AH Checklist (Uit ) AH Checklist (Uit ) Proof Theory Skill Achieved? Kow that a setece is ay cocateatio of letters or symbols that has a meaig Kow that somethig is true if it appears psychologically covicig accordig to curret kowledge Kow that somethig is false if it is ot true Kow that truth i real-life is ofte time-depedet; for eample, The presidet of America is Roald Reaga was true but is presetly false Kow that truth i mathematics is ot time-depedet Kow that ot every setece is true or false, for eample: Who is that perso? Walk! This setece is false Kow that a statemet (aka propositio) is a setece that is either true or false, for eample: All fish are orage i colour The Milky Way is a galay (F) (T) 4 is a prime umber (F) 6 is divisible by (T) Kow that a compoud statemet is oe obtaied by combiig or more statemets, especially by usig ad or or, for eample: The Milky Way is a galay ad 4 is a prime umber (F) The Milky Way is a galay or 4 is a prime umber (T) Kow that the egatio of a statemet S is the statemet ot S ( S ), ad is such that if S is true, the the egatio is false (or, if S is false, the the egatio is true) Kow that a uiversal statemet is oe that refers to all elemets of a set Kow that a eistetial statemet is oe that refers to the eistece of at least oe elemet of a set M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Kow that a proof is a logically covicig argumet that a give statemet is true Kow that a aiom (aka assumptio or hypothesis or postulate or premise) is a statemet that is take to be true (ot requirig proof) ad used before the ed of a argumet Kow that a coclusio (aka thesis) is a statemet that is reached at the ed of a argumet Kow that the statemet, If A, the B is called a (material) coditioal (aka if, the statemet or coditioal or implicatio) ad writte A B (read, A implies B ); A is called the implicat (aka atecedet) ad B the implicad (aka cosequet) Kow that a coditioal is true ecept whe A is true ad B is false (a true statemet caot imply a false oe) Kow that A ad B are equivalet statemets if A B ad B A, i.e. if A B (read, A if ad oly if B ); the statemet A B is called a bicoditioal or double implicatio Kow that the coverse of the statemet A B is B A Kow that the iverse of the statemet A B is A B Kow that the cotrapositive of the statemet A B is B A ad is equivalet to the statemet A B Kow that a eample (aka istace) is somethig that satisfies a give statemet Kow that if a eistetial statemet is true, the it ca be proved by citig a eample Prove eistetial statemets by citig a eample, such as: N such that + is eve Kow that a coutereample is a eceptio to a proposed statemet Kow that to disprove a statemet meas provig a statemet false Kow that if a uiversal statemet is false, the it ca be disproved by citig a coutereample Disprove a uiversal statemet by fidig a coutereample, such as: + is a multiple of ( N) + + 5 is prime ( N) m divisible by 4 m divisible by 4 ( m N) M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) a + b irratioal ab irratioal ( a, b N) k prime k prime ( k N) Kow that a direct proof is oe where a statemet S is proved by startig with a statemet ad assumptios ad proceedig through a chai of logical steps to reach the coclusio S Use direct proof to prove statemets about odd ad eve umbers, for eample: The square of a eve umber is eve The square of a odd umber is odd The product of a eve ad a odd umber is odd The cube of a odd umber is odd The cube of a odd umber plus the square of a eve umber is odd Use direct proof to prove statemets that ivolve a fiite umber of cases, for eample: N + is eve odd is divisible by 8 Use direct proof to prove other statemets, for eample: + + is always odd Kow that a idirect proof is a proof that ivolves egatig a statemet ad are of types Kow that proof by cotrapositive is a idirect proof techique of provig a coditioal by assumig the egatio of the cosequet ad provig the egatio of the atecedet Prove statemets by cotrapositive, for eample: odd odd Kow that proof by cotradictio is a idirect proof techique of provig a coditioal by assumig the atecedet ad the egatio of the cosequet ad reachig a cotradictio by egatig the atecedet Prove statemets by cotradictio, for eample: is irratioal M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) 5 is irratioal ( ) 4 is