALANCED whatever mathematical operation you do to one side of an equals sign must be done to other side as well. 3. gradient m passing through a . y2 ) +ve gradient dy y1 c ( x1 . b has the equation: y−b=m x−a Also see 24.5.1. y1 ) -ve gradient dx x x1 x2 The general equation of a straight line of gradient m cutting the y axis at 0.11 The Straight Line [K Singh (1) 100–108 (2) 101-110] y y2 ( x2 . c is y =m xc where the gradient m= or y2 – y1 x 2− x 1 or dy y 2 – y 1 = .1. See 20. 20. 13.1. MC ------------------------------------------------------------------------------------------------------ Contents p1 9 Notation 22 24 Computer Input .10.1. 20. 21.2. 20.3.3 dx x 2− x 1 y 1=m x 1+ c (1) y 2=m x 2+ c (2) then (1) – (2) and solve for m (then c ) Also: A straight line.2.3. back to 20.2 and 16. . ------------------------------------------------------------------------------------------------------- Contents p1 9 Notation 23 24 Computer Input ........ The solutions (roots) x 1 and x 2 of a x 2b xc=0 are given by the Quadratic Formula.... …....... B 2 see 22.................... Also: a x 2 b xc=0 b c 2 x x =0 a a b a 2 2 where 2 c b d = − a a 2 x d =0 2 If y=k x A2 B the turning point is − A .............. 2a 4a c ) a minimum turning point Geogebra quadratic MC The real solutions (roots) x 1 and x 2 of the equation a x 2b xc=0 are the value(s) of x where y=a x 2 b xc crosses the x axis....113 (2) 110-113] y x= −b 2a -ve a 2 y=a x b xc +ve a F x1 x2 Focus x ( 2 −b −(b −4 a c−1) F= ...............2) 86 ......................12 Quadratic Equations [K Singh (1.....90 & (1) 109 ..4 Geogebra back to 13........................ 2 −b √ (b – 4 a c) x= ± (2 a) (2 a) −b± b2 – 4 a c x= or 2 a Definition of a root: The value(s) of x which make y equal to zero...... a x+b y=e (1) c x+ d y= f (2) Multiply every term on both sides of (1) by c and every term on both sides of (2) by a and re-label as (3) and (4). c a x+c b y=c e (3) −a c x−a d y=−a f (5) Add (3) to (5) to eliminate x Calculate the value of y Substitute the value of y into equation (1) Calculate the value of x MC Check by substituting the values of x and y into (2) ---------------------------------------------------Graphical Solution a x+b y=e y y2 c x +d y= f y1 x1 x x2 If f x = g x then f x – g x=0 .also see 11 and 12 _________________________________________________________ Contents p1 9 Notation 24 24 Computer Input . c a x+ c b y=c e (3) a c x+a d y=a f (4) Multiply every term on both sides of (4) by -1 and re-label.13 Simultaneous Equations with 2 variables [K Singh (1) pp 90-98 (2) 90-99] General method: Write down both equations and label (1) and (2). Identity = 0 0 1 .14 Matrices [K Singh (1) pp 507 – 566 (2) 560-635] Notation: [ ] 1 0 0 . . a d −b c≠0 det A −c a MC ∣ ∣ where the Determinant det A= a b =ad −bc c d -------------------------------------------------------- Contents p1 9 Notation 25 24 Computer Input . ----------------------------------------------------------------------------------------------------- [ ] [ a 11 a 12 b 11 b12 and B= a 21 a 22 b 21 b 22 If A= then A B= and A× B= [ [ a 11b11 a 12b12 a 21b 21 a 22b22 ] ] a 11 b11a 12 b 21 a 11 b 12a 12 b 22 a 21 b11a 22 b 21 a 21 b 12a 22 b 22 ] Columns A=Rows B ------------------------------------------------------------------------------------------------------Solution of Equations 2 x 2 [ ][ ] [ ] a b c d If A X = B then X = A−1 B If A= x e = y f [ ] a b c d then the inverse matrix.. −1 A = [ ] 1 d −b .. a ij an element in the i th row and j th column. . A m×n matrix has m rows and n columns.. 0 1 0 .. . . Contents p1 9 Notation 26 24 Computer Input . Canada. 3 x 3 (and larger square matrices) Start with [ ∣ ] [ ∣ ] a 11 a12 a 13 1 0 0 a 21 a22 a 23 0 1 0 a 31 a32 a 33 0 0 1 carry out row operations to: 1 0 0 b 11 b12 b13 0 1 0 b 21 b22 b23 0 0 1 b 31 b32 b33 b 11 b12 b13 −1 where b 21 b22 b 23 = A b 31 b32 b33 [ ] Determinant of a 3 x 3 matrix ∣ ∣ a 11 a 12 a13 a a 23 a a a a det A= a 21 a 22 a 23 =a 11 22 −a 12 21 23 a 13 21 22 a32 a 33 a 31 a 33 a 31 a 32 a 31 a 32 a33 ∣ ∣ ∣ _ or use Sarrus' Rule as below ∣ ∣ ∣ ∣[ _ ∣ _ ] a 11 a 12 a 13 a 11 a 12 a 13 a 11 a 12 det A= a 21 a 22 a 23 = a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32 a 33 a 31 a 32 + + + detA=a 11 a 22 a 33 a 12 a 23 a 31 a 13 a 21 a 32 −a 31 a 22 a 13 −a 32 a 23 a 11−a 33 a 21 a 12 ----------------------------------------------------------------------------------------------Thanks to Richard Kaczkowski.Inverse Matrix. Calgary. transpose the Co-factor Matrix (rows to columns) Adjoint (Adjunct) Matrix [ ] cf (a) cf (d ) cf (g ) C = cf (b) cf (e) cf (h) = adjA cf (c) cf ( f ) cf (i ) T _________________________________________________________ Contents p1 9 Notation 27 24 Computer Input .Inverse of a 3 by 3 matrices by using co-factors [ ] a b c A= d e f g h i A−1= 1 (adjA) detA where adjA is the adjoint (adjunct) matrix of A adjA=C T where C T = the transpose of the co-factors of A Co-factors cf (a)=det [ ] [ ] [ ] e h f i cf (d )=−det b c h i cf ( g)=det b c e f [ ] [ ] [ ] cf (b)=−det d g cf (e)=det a c g i cf (h)=−det a d [ ] [ ] [ ] f i cf (c)=det d g c f cf ( f )=−det a b g h cf (i)=det a b d e Be careful of place signs! [ Co-factor Matrix cf (a) cf (b) cf (c) C= cf (d ) cf (e ) cf ( f ) cf ( g) cf (h) cf (i) + − + − + − + − + e h ] [ ] Then. y) The equation x – a2 y – b2=r 2 represents a circle centre a .1 Radian Measure r A radian: The angle θ subtended (or r made by) an arc the same θ length as the radius of a circle. r BE. r b Parametric x=a+ r cos t .1. b and radius r .4.15 The Circle A C Minor Sector Minor Segment C A D B B D Major Sector Major Segment -------------------------------------------------------y (x.1 Contents p1 9 Notation 28 24 Computer Input . y=b+r sin t a x -----------------------------------------------------------------------------------------------------15.com degrees and radians Geogebra Radians See also 16. Notice that an arc is curved. then R 2=a 2b 2 or R= a b 2 R 2 b use of Pythagoras' Theorem BE surprising uses a Pythagorean distance BE pythagorean distance Interactive proof http://www. 0o ≤o ≤180 o or 0≤≤ =tan−1 b arctan b the value of the basic angle whose tangent function − ≤≤ value is b . length R .sunsite.16 Trigonometry [K Singh (1) 167-176 (2) 171-234] 16.1 Notation for Trigonometry A Labelling of a triangle b c B C a sin the value of the sine function of the angle cos the value of the cosine function of the angle tan the value of the tangent function of the angle =sin−1 b arcsin b the value of the basic angle whose sine function − ≤≤ value is b .ubc. with hypotenuse.2 Pythagoras’ Theorem In a right angled triangle. −90o≤o≤90 o or 2 2 =cos−1 b arccos b the value of the basic angle whose cosine function value is b . and the other two sides of lengths a and b .1. −90o≤ o≤90o or 2 2 ------------------------------------------------------------------------------------------------------ 16.html --------------------------------------------------------Contents p1 9 Notation 29 24 Computer Input .ca/LivingMathematics/V001N01/UBCExamples/Py thagoras/pythagoras. ies.16.jp/math/java/trig/yogen1/yogen1.html Area Formula Area = b c sin( A) 2 ------------------------------------------------------------------------------------------------------Contents p1 9 Notation 30 24 Computer Input . of length H . SOHCAHTOA H The