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Chastity Bishop
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1 LSM Nov. 00 Cotet List Mathematics (AH). Algebra... kow ad use the otatio!, C r ad r.. kow the results = r r + + = r r r..3 kow Pascal's triagle. Pascal's triagle should be eteded up to = kow ad use the biomial theorem r r ( a + b) = a b, for r, N r= 0 r e.g. epad ( u - 3v ) 5..5 evaluate specific terms i a biomial epasio e.g. 5 i ( + 3) 7 e.g. i epress a proper ratioal fuctio as a sum of partial fractios (deomiator of degree at most 3 ad easily factorised). iclude cases where a improper ratioal fuctio is reduced to a polyomial ad a proper ratioal fuctio by divisio or otherwise.. Differetiatio... kow the meaig of the terms limit, derivative, differetiable at a poit, differetiable o a iterval, derived fuctio, secod derivative... use the otatio: f f dy,,, d d..3 recall the derivatives of 9 d y α α ratioal, si ad cos..4 kow ad use the rules for differetiatig liear sums, products, quotiets ad compositio of fuctios: ( f + g ) = f + g ; ( kf ) kf, = where k is a costat; the chai rule: ( f ( g )) = f ( g ) g ; the product rule: ( f g ) f g f g the quotiet rule:..5 kow = + ; f f g f g = g ( g ) Differetiate give fuctios which require more tha oe applicatio of oe or more of the chai rule, product rule ad the quotiet rule. the derivative of ta the defiitios ad derivatives of sec, cosec ad cot the derivatives of ( ep )..6 kow the defiitio f e ad l ( + ) f h f = lim h 0 h..7 kow the defiitio of higher derivatives f,..8 apply differetiatio to: a) rectiliear motio d y b) etrema of fuctios: the maimum ad miimum values of a cotiuous fuctio f defied o a closed iterval [a, b] ca occur at statioary poits, ed poits or poits where f' is ot defied. c) optimisatio problems d.3 Itegratio.3. kow the meaig of the terms itegrate, itegrable, itegral, idefiite itegral, defiite itegral ad costat of itegratio..3. recall stadard itegrals of α α Q, α, si ad cos ad kow the followig ( ),, af + bg d = a f d + b g d a b R b c b, f d = f d + f d a < c < b a a c b, a b a b f d = f d b a =, where = a f d F b F a F f.3.3 kow the itegrals of e,, sec..3.4 itegrate by substitutio: epressios requirig a simple substitutio cadidates are epected to itegrate simple fuctios o sight epressios where the substitutio will be give e.g. 3 cos si d, u = cos 6 si e.g. d, u = cos - 4 cos the followig special cases of substitutio f f ( a + b) d, d f.3.5 use a elemetary treatmet of the itegral as a limit usig rectagles

2 .3.6 apply itegratio to the evaluatio of areas icludig itegratio with respect to y. Other applicatios may iclude (i) volumes of simple solids of revolutio (disc/washer method) (ii) speed/time graph..4 Properties of Fuctios.4. kow the meaig of the terms fuctio, domai, rage, iverse fuctio, critical poit, statioary poit, poit of ifleio, cocavity, local maima ad miima, global maima ad miima, cotiuous, discotiuous, asymptote..4. determie the domai ad the rage of a fuctio..4.3 use the derivative tests for locatig ad idetifyig statioary poits i.e. cocave up f > 0, cocave dow 0 f <, a ecessary ad sufficiet coditio for a poit of ifleio is a chage i cocavity..5. use of the itroductio of matri ideas to orgaise a system of liear equatios.5. kow the meaig of the terms matri, elemet, row, colum, order of a matri, augmeted matri..5.3 use elemetary row operatios (EROs) Reduce to upper triagular form usig EROs..5.4 solve a 3 3 system of liear equatios usig Gaussia elimiatio o a augmeted matri.5.5 fid the solutio of a system of liear equatios A = b, where A is a square matri, iclude cases of uique solutio, o solutio (icosistecy) ad a ifiite family of solutios.5.6 kow the meaig of the term ill-coditioed.5.7 compare the solutios of related systems of two equatios i two ukows ad recogise ill-coditioig.4.4 sketch the graphs of si, cos, ta, e, l ad their iverse fuctios, simple polyomial fuctios..4.5 kow ad use the relatioship betwee the graph of y = f ad the graphs of y = kf, y = f + k, y = f ( + k ) y = f ( k) where k is a costat..5 Systems of Liear Equatios LSM Nov. 00,,.4.6 kow ad use the relatioship betwee the graph of y = f ad the graphs of y = f, y = f.4.7 give the graph of a fuctio f, sketch the graph of a related fuctio..4.8 determie whether a fuctio is symmetrical, eve or odd or either ad use these properties i graph sketchig..4.9 sketch graphs of real ratioal fuctios usig available iformatio, derived from calculus ad/or algebraic argumets, o zeros, asymptotes (vertical ad o-vertical), critical poits, symmetry.