irratioal is irratioal + is irratioal a, b R ad a + b is irratioal a or b is irratioal a, b N ad ab is irratioal a + b is irratioal Kow that the techique of proof by mathematical iductio (aka iductio) ivolves provig a statemet, deoted P() regardig the set of all atural umbers (or all atural umbers ecept a fiite subset thereof) Kow that proof by iductio ivolves verifyig (i) the Base case (usually = ) (ii) provig the Iductive step (usig the iductive hypothesis) by verifyig that P() P( + ) Prove statemets by iductio where P() is, for eample: 4 is divisible by ( N) is divisible by 7 ( N) p is odd p is odd ( N) ( + a) + a ( N) > ( N) > ( N) > ( N {,,, 4})! > ( N {,, }) M Patel (August ) 4 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Further Differetiatio Skill Achieved? Differetiate the iverse of a fuctio f usig the formula: D (f ) = ( Df ) f Kow that whe f (), the above formula is writte i Leibiz otatio as: = Differetiate iverse fuctios usig the above formula Kow that: D (si ) = D (cos ) = D (ta ) = + Differetiate fuctios such as: f () = ( + ) ta w () = ta + a () = cos () ta g () = + 4 Kow the meaigs of implicit equatio ad implicit fuctio Kow the meaig of implicit differetiatio Determie whether or ot a give poit lies o a curve defied by a implicit equatio Give the -coordiate of a poit o a curve defied by a implicit equatio, determie the y-coordiate Give the y-coordiate of a poit o a curve defied by a M Patel (August ) 5 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) implicit equatio, determie the -coordiate Fid for implicit equatios such as: y + y y 4y + = y = 4 = cot y y + 4 y + = y 5 l si cos y e + l (7y ) ta e si + cos y + e Work out the secod derivative of a implicitly defied fuctio d y Use implicit differetiatio to fid for implicit equatios such as: y = 4 Use implicit differetiatio to fid the equatio of the taget to a curve writte i implicit form, such as: y + 5 at (, ) y + 4 at = Kow that logarithmic differetiatio ivolves takig the (usually, atural) logarithm of a fuctio ad the differetiatig Use logarithmic differetiatio to differetiate fuctios of the form: M Patel (August ) 6 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) ( + ) ( + ) 4 ( + ) ( 5) (4 + 7) 4 e si e cos si e ( + ) Use logarithmic differetiatio to differetiate fuctios of the form: (si ) π e 5 cos e 4 + si ( + ) + Kow that a curve f () ca be defied parametrically by fuctios, (t) ad y (t), called parametric fuctios (aka parametric equatios) with parameter t Determie whether or ot a poit lies o a parametrically defied curve Give a pair of parametric equatios, kow that: M Patel (August ) 7 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) = dt dt Kow that the above formula is sometimes writte: = y Calculate for parametric fuctios such as: = sec θ, si θ Fid the gradiet or equatio of a taget lie to a parametrically defied curve give a poit o the curve, such as: = t + t, t t + at (, 5) Fid the gradiet or equatio of a taget lie to a parametrically defied curve give a value for the parameter, such as: = 5 cos θ, 5 si θ whe θ = π 4 Give a pair of parametric equatios, kow the formulae for calculatig the secod derivative: d y y y = = d y dt Work out d y for parametric fuctios such as: = cos t, si t Ivestigate poits of ifleio for parametric fuctios such as: = t t, t t, t 5 t, = t M Patel (August ) 8 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Applicatios of Differetiatio Skill Achieved? Kow that plaar motio meas motio i dimesios, described i Cartesia coordiates by fuctios of time (t) ad y (t) (the depedece o t usually beig suppressed) Kow that the displacemet of a particle at time t i a plae is described by the displacemet vector : s (t) def = ( (t), y (t) ) = (t) i + y (t) j Calculate the magitude of displacemet, aka distace (from the origi), usig: s (t) def = + y Kow that the velocity of a particle at time t i a plae is described by the velocity vector : def ds v (t) = = ( (t),y (t ) = (t) i + y (t) j dt Calculate the velocity vector give the displacemet vector Calculate the magitude of velocity, aka speed, at ay istat of time t usig: v (t) def = + y Calculate the directio of motio (aka directio of velocity), θ, at ay istat of time t usig: ta θ = y where θ is the agle betwee i ad v Kow that the acceleratio of a particle at time