other two sides have lengths A (adjacent.3.co.html Cosine Rule (b 2 +c 2 – a 2) cos( A)= (2b c) or a 2=b 2+ c 2 – 2b c cos( A) http://www.1 Sine and Cosine Rules and Area Formula [K Singh (1) 187-192 (2) 195-191] A In any triangle ABC. or next to angle ) and O (opposite to angle ) then O θ A MC sin (θ)= O H cos (θ)= A H tan(θ)= O A see also 18.ies.1 and 11 -----------------------------------------------------------------------------------------------------16. (which is the longest side). where A is the angle at A. with hypotenuse. B is the angle at B and C is the angle at C the following hold: b c B a Sine Rule C a b c = = sin ( A) sin (B) sin(C) sin ( A) sin (B) sin(C) = = a b c or http://www.3 The Triangle In a right angled triangle.co.jp/math/java/trig/seigen/seigen. no units .202 (2) 181.e.16.4 Trigonometric Graphs and Equations [K Singh (1) 177.horizontal axis is usually time. y=sin (t) y t Calculator answer y=cos(t ) Calculator answer Contents p1 9 Notation 31 24 Computer Input .210] MC Radians i. Trigonometric Graphs .jp/math/java/trig/graphSinX/graphSinX.degrees y=sin (x o) Calculator answer Geogebra Sine wave slider http://www.co.jp/math/java/trig/graphCosX/graphCosX.co.ies.html y=cos (x o) Calculator answer Geogebra Cosine wave slider Contents p1 http://www.html 9 Notation 32 24 Computer Input .ies. 210. 150.y=tan (x o) Calculator answer -----------------------------------------------------------------------------------------------------16.4.2 ------------------------------------------------------------------------------------------------------ Contents p1 9 Notation 33 24 Computer Input . 225. 30. 330.Radians Conversion [K Singh (1) 192-195 (2) 201-204] 0. 300 315.1 Degrees . 120. 180. 270. 360 0 2 3 5 7 5 4 3 5 7 11 2 6 4 3 2 3 4 6 6 4 3 2 3 4 6 Degrees to radians o x ÷180×= rad r r r Radians to degrees rad ÷×180=x o =1 radian Geogebra Radians BE degrees and radians see 6. 135. 90. 240.1. 45. 60. 6.2 Sinusoidal Wave [K Singh (1) 195-202 (2) 204-212] V =R sin t R t see 20.4.16. 5. Bedford College + + 1 sin(θ) θ cos(θ) θ 0 (-) + (-) (-) Unit Circle ------------------------------------------------------------------------------------------------------- Odd Function Even Function Saw Tooth Square Wave Contents p1 9 Notation 34 24 Computer Input .4 Period = [ 2 Frequency = 2 ] = phase shift ------------------------------------------------------------------------------------------------------ = phase angle Thanks to Mark Perkins. 16. cosec( A)= .5 Trigonometric Identities [K Singh (1) 203-213 (2) 212-223] tan( A)= sin ( A) cos( A) cot ( A)= 1 cos ( A) = . (the cotangent of A ) tan ( A) sin ( A) -------------------------------------------------------- sec( A)= 1 1 . (secant of A ). (cosecant of A ) cos( A) sin ( A) -------------------------------------------------------2 sin 2 ( A)+ cos 2 ( A)=1 2 entered as ( sin ( A) ) + ( cos( A) ) -------------------------------------------------------(an ODD function) sin (−θ)=−sin (θ) (an EVEN function) cos −=cos -----------------------------------------------------------------------------------------------16.6 Multiple / double angles [K Singh (1) 213-222 (2) 223-234] sin A B=sin A cos Bcos Asin B sin (2 A)=2 sin A cos A sin A – B=sin Acos B – cos Asin B cos A B=cos Acos B – sin Asin B cos 2 A=cos 2 A – sin 2 A =2 cos2 A−1 1 cos 2 A= cos 2 A1 2 cos 2 A=1−2sin 2 A 1 sin 2 A= 1−cos 2 A 2 cos A – B=cos Acos Bsin Asin B tan AB= tan Atan B 1 – tan A tan B tan A− B= tan A−tan B 1tan A tan B tan 2 A= 2 tan A 1 – tan 2 A -------------------------------------------------------- Contents p1 9 Notation 35 24 Computer Input . 16.7 Products to Sums 1 sin A cos B= sin ABsin A− B 2 1 cos Asin B= sin AB−sin A−B 2 1 cos Acos B= cos A Bcos A− B 2 1 sin Asin B= cos A−B−cos A B 2 --------------------------------------------------------Sums to Products sin Asin B=2sin A B A– B cos 2 2 sin A−sin B=2 cos AB A– B sin 2 2 cos Acos B=2cos cos A−cos B=−2 sin AB A– B cos 2 2 A B A– B sin 2 2 ----------------------------------------------------------------------------------------------------- Contents p1 9 Notation 36 24 Computer Input . z a complex number z=a j b (or x y i ) r a complex number in polar form z complex conjugate of the complex number If z=a j b then the complex conjugate z=a – j b or if z=r then the complex conjugate z=r − z=a j b=rcos j sin =r =r e j where j 2=−1 --------------------------------------------------------Im Modulus. a .17 Complex Numbers [K Singh (1) 463-506 (2) 513-559] Notation for Complex Numbers BE . r =∣z∣= a 2b 2 jb (or magnitude) see 17.htm ____________________________________________________________ Contents p1 9 Notation 37 24 Computer Input .imaginary numbers −1 j symbol representing (defined as j2 =−1 ) .2 r =arg z =tan−1 Argument.com/home.justinmullins.1 Co-ordinate conversion MC --------------------------------------------------------De Moivre's Theorem (r θ)n=r n (nθ)=r n (cos(nθ)+ j sin (nθ)) r = r 2 http://www.Complex arithmetic . ( i used on most calculators) a j b a complex number in Cartesian (or Rectangular) form ( x y i on a calculator). 17. b∈ℝ . j b imaginary part.better explained Addition a jbc j d =ac j bd Multiplication a jbc jd Division a jbc− jd c jd c− jd Polar Multiplication z 1 z 2=r 1 1×r 2 2=r 1 r 2 12 Polar Division z 1 r 1 1 r 1 = = 1−2 z 2 r 2 2 r 2 See also: 18.2. b a θ a (or angle) Re Argand Diagram BE . . r a vector in polar form (where r=∣v∣ ) ) a vector in Component form (Rectangular Form) ∣v∣ modulus or magnitude of vector v .b) b a r bj θ x A vector v= A point a .1 Co-ordinate Conversion Scalar Product a×b=∣a∣∣b∣cos(θ) θ b Dot Product a⋅b=a 1 b 1a 2 b 2a 3 b 3 .18 Vectors Notation for Graphs and Vectors [K Singh (1) 567-600 (2) 636-671] x . . y the co-ordinates of a point. a b --------------------------------------------------------- Vectors y x (a.2 a see also 18. ------------------------------------------------------------------------------------------------------Contents p1 9 Notation 38 24 Computer Input . where x is the distance from the y axis and y is the distance from the x axis v a vector. b ai a or v=r b MC a c ac = b d bd Vector Addition Geogebra and 18. Always underlined in written work AB a vector a i b j a vector in Cartesian form (Rectangular form) where i is a horizontal 1 unit vector and j is a vertical 1 unit vector. b1 a1 b a where a= 2 and b= 2 a3 b3 .. ( x . y) P to R r . θ ) = x y out Casio S-VPAM and new Texet θ Edit keystrokes for your calculator R to P SHIFT Pol( x SHIFT . y out Sharp WriteView . Edit keystrokes for your calculator R to P x 2ndF .3 Casio Natural Display and Texet EV-S Edit keystrokes for your calculator R to P SHIFT Pol( x SHIFT . y ) = r out P to R SHIFT Rec( r SHIFT .L.θ out θ 2ndF x y x. ) = x out RCL tan y out y 2ndF r r .1 Co-ordinate Conversion using Scientific Calculators R to P x to r ( x jy to r ) y Rectangular to Polar P to R x ( r to x jy ) y r to Polar to Rectangular see also 16.A. y) Sharp ADVANCED D. R to P x ( x .18. y ) = r out RCL tan out P to R SHIFT Rec( r SHIFT . 2ndF → x y x out y out or MATH ⋅ y out 2 x out 2ndF Old Casio fx & VPAM R to P x SHIFT RP y = r out SHIFT X Y out P to R r SHIFT PR = x out SHIFT X Y y out Contents p1 9 Notation 39 24 Computer Input . y 2ndF →rθ r out out or MATH 1 r out 2ndF ⋅ out P to R r 2ndF . out R to P OPTN ▶ F2 ▶ ▶ Rec( r .Texet . y ) EXE r . y ) EXE r out ALPHA J EXE out P to R SHIFT Rec( r SHIFT . ) EXE x ALPHA J EXE y Casio Graphics (7 series) R to P OPTN ▶ F2 ▶ ▶ Pol( x . 