t i a plae is described by the acceleratio vector : def dv a (t) = = (ẋ (t),ẏ (t ) = ẋ (t) i + ẏ (t) j dt Calculate the acceleratio vector give the velocity vector or displacemet vector M Patel (August ) 9 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Calculate the magitude of acceleratio usig: a (t) def = + y Calculate the directio of acceleratio, η, at ay istat of time t usig: ta η = y where η is the agle betwee ẋ i ad a Kow that related rates of chage refers to whe y is a fuctio of ad both ad y are each fuctios of a third variable u Kow that related rates of chage are liked via the chai rule: = du du = du du Kow that related rates of chage problems may ivolve use of: = Solve problems ivolvig related rates of chage, for eample, if a spherical balloo is iflated at a costat rate of 4 cubic cetimetres per secod, fid (i) the rate at which the radius is icreasig whe the radius is 8 cm (ii) the rate at which the radius is icreasig after 5 secods Kow that i related rates of chage problems, the relatioship betwee ad y may be a implicit oe M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Further Itegratio Skill Achieved? Kow the stadard itegrals: = a = a si a + C = = ta a + + C a + a a Kow the special cases of the above stadard itegrals: = si + C = ta + C + Kow that ratioal fuctios ca be itegrated usig the techique of itegratio by partial fractios Use itegratio by partial fractios to fid or evaluate itegrals such as: 6 k + 6 4 + 6 9 6 ( 6) 4 M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) + ( + 5) + 5 ( + )( + )( + ) Kow that itegratio by parts is the techique that is used to itegrate some products of fuctios Kow the itegratio by parts formula : u ( D v ) = u v ( D u ) v Kow that itegratio by parts ivolves differetiatig oe fuctio, u, ad itegratig the other, Dv Use itegratio by parts to fid or evaluate itegrals such as: π 4 si 4 l ( + ) e si π 4 4 cos ta M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) l ta Use a double applicatio of itegratio by parts to fid itegrals, for eample: si 8 si 4 cos Use itegratio by parts to fid itegrals by a cyclical procedure, for eample: e si e cos Kow the meaig of reductio formula Obtai reductio formulae for itegrals, for eample: a e I = a a e a I (a R {}) e I = e + I si I = si cos + I cos I = cos si + I Obtai the geeral solutio of simple differetial equatios such as: M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) = 5 + = si ( N) = 7 4 = cot 5 Obtai particular solutios of simple differetial equatios, give iitial coditios Solve simple differetial equatios i practical cotets Kow that a separable differetial equatio is oe that ca be writte i the form: = f () g (y) Obtai the geeral solutio of separable DEs, for eample: dm dt = km = y dv dt = V ( V ) y = 4 dg dt = 5k G 5 (k costat) e y = = ( + y) + Give iitial coditios, obtai a particular solutio of a separable DE M Patel (August ) 4 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Comple Numbers Skill Achieved? Kow that a comple umber is a umber of the form (aka Cartesia form), z = + iy where, y R ad i = ; is called the real part of z (Re(z)) ad y the imagiary part of z (Im(z)) Kow that the set of all comple umbers is defied as: C def = { + i y :, y R, i = } Kow that comple umbers are equal if their real parts are equal ad their imagiary parts are equal, ad coversely Kow that: a = i a Solve ay quadratic equatio, for eample: z z + 5 = Add or subtract comple umbers by addig or subtractig the correspodig real parts ad the correspodig imagiary parts: (a + ib ) ± (c + id ) = (a ± c) + i (b ± d ) Multiply comple umbers accordig to the rule: (a + ib ) (c + id ) = (ac bd ) + i (ad + bc ) Kow that the comple cojugate of z = + iy is defied as: z def = iy Kow that the comple cojugate satisfies: z z = + y Use the comple cojugate to divide ay comple umbers Calculate the square root of ay comple umber Kow that a comple umber z ca be represeted as a poit P (or coordiate or vector) i the comple plae (aka Argad plae); with P plotted, the result is called a Argad diagram (aka Wessel diagram) Kow that the horizotal ais i the comple plae is called the M Patel (August ) 5 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) real ais, while the vertical ais is called the imagiary ais Plot a comple umber writte i Cartesia form i