2ndF CPLX R to P x a y b 2ndF a r out b out P to R r a b 2ndF b x out b y out Texas .36X R to P x x↔ y y 3rd RP r out x↔ y out P to R r x↔ y 2nd PR x out x↔ y y out Contents p1 9 Notation 40 24 Computer Input . ) EXE x.albert 2 R to P x INV x y y RP r out INV xy out P to R r INV x y PR x out INV xy y out Casio Graphics (1) R to P SHIFT Pol( x SHIFT . ) EXE x out ALPHA J EXE y out Casio Graphics (2) R to P FUNC 4 MATH 4 COORD 1 Pol( x .y out Old Texet and old Sharp and some £1 calculators You must be in Complex Number mode. y ) EXE r ALPHA J EXE P to R FUNC 4 MATH 4 COORD 1 Rec( r . Texas Graphics (TI 83) R to P P to R 2nd Angle 2nd Angle 2nd Angle 2nd Angle R Pr ( R P ( x.y ) ENTER r out x. ) ENTER x out ) ENTER y out Sharp Graphics R to P P to R MATH (D)CONV x .y ) ENTER out P Rx( P R y( r .2 Graphical Vector Addition Contents p1 9 Notation 41 24 Computer Input . y ) ENTER out MATH (D)CONV (5) r x ( r . ) ENTER y out (3) y( Insert the keystrokes for your calculator here (if different from above) R to P P to R -----------------------------------------------------------------------------------------------------Degrees to Radians ÷180× Radians to degrees ÷×180 _____________________________________________________________ 18. r . ) ENTER x out MATH (D)CONV (6) r r . y ) ENTER r out (D)CONV xy r ( (4) xy ( MATH x . 1 Indices and Logarithms [K Singh (1.1 Rules of Indices: MC 1. log a N = A B =log (A)– log( B) log b N log b a ------------------------------------------------------------------------------------------------------ exp (x)=e( x) log e ( x)=ln( x) log 10 ( x)=lg (x) -----------------------------------------------------------------------------------------------------Contents p1 9 Notation 42 24 Computer Input . log ( An ) =n log (A) 4. a m×a n =a mn 2.2) 7-11.19 Functions 19. (1) 223-245 (2) 235-259] notation 9.5 a 1=a a=b⇔ bn =a and 2 a= a n -----------------------------------------------------------------------------------------------------Definition of logarithms If N =a n then n=log a ( N ) --------------------------------------------------------Rules of logarithms: MC 1. log A× B =log (A)+ log (B) 2. am an 3. a = a 5. a m n =a mn 4. k a−n = =a m−n m n n a = a 1 n m n k an Also. 1 2 a 0=1 x= x = x 0.1. log 3. .. 358-369] x x2 x3 x 4 x5 x6 x7 e =1 ..demystifying the natural logarithm --------------------------------------------------------Hyperbolic Functions . 3! 5! 7 ! − jx 2 4 6 x x x =1− − ...... 2! 4! 6! x−1 x−12 x−13 ln x= – −.2 Infinite Series and Hyberbolic Functions [K Singh (1) pp 246-346. 1! 2! 3 ! 4 ! 5! 6! 7 ! x for ∣x∣∞ BE exponential functions better explained jx − jx e −e sin x= j2 jx e e cos x= 2 3 5 7 x x x =x− − . 1 2 3 for ∣x∣∞ for ∣x∣∞ for 0 x≤2 BE.19. 2! 4! 6! “shine x” e x −e −x tanh x= x −x e e “cosh x” “thaan x” ______________________________________________________________ y = cosh x y = ex y=x y = sinh x y = ln x y = tanh x y = tanh x y = sinh x ke ax slider k lna x slider Contents p1 9 Notation 43 24 Computer Input ..definitions [K Singh (1) 246-247 (2) 259-260] pronunciation MC x sinh x= −x e −e 2 x 3 =x −x e e cosh x= 2 5 7 x x x .... 3! 5! 7! 2 4 6 x x x =1 . 338-346 (2) 259-270. 19.3 Exponential and Logarithmic Graphs Contents p1 9 Notation 44 24 Computer Input . 19.4 Graphs of Common Functions y=a x 3b x 2c xd y=a x 4b x 3c x 2d x f a y= b x y=x 2 and y= x y=k 1−e− t Contents p1 y=k e− tb 9 Notation 45 24 Computer Input . gentle introduction to learning Calculus Contents p1 9 Notation discovring pi . time. ( ∂ “partial d”) x a small change (increment) in x . The f '' x the second derivative of f x . (Newtonian mechanics) ∂z ∂x the partial derivative of z w.t.r.20 Calculus 20. (Newtonian mechanics) D u the first derivative of u d2 y dx 2 the second derivative of y w.1.r. x . ( f 2 x is also used) v¨ the second derivative of v w. (Euler) v˙ the first derivative of v w.betterexplained.r.t x . time.t. ( “delta”) dy dy of dx dx -----------------------------------------------------------------------------------------------------Integration ∫ the integral sign (Summa) ∫ f x dx the indefinite integral of f x (the anti-differential of f x ) b ∫ f x dx the definite integral of f x from x =a to x=b a the area under f x between x=a and x=b F x the primitive of f x ( ∫ f x dx without the c ) L[ f t] the Laplace operator (with parameter s ) ------------------------------------------------------------------------------------------------------BE .1 Notation for Calculus see also section 9 Differentiation dy dx the first derivative of y where y is a function of x (Leibniz) Also see 11 f ' x the first derivative of f x .t.r.com 46 24 Computer Input . (as above). 20.2 Differential Calculus .358 (2) 271-398] dy or f ' x dx y or f x See 11 .Derivatives dy dx [K Singh (1) pp 258 . 11 ________________________________________________ x n nx MC n−1 sin x cos x cos x −sin x ex ex ln x 1 x ________________________________________________ k 0 k xn k n x n−1 sin (a x) a cos(a x) cos (a x) −a sin (a x) e (a x) a e( a x) ln (a x) a 1 = ax x ________________________________________________ k a xbn k n a a xbn−1 k sin a xb k a cos a xb k cosa xb −k a sin a xb k tan a xb k a sec 2 a xb= k e axb k a e ax b k ln a xb ka a xb ka cos a xb 2 e x gradient slider ________________________________________________ Contents p1 9 Notation 47 24 Computer Input . Further Standard Derivatives dy or f ' x dx y or f x ______________________________________________ f ' x f x ln [ f x] sin−1 x a 1 . 2 a −x 2 1 2 2 2 . y=k x n c dy =k n x n−1 dx y c x -----------------------------------------------------------------------------------------------------Contents p1 9 Notation 48 24 Computer Input . 2 2 sinh a xb a cosha xb cosh a xb a sinh a xb tanh a xb a sech (a x+ b) −1 sinh −1 cosh −1 tanh 2 1 2 x a x a x a x −a x a a . x 2a 2 x 2a 2 _____________________________________________________________ Differentiation as a gradient function (tangent to a curve). 2 2 a – x x 2a 2 x a −1 a2 – x x 2a 2 x a a a x2 −1 cos tan −1 . ... MC dy dx d2 y dx2 + 0 − − – 0 + + 0 + + 0 − ? − ? ------------------------------------------------------------------------------------------------------20...e..... k a constant -------------------------------------------------------Function of a function rule D [ f g x]= f ' g x×g ' x dy dy du = × dx du dx MC -------------------------------------------------------If u and v are functions of x then: D uv= Addition Rule du dv =u ' v ' dx dx .............- D uv=v Product Rule du dv u =v u ' u v ' dx dx MC ........2 Differentiation Rules [K Singh (2) 274–285 (2) 286-302] For D read differentiate D [k f x]=k f ' x .1 Maxima and Minima (Stationary Points) [K Singh (1) 308-335 (2) 327-354] If y= f x then at any turning point or stationary point dy = f ' x=0 dx Determine the nature (max..........................2..20.- du dv –u u dx dx vu ' – uv ' D = = 2 v v v2 Quotient Rule v MC ------------------------------------------------------------------------------------------------------- Contents p1 9 Notation 49 24 Computer Input ......... close to turning point)....... min or saddle) of the turning points by evaluating gradients locally (i.......2. 20.2.3 Formula for the Newton-Raphson Iterative Process [K Singh (1) pp 352 - 356 (2) 389-398] f x =0 with guess value x 0 (from graph) Set ∣ Test for Convergence xn f x 0 f ' ' x 0 f x n [ f ' x 0]2 ∣ 1 see 11 see 9 - modulus f x n f ' x n (where f ' x n ≠0 ) f ' xn x n1= x n – f x =0 when x n1=x n to the precision required. http://archives.math.utk.edu/visual.calculus/3/newton.5/1.html ------------------------------------------------------------------------------------------------------20.2.4 Partial Differentiation [K Singh (1) 695-725, (2) 772-805] If z= f x , y then a small change in x , named x (delta x) and a small change in y , named y etc. will cause a small change in z , named z ∂z ∂z ∂z x y... where is the partial derivative of z ∂x ∂y ∂x ∂z w.r.t. x and is the partial derivative of z w.r.t y . see 9 ∂y such that z ≃ ------------------------------------------------------------------------------------------------------20.2.5 Implicit Differentiation [K Singh (1) 298-306 (2) 315-325] ∂z ∂x If z = f x , y then dy = dx ∂z ∂y Also dy = 1 dx dx dy ------------------------------------------------------------------------------------------------------20.2.6 Parametric Differentiation y=g t dy =g ' t and dt dy dt dx dy g ' t dy f ' t , ≠0 = = or MC dx dx f ' t dt dx dt ______________________________________________________________ If x= f t dx = f ' t dt [K Singh (1) 291-296 (2) 308-315] and Contents p1 9 Notation 50 24 Computer Input 20.3 Integral Calculus - Integrals [K Singh (1) 359-462 (2) 399-512] dy or f x dx ∫ f x dx or F x + c ____________________________________________________ x y or x n1 n1 −cos x sin x ex n sin x cos x ex 1 = x−1 x ln x n≠−1 (when n=−1 ) ____________________________________________________ k kx kx n1 n≠−1 n1 −cos(a x) a sin (a x ) a (ax ) e a k xn sin (a x) cos (a x) e (a x) k =k x−1 x k ln x (where n=−1 ) ___________________________________________________ k a xb k a xbn1 n1a −k cos a xb a k sin a xb a k tan a xb a a xb ke a k ln a xb a n k sin a xb k cos a xb k sec2 a xb k e a xb k a xb n≠−1 n=−1 _____________________________________________________ Contents p1 9 Notation 51 24 Computer Input ∫ Further Standard Integrals dy or f x dx or F x + c ________________________________________________________ dy dx y y or ∫ f x dx f ' x f x ln f x 1 , x 2a 2 2 a −x 1 2 2 a x sin−1 2 ln y x a 1 x tan−1 a a 1 cosh a xb a 1 sinh a xb a 1 tanh a xb a x sinh−1 or ln x x 2a 2 a x cosh−1 or ln x x 2−a 2 a a x 1 x 1 tanh−1 ln or a a 2 a a – x x−a −1 x 1 coth −1 ln or a a 2 a xa sinh a xb cosh a xb sech 2 a xb 1 , x 2a 2 2 x a 1 , x 2a 2 2 2 x −a 1 , x 2a 2 2 2 a −x 1 , x 2a 2 2 2 x −a 2 ∣ ∣ ∣ ∣ ______________________________________________________________ Addition Rule ∫ f x g x dx=∫ f x dx∫ g x dx ------------------------------------------------------------------------------------------------------20.3.1 Integration by Substitution [K Singh (1) 368 (2) 414] ∫ f g x dx ∫ f u du MC where u=g x then x=b Note change of limits ∫ du du = g ' x and dx= g ' x dx u when x=b f g xdx to x=a ∫ f udu u when x=a du is a function of u or du ∈ℝ Notes and exercises -------------------------------------------------------------------------------------------------------20.3.2 Integration by Parts [K Singh (1) 388-395 (2) 432-440] ∫ u dv=u v−∫ v du see 20.6 MC Notes and exercises -----------------------------------------------------------------------------------------------------Contents p1 9 Notation 52 24 Computer Input 20.3.3 Indefinite Integration dy = f x dx dy= f x dx ∫ 1dy=∫ f x dx y=F xc MC -----------------------------------------------------------------------------------------------------y 20.3.4 Area under a Curve y = f(x) - Definite Integration [K Singh (1) 442 (2) 489] b ∫ f x dx a b =[ F x c ] a = F bc – F ac F(b) - F(a) a x b MC, MC http://surendranath.tripod.com/Applets/Math/IntArea/IntAreaApplet.html Hyperlink to interactive demo of areas by integration Procedure y Plot between limits - a and b Check for roots ( R1 , R 2 .. R n ) and evaluate See Newton Raphson 20.2.3 Integrate between left limit, a , and R1 then between R1 and R 2 and so on to last root R n and right limit b Add moduli of areas. (areas all +ve) +ve +ve a R1 -ve R2 b x ------------------------------------------------------------------------------------------------------20.3.5 Mean Value [K Singh (1) p 445 (2) 492] If y= f x then y , the mean (or average) value of y over the interval x =a to x=b is y = f(x) y b y= y 1 ∫ y dx b−a a a x b -------------------------------------------------------20.3.6 Root Mean Square (RMS) b 1 2 y rms= y dx ∫ b−a a where y= f x ------------------------------------------------------------------------------------------------------Contents p1 9 Notation 53 24 Computer Input y) a x b a -----------------------------------------------------------------------------------------------------20.7 Volume of Revolution around the x axis [J Bird 207-208] MC b 2 V = ∫ y dx where y= f x a ------------------------------------------------------------------------------------------------------20.3.3. y = f(x) b b ∫ x y dx x= y a and b ∫ y dx y= 1 2 y dx ∫ 2 a x b y ∫ y dx a Centroid = (x. y .8 Centroid [J Bird 208 .3.9 Partial Fractions [K Singh (1) 396-410 (2) 440-455] f x A B ≡ see 10 xa xb x a x b f x A B C ≡ 2 2 xa xb xa xa xb f x Ax B C ≡ 2 2 x a xb x a x a xb 2 MC ----------------------------------------------------------------------------------------------------- Contents p1 9 Notation 54 24 Computer Input .20.210] The centroid of the area of a lamina bounded by a curve y= f x and limits x =a and x=b has co-ordinates x . 1 Simpson's Rule y w= y = f(x) b−a n yn y1 y2 y3 yn-1 yn x1 x2 x3 a b x xn-1 xn w b w ∫ f x dx≈ Area≈ 3 y 14 y 22 y 3…2 y n−14 y n y n1 a ( n is even) b ∫ f x dx≈ w3 [ firstlast4 ∑ evens 2 ∑ odds ] a Multiplier m Product m y n 1× y 1 1 1 4× y 2 2 4 2× y 3 3 2 . .3. .. .10. . . .3.2 b xn a aw a2w yn y1 y2 y3 Trapezium Method w ∫ f xdx≈ y 12 y 22 y 3. 4 n y n1 1× y n1 1 n1 b Sum = ×w = ÷3 = --------------------------------------------------------- n 20.3.10 Approximation of Definite Integrals [K Singh (1) 434 (2) 481] 20. . . ..10. . . ..20. . y n−1 2× y n−1 .. 2 n−1 yn 4× y n .. .2 y n y n1 2 a ------------------------------------------------------------------------------------------------------- Contents p1 9 Notation 55 24 Computer Input . 4 Laplace Transforms [J Bird 582 – 604] Table of Laplace Transforms L[ f (t )] is defined by ∞ ∫ f t e−st dt 0 f t 1 2 t 3 tn 4 e−a t 5 1−e−a t 6 t e−a t 7 t n e−a t 8 sin (ω t ) 9 cos (ω t ) 10 1−cos(ω t) 11 ω t sin (ω t ) 12 sin (ω t)−ω t cos (ω t ) 13 e−a t sin(ω t ) 14 e− a t cos(ω t) a e−a t (cos (ω t)− ω sin (ω t )) sin (ω t+ ϕ) 16 17 a e−a t + ω sin (ω t)−cos(ω t) Contents p1 9 Notation and is written as F s L[ f t ] 1 15 L [ f (t)] 56 1 L[0]=0 s 1 s2 n! s n 1 1 sa a ssa 1 sa2 n! san1 2 s 2 s 2 2 s ω2 s(s 2+ ω2) 2 2 s s 22 2 23 s 22 2 2 2 see 13 sa sa 2 2 sa s 2 2 sa ssin cos 2 2 s a 22 sa s 2 2 24 Computer Input .20. 