the Argad plae Plot the comple cojugate of a give comple umber i Cartesia form i the Argad plae Kow that the modulus of z is the distace from the origi to P ad defied by: r z def = + y Calculate the modulus of ay comple umber Kow that the agle i the iterval ( π, π] from the positive ais to the ray joiig the origi to P is called the (pricipal) argumet of z ad defied by: def y θ arg z = ta Kow that a comple umber has ifiitely may argumets, but oly pricipal argumet, the types of argumet beig related by: Arg z def = { arg z + π : Z } Calculate a argumet ad the pricipal argumet of ay comple umber Kow that, from a Argad diagram: = r cos θ, r si θ Kow that a comple umber ca be writte i polar form : z = r (cos θ + i si θ ) r cis θ Plot a comple umber writte i polar form i the comple plae Give a comple umber i Cartesia form, write it i polar form Give a comple umber i polar form, write it i Cartesia form Idetify, describe ad sketch loci i the comple plae, for eample: z = 6 z 4 z = z + i = M Patel (August ) 6 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) z + 4i = z = z + i π arg z = Kow that whe multiplyig comple umbers z ad w, the followig results hold: zw = z w, Arg zw = Arg z + Arg w Kow that whe dividig comple umbers z ad w, the followig results hold: z w = z w, Arg z w = Arg z Arg w Multiply ad divide or more comple umbers i polar form usig the above rules for the modulus ad argumet Kow de Moivre s Theorem (for k R): z = r (cos θ + i si θ ) k z = k r (cos kθ + i si kθ ) Use de Moivre s Theorem to evaluate powers of a comple umber writte i polar form, writig the aswer i Cartesia form By cosiderig the biomial epasio of (cos θ + i si θ ) ( N), use de Moivre s Theorem to obtai epressios for cos θ ad si θ, i particular, cos θ, si θ, cos 4θ ad si 4θ Kow that if w = r (cos θ + i si θ ), the the solutios of the equatio z = w are give by: z = k r cos θ + πk θ + πk + si (k =,,,, ) Kow that plottig the solutios z = w with w give as above o a Argad diagram illustrates that the roots are equally spaced o a circle cetre (, ), radius r, with the agle betwee ay cosecutive solutios beig π Fid (ad plot) the roots of ay comple umber usig the above formula M Patel (August ) 7 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Kow that solvig the equatio z = gives the roots of uity Kow that the roots of uity ( > ) satisfy the equatio: zk k = = Verify that a give comple umber is a root of a cubic or quartic Kow that a repeated root (occurrig m times) of a polyomial p is called a root of multiplicity m Kow that the Fudametal Theorem of Algebra states that every (o-costat) polyomial with comple coefficiets has at least oe comple root Kow that the Fudametal Theorem of Algebra implies that every polyomial of degree ( ) with comple coefficiets has eactly comple roots (icludig multiplicities) Kow that a polyomial p of degree at least with comple coefficiets ca be factorised ito a product of liear factors: p (z) = r = ( z z ) Kow that if a polyomial p of degree with all coefficiets real has a o-real root, the the cojugate of this root is also a root of p Kow that a polyomial of degree with all coefficiets real ca be factorised ito a product of real liear factors ad real irreducible quadratic factors: p (z) = t ( z d ) r r = ( t )/ s = r a z b z c ( + + ) s s s Solve cubic ad quartic equatios which have all coefficiets real, for eample: 4 z + 4z + z + 4z + = z + z 5z + 5 = z 8z + 8 = Factorise a cubic or quartic ito a product of liear factors Factorise a polyomial with all coefficiets real ito a product of real liear factors ad real irreducible quadratic factors M Patel (August ) 8 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) Sequeces ad Series Skill Achieved? Kow that a series (aka ifiite series) is the terms of a sequece added together Kow that the sum to terms (aka sum of the first terms aka th partial sum) of a sequece is: S def = = r Calculate the sum to terms of a give sequece Give a formula for S, calculate u, u etc. usig the prescriptio: u r u = S + Kow that the sum to ifiity (aka ifiite sum) of a sequece is the limit (if it eists) as of the th partial sums, i.e. : S S def = lim S Kow that a ifiite series coverges (aka is summable) if S eists; otherwise, the series diverges Kow that ot every sequece has a sum to ifiity Kow that