1 Diff Eq Second order differential equation: [ ] 2 d y dy L =s 2 L[ y ]– s y0− y ' 0 where y ' 0 is the value of at t =0 2 dt dt MC Efunda Calculator Efunda .1.5 Approximate numerical solution of differential equations [K Singh (1) 630-655 (2) 703-729] and section 26.Laplace ----------------------------------------------------------------------------------------------------- 20.Differential Equations _____________________________________________________________ Contents p1 9 Notation 57 24 Computer Input .L[ f t] 2 s − 2 s 2 2 s − 2 2 sa − sa 2 2 sa − f t 18 sinh (βt ) 19 cosh(βt ) 20 e− a t sinh(β t) 21 e−a t cosh(βt ) First order differential equation: L [ ] dy =s L[ y ] – y 0 where y 0 is the value of y at t=0 dt see also 26. x0 y0 ( y ' )0 ( ) dy dx y 1= y 0+ h( y ' )0 Plot the graph of y against x from values in first 2 columns. y 0 ) is the boundary.1 Eulers’ method y 1= y 0h y ' 0 Range x=ahb 11 where h is the step size a ( = x 0 ) and b are limits and (x 0 .3. and Spreadsheet Method 24.2 – Runge-Kutta.2 -----------------------------------------------------------------------------------------------------See also [K Singh (1) 601-693 (2) 672-771] . See also 24. 4.html --------------------------------------------------------- Contents p1 9 Notation 58 24 Computer Input .657] and next page and 24 and 24.2 a n= b n= 2 T 2 T T 2 ∫ f t cosn t dt n=1. 3… −T 2 T 2 ∫ −T 2 Alternatively written as: f t =a 0c1 sin t 1c 2 sin 2 t 2 …c n sin n t n see 16.6 Fourier Series.2 a 0 constant. 3… f t sin n t dt n=1... 2. c n = a 2nb 2n and α n=tan −1 ( ) an bn f t = constant + first harmonic + second harmonic + .20..falstad.com/fourier/index.3.1 For period T . b n constants b1 sin t b 2 sin 2 t b3 sin 3 t … or ∞ f t =a 0∑ a n cosn t b n sin n t n=1 where a 0= 1 T T 2 ∫ f t dt mean value of f t over period T −T 2 see 20.. 2. See Fourier series applet http://www.. [J Bird pp 611 . (determine from a graph) Fundamental angular frequency = 2 T f t =a 0a 1 cos t a 2 cos 2 t a 3 cos3 t … a n . the smallest period of f t . Calculate Sum Start with 6 terms ( n from 1 to 6) but you may need more.1 Fourier Series .6. and 2 T T -----------------------------------------------------------------------------------------------!! use (type as w ) in input. -----------------------------------------------------------------------------------------------Substitute in the value for w -----------------------------------------------------------------------------------------------Trial plot -----------------------------------------------------------------------------------------------a 0 By observation OR Input 1 f t T For piecewise functions Integrate between −T T and 2 2 as above. T 2 1 .20. Close wxMaxima and start again F6 for text -----------------------------------------------------------------------------------------------Write down the values of T . not a number. Add the parts of a 0 ----------------------------------------------------------------------------------------------Add a 0 to the Sum -----------------------------------------------------------------------------------------------Plot You will have to adjust horizontal range to be able to see the result. ______________________________________________________________ Contents p1 9 Notation 59 24 Computer Input . .wxMaxima method. ------------------------------------------------------------------------------------------------ an 2 f t cosn w t T −T T Integrate between and 2 2 Input For piecewise functions −T T and 0 and 0 and 2 2 or smaller intervals Add the parts of a n --------------------------------------------------- bn 2 f tsin n w t T −T T Integrate between and 2 2 Input For piecewise functions as above Add the parts of b n -----------------------------------------------------------------------------------------------a n cos n w t bn sin n w t Make up the sum -----------------------------------------------------------------------------------------------Sum Calculus. 1 Notation for Statistics n sample size x xi a sample statistic (a data value) OR the variate X a population statistic x the arithmetic mean point of a sample set of data s standard deviation of a sample the mean value of a population standard deviation of a population ∑ the sum of all terms immediately following f frequency Q quartile. Q 1 : the middle value between minimum and Q 2 .21 Statistics [K Singh (1) 726-796 9 (2) 806-887] 21. Lower. Q 2 : the middle value. ----------------------------------------------------------------------------------------------Contents p1 9 Notation 60 24 Computer Input . P= X −x the probability that the population statistic equals the sample statistic x! x× x−1× x−2× x −3×…×1. Q 3 upper) df degrees of freedom n−1 of a sample. x ∈ℕ Range maximum value – minimum value Quartiles in a set of ordered data. ( Q 1 lower.1. Percentile: the k th percentile is in position k 1 ×n . min Q1 Q2 Q3 max Median. Q 3 : the middle value between Q 2 and the maximum. 100 2 Mode in a set of data the mode is the most frequently occurring value. Upper. Q 2 median. . . x = ∑fx ∑f or x= ∑ xi where x i is the variate. . . 2 ∑ f x−x 2= ∑ f xi = ∑ f xi = x = s= n ∑ f x−x 2 = n−1 -------------------------------------------------------Coefficient of Variation s ×100 x of a sample (as a %) ------------------------------------------------------------------------------------------------------ SIR= Semi-interquartile Range Q 3−Q 1 2 ------------------------------------------------------------------------------------------------------ Contents p1 9 Notation 61 24 Computer Input .21.averages -------------------------------------------------------Population Standard Deviation = ∑ x i – x2 = ∑ f d2 ∑f ∑ xi – x2 n d = xi – x -------------------------------------------------------Sample Standard Deviation s= n−1 where n is the sample size -------------------------------------------------------Table for the calculation of Sample Mean and Standard Deviation xi f f xi x− x f x−x . . .2 Statistical Formulae Mean. . . n f is frequency n is the sample size BE . . see 11 and 24.2 T Test 1 sample Standard Error of the Mean SE x= T test (1 sample test) t= s n x− SE x --------------------------------------------------------2 sample for n30 ( d f = n 1n 2 – 2 ) s1 s 2 n1 n2 x − x − 1− 2 t= 1 2 SE x 1− x 2 SE x 1−x 2= Standard Error of Mean T test (2 sample test) --------------------------------------------------------2 sample for n30 2 2 n 1 – 1 s 1n2 – 1 s 2 sp= n1n 2−2 1 2 SE x 1−x 2=s p n1 n 2 Pooled Standard Deviation Standard Error of Mean ------------------------------------------------------------------------------------------------------Contents p1 9 Notation 62 24 Computer Input .the probability of the occurrence of a rare event e − x P X = x= x! Geogbra Poisson slider ------------------------------------------------------------------------------------------------------21.1 Regression Line .2.1 For the line y=ab x where b is the gradient and a is the y intercept and n is the number of pairs of values. a= ∑ y –b∑ x b= n n ∑ xy – ∑ x ∑ y 2 n∑ x – ∑ x 2 -------------------------------------------------------Product moment coefficient of Correlation (r value) r= n ∑ xy – ∑ x ∑ y n∑ x2 – ∑ x 2 n ∑ y 2 − ∑ y 2 −1≤r≤1 ------------------------------------------------------------------------------------------------------ Z= Z Scores x− ------------------------------------------------------------------------------------------------------Poisson Distribution .2.21.3. 8438 0.7794 0.001866 3.8106 0.9793 0.4 0.6844 0.1 2.5478 0.00000854 6.8485 0.9973 0.8907 0.9177 0.9987 0.6591 0.9162 0.6255 0.7157 0.9 1.6 0.9778 0.9382 0.00621 3.9964 0.9564 0.5910 0.5 9.0 6.6700 0.09 0.8599 0.9980 0.8 0.9971 0.9966 0.9418 0.0001591 4.9988 0.9732 0.9854 0.000002112 8.9962 0. http://www.7291 0.21.9929 0.9881 0.00001335 6.9961 0.6 0.6554 0.9699 0.000005413 7.9767 0.9817 0.8315 0.049E-21 These tables are public domain.05 0.2.6331 0.9744 0.9082 0.00007235 4.9890 0.2 0.5 3.7224 0.03 0.7389 0.9099 0.9788 0.9545 0.00820 3.5636 0.8186 0.5 1.6 1.6443 0.9633 0.9878 0.8 1.9970 0.8133 0.9 3.9713 0.5557 0.9525 0.9783 0.ca/~knight/utility/NormTble.9846 0.016E-11 2.129E-19 2.9357 0.7910 0.0001078 4.9808 0.7673 0.00003167 Z 5.7967 0.8997 0.9948 0.867E-7 2.9830 0.8708 0.9032 0.8925 0.9265 0.0 P{Z to ∞ } 0.9279 0.9641 0.7357 0.9985 0.8686 0.7454 0.7704 0.6406 0.8289 0.9953 0.2 0.5596 0.9495 0.9868 0.9292 0.5239 0.6217 0.0003369 4.8790 0.000001300 8.933E-7 9.5714 0.9429 0.3 0.2 1.9901 0.9979 0.5 0.9573 0.9441 0.0006871 4.6179 0.7734 0.9515 0.9956 0.8 2.9974 0.5 1.6141 0.9864 0.9834 0.9969 0.8849 0.9957 0.8869 0.5793 0.5 2.8212 0.9940 0.8413 0.01390 3.9985 0.9649 0.004661 3.01786 3.02275 Z 3.480E-18 2.8980 0.6664 0.4 0.9990 0 Far Right Tail Probabilities 00.9946 0.9984 0.5832 0.9916 0.5120 0.9976 0.9990 0.7190 0.9951 0.9842 0.4 2.9370 0.9943 0.2 0.9857 0.0 1.9955 0.6026 0.8238 0. William Knight Contents p1 9 Notation 63 24 Computer Input .08 z0.5359 0.9236 0.8888 0.1 1.8159 0.9484 0.9222 0.9934 0.7422 0.9941 0.8 7.899E-8 2.9049 0.6808 0.7580 0.0002326 4.02 0.9987 0.7 0.5948 0.9965 0.9066 0.9906 0.8 0.7823 0.8051 0.6103 0.9909 0.9251 0.6517 0.9938 0.3 1.4 1.8264 0.5279 0.9616 0.9893 0.9959 0.9147 0.00004810 4.3 2.9974 0.7939 0.9952 0.06 0.9599 0.9826 0.9505 0.792E-7 9.9535 0.0 1.9693 0.3.8749 0.7324 0.9115 0.0004834 4.9960 0.0 9.8643 0.htm They are produced by APL programs written by the author.9981 0.0 P{Z to ∞ } 0.9706 0.07 0.9972 0.9 4.7 0.9015 0.9925 0.5 0.4 0.1 0.9920 0.7995 0.0 1.9591 0.7881 0.9554 0.9798 0.5000 0.7 2.0 0.9949 0.9345 0.7 0.3 Statistical Tables 21.9821 0.1 0.2 2.8340 0.8729 0.0 P{Z to ∞ } 0.5675 0.7 1.5 4.8621 0.9979 0.9463 0.9963 0.6915 0.1 0.9207 0.9474 0.9911 0.9987 21.7486 0.7764 0.6736 0.9918 0.00002066 5.2 0.0 P{Z to ∞ } 2.6064 0.7517 0.9812 0.9319 0.04 0.8554 0.5871 0.191E-14 2.9608 0.9772 0.5753 0.9936 0.8810 0.2.9726 0.5040 0.8665 0.3 0.9896 0.5517 0.8461 0.9988 0.9875 0.5080 0.2.5987 0.9664 0.002555 3.6368 0.9671 0.280E-12 2.9989 0.8944 0.unb.9968 0.7611 0.9932 0.9887 0.8577 0.001350 Z 4.8531 0.5319 0.7088 0.00 0.9982 0.9394 0.9884 0.9 2.8 0.7019 0.8365 0.1 0.7852 0.5160 0.221E-16 2.0 0.7123 0.9981 0.6 0.5 0.9927 0.01 0.8830 0.9975 0.5398 0.0009676 4.9656 0.2 0.9838 0.7 0.7054 0.7549 0.8508 0.math.866E-10 2.9904 0.9406 0.9986 0.8078 0.8389 0.9913 0.9967 0.5199 0.9861 0.9978 0.4 0.9738 0.6985 0.6 0.9761 0.9931 0.9983 0.9898 0.9750 0.9803 0.0 2.9582 0.6480 0.6628 0.9625 0.6879 0.5438 0.9945 0.9192 0.5 1.1 Normal Distribution Z Probability Content from −∞ to Z Z 0.9131 0.8023 0.3 0.5 0.8962 0.9922 0.9719 0.9850 0.9756 0.9306 0.7257 0.9982 0.9989 0.3.6293 0.6950 0.9678 0.003467 3.6 2.9977 0.7642 0.9871 0.9977 0.9 0.9984 0.6772 0.01072 3.000003398 7.9332 0.9686 0.8770 0.9986 1 Z Z 2.3 0.9452 0.9 0. 807 1.328 1.145 2.021 2.html Contents p1 9 Notation 64 24 Computer Input .edu/vbissonnette/tables/tables.310 1.706 2.761 2.717 2.415 1.860 2.447 3.782 2.262 3.397 1.169 1.980 2.000 2.060 2.761 2.699 2. Bissonnette Reproduced with permission http://facultyweb.895 2.812 2.771 2.684 2.660 1.753 2.383 1.671 2.796 2.725 2.684 2.658 1.364 120 1.860 2.356 1.714 2.567 18 1.998 8 1.729 2.056 2.717 2.341 1.042 2.703 2.639 1.21.032 0.069 2.706 2.878 1.718 12 1.358 140 1.492 25 1.485 26 1.779 1.861 1.2.763 1.374 100 1.337 1.833 2.333 1.01 1.947 1.708 2.476 2.833 2.318 1.berry.660 1.960 2.819 1.319 1.045 2.943 2.499 1.583 17 1.316 1.539 20 1.664 2.403 60 1.821 10 1.746 2.831 1.658 2.797 1.365 6 1.552 19 1.796 2.711 2.645 1.943 3.898 1.327 d f = n 1−1n 2−1=n 1n2 – 2 Copyright (c) 2000 Victor L.895 2.664 1.676 2.311 1.750 1.479 27 1.179 3.787 1.626 1.650 14 1.703 2.015 2.131 2.080 2.729 2.314 1.390 80 1.977 1.740 2.500 24 1.678 1.3 Critical Values of the t Distribution 2-tailed testing 2 sample test 1-tailed testing 0.746 2.086 2.764 11 1.984 2.289 1.282 1.896 9 1.660 2.1 5 2.704 1.363 1.05 0.714 2.701 2.753 2.323 1.462 30 1.697 2.350 1.306 3.617 1.721 2.372 1.303 1.756 1.707 1.009 2.315 1.074 2.015 3.528 21 1.711 2.325 1.467 29 1.571 4.064 2.457 40 1.101 2.771 1.518 22 1.110 2.734 2.3.228 3.473 28 1.602 16 1.012 1.292 1.734 2.721 2.106 1.120 2.676 2.845 1.624 15 1.671 2.681 13 1.330 1.699 2.990 2.093 2.365 3.345 1.725 2.921 1.313 1.143 7 1.508 23 1.701 2.708 2.771 2.055 1.290 1.160 3.250 1.1 0.01 df 0.299 1.423 50 1.697 2.645 2.812 2.740 2.048 2.440 1.321 1.355 1.05 0.576 1.201 3.296 1.052 2.782 2. permutations and combinations Permutations Repetition allowed n P r =nr No repetition n Pr= n! n−r ! order does matter No repetition n Cr= n! r !n−r ! order doesn’t matter Repetition allowed n Cr= nr−1! r !r−1! order doesn’t matter order does matter Combinations ______________________________________________________________ Thanks to Gillian Cunningham.. Contents p1 9 Notation 65 24 Computer Input .2.5 Binomial Theorem MC n x yn= ∑ k=0 x yn= x n n n−k k x y k where n! n = k k ! n – k ! n! n! n! x n−1 y 1 x n−2 y 2.7 % Geogebra Normal Dist slider Geogebra Skewed Dist ---------------------------------------------------------------------------21.2. Aberdeen College..2.6 Permutations and Combinations The number of ways of selecting r objects from a total of n BE .4 Normal Distribution Curve y= 1 2π e −x−μ 2 2 2 ±1sd≈68% ±2sd≈95 % ±3 sd≈99.21. x 1 y n−1 y n 1!n−1! 2 ! n−2! n−1! 1! ------------------------------------------------------------------------------------------------------21. (usually written as r ) d Discount rate (per time period) expressed as a fraction.visual guide to interest rates Efunda Calculator ______________________________________________________________ Contents p1 9 Notation 66 24 Computer Input .22 Financial Mathematics Notation for Financial Mathematics i Interest rate (per time period) expressed as a fraction. n Number of time periods (sometimes written as i ) P Principal A Accrued amount a Amount Sn Sum to the n th term (of a geometric progression) NPV Net Present Value (of an accrued amount) irr Internal Rate of Return (when NPV =0 ) ------------------------------------------------------------------------------------------------------Financial Mathematics Formulae r=1i A= P 1i n a r n – 1 S n= r−1 A= P 1 – d n or a1−r n S n= 1−r a 1 – r−n P= r−1 (annuities) BE . net/wiki/index.sourceforge.23 Recommended Computer Programs wxMaxima free (Open Source) MS Windows and Linux http://wxmaxima.info/forum/default. you do not need to download it although you need to be running Java Runtime Environment (free download). Taylor series.com/manual.geogebra. ----------------------------------------------------------------------------------------------Geogebra free (Open Source) MS Windows and Linux http://www. including differentiation. Maxima can plot functions and data in two and three dimensions. An expression in the algebra window corresponds to an object in the geometry window and vice versa. Also.aspx?g=posts&t=1158 Looks and works like Mathcad. algebra and calculus. You are not allowed implicit multiplication. integration. Mathcad Notes --------------------------------------------------------------------------------------------------SMath free (Closed Source) MS Windows and Linux http://en.e. It is being constantly updated.com/node/18166 (portable application) A open source free download computer algebra system.dk/graph/ --------------------------------------------------------------------------------------------------Casio Calculator Manuals (in pdf format) http://support. Laplace transforms and ordinary differential equations.padowan. http://www. GeoGebra is a dynamic mathematics software that joins geometry. ---------------------------------------------------------------------------------------------------Mathcad ( £1000 approx.casio.24.0 (or later version) http://portableapps.php/Main_Page Windows: download maxima 5.php?rgn=2&cid=004 _______________________________________________________________________ Contents p1 9 Notation 67 24 Computer Input .) MS Windows This is the tool of choice for most engineering mathematics.org This program can be accessed over the web i.smath. 