the symbol a is traditioally used to deote the first term of a sequece Kow that a arithmetic sequece is oe i which the differece of ay successive terms is the same, this latter beig called the commo differece (d) Show that a give sequece of umbers or epressios forms a arithmetic sequece Kow that the th term of a arithmetic sequece is give by: u = a + ( ) d (a R, d R {}) Give a, ad d for a arithmetic sequece, calculate u Give a, ad u for a arithmetic sequece, calculate d Give a, u ad d for a arithmetic sequece, calculate Give u, ad d for a arithmetic sequece, calculate a Give specific terms of a arithmetic sequece, fid the first term ad commo differece Kow that the sum to terms of a arithmetic series is give by: M Patel (August ) 9 St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) S = ( a + ( ) d ) Give a, ad d for a arithmetic sequece, calculate S Give a, ad S for a arithmetic sequece, calculate d Give a, S ad d for a arithmetic sequece, calculate Give S, ad d for a arithmetic sequece, calculate a Obtai sums of arithmetic series, such as: 8 + + 4 + + 56 Kow that the sum to terms of a arithmetic sequece ca always be writte i the form: S = P + Q (P R {}, Q R) Give a formula for S for a arithmetic series, calculate u, u etc. Kow that o arithmetic series has a sum to ifiity Solve cotetual problems ivolvig arithmetic sequeces ad series Kow that a geometric sequece is oe i which the ratio of ay successive terms is the same, this latter beig called the commo ratio (r) Kow that the th term of a geometric sequece is give by: u = a r (a R {}, r R {, }) Show that a give sequece of umbers or epressios forms a arithmetic sequece Give a, ad r for a geometric sequece, calculate u Give a, ad u for a geometric sequece, calculate r Give a, u ad r for a geometric sequece, calculate Give u, ad r for a geometric sequece, calculate a Give specific terms of a geometric sequece, fid the first term ad commo ratio Kow that the sum to terms of a geometric series is give by: a ( r ) S = r Give a, ad r for a geometric sequece, calculate S Give S, ad r for a geometric sequece, calculate a Give a, S ad r for a geometric sequece, calculate Give a, ad S for a geometric sequece, calculate r Obtai sums of geometric series, such as: M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) 5 + 8 (to 8 terms) Kow that a geometric series may or may ot have a sum to ifiity Kow that S eists for a geometric series if r < Kow that the sum to ifiity of a geometric series is give by: S = a r Give a ad r for a geometric sequece, calculate S Give S ad r for a geometric sequece, calculate a Give S ad a for a geometric sequece, calculate r Epress a recurrig decimal as a geometric series ad as a fractio Kow that a power series is a epressio of the form: i a i i = = a + a + a + a + Kow that if <, the ( a i R) ( ) = = + + + + def = Epad as a power series other reciprocals of biomial epressios, statig the rage of values for which the epasio is valid, for eample: i = i + = + + = i = i i ( ) = + + 4 + 6 + = i = i + = + 9 7 + = i = i i i ( ) Epad as a geometric series (statig the rage of validity of the epasio), more complicated reciprocals of biomials, for eample: ( + 4) (si cos ) M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) (cos ) Kow that: e def = lim + = +! Kow that: +! + = b = b! e def = lim + Fid other limits usig the above limit, for eample: lim + 5 Kow that if a power series teds to a limit, the the power series ca be differetiated ad the differetiated series teds to the derivative of the limit Differetiate a power series ad fid a formula for the limit, for eample: = 5 e d ( + + ) = + Kow that if a power series teds to a limit, the the power series ca be itegrated ad the itegrated series teds to the itegral of the limit Itegrate a power series ad fid a formula for the limit, for eample: ( + + ) = l ( + ) Solve cotetual problems ivolvig arithmetic sequeces ad series Kow the results: = = r r r = = ( + ) Eplore other results such as: M Patel (August ) St. Machar Academy
AH Checklist (Uit ) AH Checklist (Uit ) r r = = r r = = 4 ( + ) ( + ) 6 ( + ) Evaluate fiite sums usig a combiatio of the above fiite sums, for istace: k = ( k ) r = (4 6 r) Evaluate fiite sums that do t start at, for eample: k = (7 k ) 7 r = 5 ( r + ) M Patel (August ) St. Machar Academy