5e 2t3sin 4 typed as 5∗% e ^ 2∗t 3∗sin % pi/ 4 The % sign designates special functions. (numerical values of letters) Maxima is a system for the manipulation of symbolic and numerical expressions. Notes here SMath Primer --------------------------------------------------------------------------------------------------Graph free (Open Source) MS Windows A useful graphing tool which is easy to use. Notes available. 7 ln CTRL g 10 CTRL g*0.5) e x (1) As all programs work in radians by default you must change every input into degrees (if you have to work in degrees).7 sin−1 0. The same is true for e .5 means arcsin(0..5) asin(0.7 asin(0. Most of this also applies to spreadsheets and online maths sites. Back to 2 Web Sites ------------------------------------------------------------------------------------------------------ Contents p1 9 Notation 68 24 Computer Input .5) ^ ^ ^ sqrt() (also on drop down list) \ sqrt( ) n 1 1 ^ ^ x x Calculator toolbar o o 5sin x 30 5*sin(x/180*%pi+ o symbol from 5sin x deg 30 deg 30/180*%pi) drop down list e from drop down list then ^ or exp( ) ln pi 10 pi *0.7 %e^( ) or exp( ) log %pi 10*%pi*0. (2) Also available on toolbars. Calculator key × ÷ x2 Computer (Keyboard) entry Geogebra (3) Mathcad (2) wxMaxima (5) ∗ ∗ (Shift 8) ∗ / / / ^2 ^2 ^2 (Shift 6 then 2) (Shift 6 then 2) x ▄ or ^ or x y or y x x 5sin x o 30o (1) e x ln π 10××0. Spreadsheet programs are not recommended (except for statistical calculations)..24 Computer Input wxMaxima and Geogebra are recommended .5 (5) In wxMaxima typing pi will produce π as a variable NOT 3. (3) Only x allowed as variable (4) See also 17.5) asin(0.1415. 2) Equations.[style. y) dx 20. --------------------------------------------------------24. Paste onto a worksheet ------------------------------------------------------------------------------------------------------ Assign f(x):=3*x (means f x =3x ) w:3. Initial value problem (1) or (2).points]) you can replace points with line. Do not use ∗ ------------------------------------------------------------------------------------------------------Newton Raphson load(newton1) newton f x . x .7 ) ------------------------------------------------------------------------------------------------------- [ ][ ] ⋅ Matrix multiplication Use .1.edu/~aaronts/maximatutorial.x. until outputs are identical to significant figures required --------------------------------------------------------24.24.7 (means w=3.2 Runge-Kutta rk f x .usu. Solve ODE. [ x . x end .8.edu/cee/faculty/gurro/Maxima.1 and then p=0.5 To plot result: wxplot2d([discrete.1 wxMaxima Input Note: From version 0. 3 x is always typed as 3∗x -------------------------------------------------------------------------------------------------------- Zoom in Alt I Insert Text Box CTRL 1 (or F6) Zoom out Alt O -----------------------------------------------------------------------------------------------Copy as an Image to a Edit .4 (2nd page) dy dx typed as d2 y typed as d x2 ‘diff(y. h] where dy = f (x .1 Differential Equations see also 20. Start with precision p=0.1 use Shift+Enter to enter expressions to change behaviour go to Edit: Configure See wxMaxima Introduction at Notes and exercises See http://www.01 etc.neng. p . y .csulb..html but put in expression first! and Maxima by Example http://www. See http://www.hawaii.. %o# is a previous output line.pdf a simple introduction.math. y .Select All Spreadsheet File Right click – Copy as Image. Equations.1. y 0. ------------------------------------------------------------------------------------------------------Contents p1 9 Notation 69 24 Computer Input .x) note the apostrophe ‘ before diff ‘diff(y.%o#]. x 0. x 0.edu/~woollett/ -----------------------------------------------------------------------------------------------Note: Implicit multiplication is NOT allowed. It is also available as a live program .stfx. --------------------------------------------------------Symbolic Maths f x = use Boolean (bold) equals symbolic units Implicit multiplication: This is allowed but only with variables that cannot be confused with units. When editing expressions use the Ins key to change from editing to the left to editing to the right of cursor.ca/bliengme/SMath/SmathPrimer.2 Mathcad Input Applied Maths Definition of variables and functions variable := number and units (:= use colon :) Example: x:3kg will read as x :=3 kg and a:5 m/s^2 as a :=5 m s2 Function f x := function in terms of x Example: f(x): x*a will be interpreted as f x:= x⋅a = gives numerical answer Example f x = will produce the answer 15 N ▄ You can type a different unit in place of the box and the number will change to satisfy the units chosen. For example.pdf This is a truly remarkable program that looks and works like Mathcad and should be of use to engineers in particular. Also see Mathcad Notes _____________________________________________________________ 24.no download required although that particular version may have a few bugs. 3 x is fine but 3 s must be typed as 3∗s .24. Contents p1 9 Notation 70 24 Computer Input .3 SMath This entry will be expanded but at the moment have a look at the SMath Primer by Bernard V Liengme http://people. 24.2. y 0 ) A 1 B C step size h D 2 x y dy/dx y1=y0+hX(dy/dx) 3 x0 y0 f (x 0 . dx step size h and boundary (x 0 .4. HND Electrical Contents p1 9 Notation 71 24 Computer Input .1 (a) Data presented as x y x1 y1 x2 y2 x3 y3 x4 y4 etc etc Basic Procedure:Put x data in column A and y data in column B All data Insert Chart (or chart symbol) XY (Scatter) Give graph and axes titles (remove Legend) As New Sheet (optional) (Excel)) Format Plot Area and Left Click to white (Excel)) Add Trendline Choose most appropriate (best fit to data) Display equation on chart (best fit formula) Display R 2 value on chart ( 0.1. y) .2 _____________________________________________________________ Thanks to Andrew Henderson.95≤R2 ≤1 ) Change type to get R 2 nearer 1 (perfect fit) Highlight Select Chart Type Titles (Chart Location (Right click on plot Right click on data point Type Options Options Rt click on trendline ------------------------------------------------------------------------------------------------------24.4 Spreadsheet procedures 24.5 Runge Kutta 24.4.1 Find the ‘best fit’ formula for a set of data see 11 and 24 and 21. Back to Euler's method 20.2 Euler's Method dy = f ( x . y 0 ) =B3+$C$1*C3 4 =A3+$C$1 =D3 5 Fill down from C3 & D3 to C4 & D4 Fill down all columns from here Format cells to give 5 dp $ signs required to retain absolute column and row Input values in C1 and A3 and B3 and the formula in C3 Plot columns A and B as a scatter graph with column A as x axis. This page is intentionally not blank Contents p1 9 Notation 72 24 Computer Input . This page is intentionally not blank Contents p1 9 Notation 73 24 Computer Input . uk/publications/uncertaintyguide/uncertainty-of-measurement-guides UKAS M3003 http://www. Aberdeen College PID Proportional/Integral/Differential Control: See http://dexautomation. See: http://www.25 Calibration Error Output Output IDEAL Input = Output Zero Error Input Input Output Output Linearity Error Span Error First and Last Values coincide Input Input Output Output Linearity Error Zero-Span Error First and Last Values coincide Input Input ------------------------------------------------------------------------------------------------------Thanks to Olaniyi Olaosebikan.com/library/Technical-Information/Pubs-Technical-Articles/PubsList/M3003.com/pidtutorial.pdf Contents p1 9 Notation 74 24 Computer Input .co.npl.ukas.php Measurement and Uncertainty National Physical Laboratory. 1 0.8 2-4 4 30 69 50 . polypropylene at STP at STP at room temp at STP at room temp at STP at STP at STP Aluminium (Al) 660 2700 Steel (Fe+C) 1525-1540 7850 Copper (Cu) 1085 8920 .01 – 0.wikipedia.1. Material Rubber HDPE Nylon MDF High strength concrete under compression Aluminium Glass Brass / Bronze Titanium alloy Carbon Fibre reinforced plastic (70/30) Steel Silicon Carbide Tungsten Carbide Single-walled carbon nanotube Diamond http://en.org/wiki/Melting_points_of_the_elements_%28data_page%29 26.org/wiki/Density http://en.2 Young's Modulus.1400 Notes at Standard Temperature and Pressure for PETE/ PVC.wikipedia.approximate.wikipedia.650 1000+ 1220 24 Computer Input .8960 Silver (Ag) 962 10500 Lead (Pb) 327 11340 Mercury (Hg) -39 13546 Tungsten (W) 3422 19300 Gold (Au) 1064 19320 http://en.org/wiki/Young%27s_modulus Contents p1 9 Notation 75 Young's Modulus GN/m2 0.1.120 181 200 450 450 .1 Properties of Materials Material Melting point oC Water (Fresh) (H2O) 0 Density kg/m3 1000 Plastics 850 .26 Mechanical Tables 26.90 100 .125 105 . 98 28.2 192.88 50.39 69.00 Nitrogen Oxygen Fluorine 15 16 17 P S Cl 30.01 16. Introductory Chemistry.0 238.99 24.0 200.Ununsept ium ium -ium 68 69 70 Er Tm Yb 167.9 162. Reproduced by permission of Bob Bruner http://bbruner.4 107. most stable.2 Caesium Barium Lanthanum Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead 87 88 **89 104 105 106 107 108 109 110 111 112 113 114 Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg Cn Uut Fl (223) 226.9 183.1 197.3 Xenon 86 Rn (222) Radon 118 Uuo (294) Ununoctium 71 Lu 175.com/ 6. 2004.008 Hydrogen 9 Notation 5A 24 Computer Input 7A 7 8 9 N O F 14.Darm.9 173.01 Lithium Beryllium Boron Carbon 11 12 13 14 Na Mg Al Si 22.2 190.6 204.9 144.91 91.9 Antimony Tellurium Iodine 83 84 85 Bi Po At 209.0 157.27 Periodic Table of The Elements 1A 1 2 3 4 5 6 7 2A 3A 4A 1 3B 4B 5B 6B 7B 8B 8B 8B 1B 2B H 1. 2012 .Ununtrium Flerovium fordium ium stadtium -ium ium 58 59 60 61 62 63 64 65 66 67 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho * Lanthanide 140.0 (209) (210) Bismuth Polonium Astatine 115 116 117 Uup Lv Uus (288) (289) (293) Ununpent.94 55.96 79.8 118.00 19.62 88.97 32.31 26.18 Neon 18 Ar 39.0 (261) (262) (263) (262) (265) (268) (269) (272) (277) (284) (283) Francium Radium Actinium Ruther.69 63.Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium dymium 90 91 92 93 94 95 96 97 98 99 ** Actinide Th Pa U Np Pu Am Cm Bk Cf Es Series 232.periodicvideos.91 95.92 78.80 Krypton 54 Xe 131.08 44.4 152.0 (244) (243) (247) (247) (251) (252) Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium KEY Atomic number Symbol Atomic mass *** Name of the element Contents p1 1 H 1.003 Helium 10 Ne 20.5 164.47 87.Roentgen Copernic.0 Lutetium 100 101 102 103 Fm Md No Lr (257) (258) (259) (260) Fermium Mendelevium Nobelium Lawrencium Based on: Cracolice & Peters.org/ 76 6A June 2.0 237. *** or mass number of most common.008 Hydrogen 3 4 5 6 Li Be B C See the Periodic Table of Videos at http://www.45 Phosphorus Sulphur Chlorine 33 34 35 As Se Br 74.9 178.81 12.2 (145) 150.95 Argon 36 Kr 83.9 106.96 47.941 9.9 186.9 Series Cerium Praseo.93 58.012 10. or first-discovered isotope.10 40.5 180.3 138.9 112.0 Erbium Thulium Ytterbium 8A 2 He 4.94 (98) 101.3 158.00 54.90 Arsenic Selenium Bromine 51 52 53 Sb Te I 121.8 127.09 Sodium Magnesium Aluminium Silicon 19 20 21 22 23 24 25 26 27 28 29 30 31 32 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge 39.7 Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin 55 56 *57 72 73 74 75 76 77 78 79 80 81 82 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb 132.1 102.Dubnium Seaborgium Bohrium Hassium Meitner.22 92.6 126.2 195.4 114. 2/e.94 52. Fall 1997.55 65.0 231. Thanks to a student for starting this.9 137.07 35.3 168.61 Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn 85.Livermor.4 207.1 140.85 58.0 227.72 72. 9×10−9 100×10−9 220×10−9 35000×10−9 Material Vacuum Air Diamond (C) Salt (NaCl) Graphite (C) Silicon (Si) Dielectric Constant r 1 1.9×10−9 17.68 Permeability Values for some Common Materials Material Permeability (H/m) Electrical Steel Ferrite (Nickel Zinc) (Ni Zn) Ferrite (Manganese Zinc) (Mn Zn) Steel Nickel (Ni) Aluminium (Al) 5000×10−6 20 – 800×10−6 800×10−6 875×10−6 125×10−6 1.2×10−9 24.Commonly used prefixes meaning multiple prefix symbol ×1000000000000000 ×1000000000000 ×1000000000 ×1000000 ×1000 ×1 ×1015 ×1012 ×109 ×106 ×103 ×100 Peta kilo P T G M k ÷1000 ×10 −3 milli m ÷1000000 ×10 −6 micro ÷1000000000 ×10−9 nano n ÷1000000000000 ×10−12 pico p Tera Giga Mega 28 Electrical Tables Table of Resistivities Relative Static Permittivity Material Resistivity m at 20o C Silver (Ag) Copper (Cu) Gold (Au) Tungsten (W) Nickel (Ni) Iron (Fe) Lead (Pb) Carbon (C) 15.15 10 .15 11.10 3 .0×10−9 69.26×10−6 Thanks to Satej Shirodkar.5 . Contents p1 9 Notation 77 24 Computer Input .SI Units . Aberdeen College.4×10−9 56.00054 5. angular velocity Pi _____________________________________________________________ This work.org/licenses/by-nc-sa/3. normal stress Tau shear stress. eigenvalues Mu micro (10-6 ). coefficient of friction Nu velocity Xi damping coefficient X Omicron Rho PRODUCT. thermal conductivity. San Francisco... permittivity Zeta impedance. potential energy. angles Psi helix angle (gears). (Calculus) Epsilon linear strain. 171 Second Street. phase difference Omega RESISTANCE.141592654.0 Unported License. Aberdeen College is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0/ (click on icon below) or send a letter to Creative Commons. resistivity Sigma SUM. time constant Upsilon admittance Phi angles. viscosity Theta angles. USA. C= d density. Suite 300.. to be attributed to Peter K Nicol..29 THE GREEK ALPHABET UPPER lower CASE case Pronunciation Examples of use A B E Z H I K M N O P T Alpha angles. 3. temperature. To view a copy of this license. 94105. torque. Contents p1 9 Notation 78 24 Computer Input . volume strain Iota inertia Kappa compressibility Lambda wavelength. angular acceleration Beta angles Gamma shear strain. golden ratio Chi PEARSON’S 2 TEST .. kinematic viscosity Delta DIFFERENTIAL. heat capacity. California.. the change in. visit http://creativecommons. standard deviation. flux. damping ratio Eta efficiency.