3 Mathematics (AH).3 Comple Numbers. Further Differetiatio.. kow the derivatives of si, cos, ta.. differetiate ay iverse fuctio usig the techique: = = =,., y f f y f f y etc ad kow the correspodig result dy = d d dy..3 uderstad how a equatio f (, y) = 0 defies y implicitly as oe (or more) fuctio(s) of...4 use implicit differetiatio to fid first ad secod derivatives..5 use logarithmic differetiatio, recogisig whe it is appropriate i eteded products ad quotiets ad idices ivolvig the variable...6 uderstad how a fuctio ca be defied parametrically...7 uderstad simple applicatios of parametrically defied fuctios e.g. + y = r, = r cos θ, y = r si..8 use parametric differetiatio to fid first ad secod derivatives, ad apply to motio i a plae..9 apply differetiatio to related rates i problems where the fuctioal relatioship is give eplicitly or implicitly..0 solve practical related rates by first establishig a fuctioal relatioship betwee appropriate variables θ. Further Itegratio.. kow the itegrals of, ; use the + substitutio = at to itegrate fuctios of the form, a a + itegrate ratioal fuctios, both proper ad improper, by meas of partial fractios; the degree of the deomiator beig 3 ; the deomiator may iclude: (i) (ii) (iii) two separate or repeated liear factors three liear factors with costat umerator ad with o-costat umerator a liear factor ad a irreducible quadratic factor of the form.. itegrate by parts with oe applicatio. + a..3 itegrate by parts ivolvig repeated applicatios..4 kow the defiitio of a differetial equatio ad the meaig of the terms liear, order, geeral solutio, arbitrary costats, particular solutio, iitial coditio..5 solve first order differetial equatios (variables separable)..6 formulate a simple statemet ivolvig rate of chage as a simple separable first order differetial equatio, icludig the fidig of a curve i the plae, give the equatio of the taget at (, y), which passes through a give poit..7 kow the laws of growth ad decay: applicatios i practical cotets.3. kow the defiitio of i as a solutio of + = 0, so that i =.3. kow the defiitio of the set of comple umbers as { :, } C = a + ib a b R.3.3 kow the defiitio of real ad imagiary parts.3.4 kow the terms comple plae, Argad diagram.3.5 plot comple umbers as poits i the comple plae.3.6 perform algebraic operatios o comple umbers: equality (equatig real ad imagiary parts), additio, subtractio, multiplicatio ad divisio.3.7 evaluate the modulus, argumet ad cojugate of comple umbers.3.8 covert betwee Cartesia ad polar form..3.9 kow the fudametal theorem of algebra ad the cojugate roots property.3.0 factorise polyomials with real coefficiets.3. solve simple equatios ivolvig a comple variable by equatig real ad imagiary parts e.g. solve z + i = z +, solve z = z.3. iterpret geometrically certai equatios or iequalities i the comple plae e.g. z = ; z a = b; z = z i ; z a > b.3.3 kow ad use de Moivre's theorem with positive iteger idices ad fractioal idices.3.4 apply de Moivre's theorem to multiple agle trigoometric formulae..3.5 apply de Moivre's theorem to fid th roots of uity. LSM Nov. 00

4 .4 Sequeces ad Series.4. kow the meaig of the terms ifiite sequece, ifiite series, th term, sum to terms (partial sum), limit, sum to ifiity (limit to ifiity of the sequece of partial sums), commo differece, arithmetic sequece, commo ratio, geometric sequece, recurrece relatio.4. kow ad use the formulae u = a + d ad S = [ a + ( ) d ] for the th term ad the sum to terms of a arithmetic series, respectively.4.3 kow ad use the formulae ( r ) u = ar ad a S =, r, for the th term ad the sum to r terms of a geometric series, respectively.4.4 kow ad use the coditio o r for the sum to ifiity to eist a ad the formula S = for the sum to ifiity of a r geometric series where r <.5 Elemetary Number Theory ad Methods of Proof.5. uderstad the ature of mathematical proof.5. uderstad ad make use of the otatios =>, <= ad <=> kow the correspodig termiology implies, implied by, equivalece.5.3 kow the terms atural umber, prime umber, ratioal umber, irratioal umber.5.4 kow ad use the fudametal theorem of arithmetic.5.5 disprove a cojecture by providig a couter-eample.5.6 use proof by cotradictio i simple eamples.5.7 use proof by mathematical iductio i simple eamples.5.8 prove the followig results r = ( + ) r = the biomial theorem for positive itegers; de Moivre's theorem for positive itegers ;.4.5 epad r as a geometric series ad ted to a + b.4.6 kow the sequece + ad its limit..4.7 kow ad use the otatio..4.8 kow the formula r = ( + ) r = simple sums e.g. ad apply it to ar + b = a r + b r = r = r = LSM Nov. 00

5 3. Vectors 3.. kow the meaig of the terms positio vector, uit vector, scalar triple product, vector product, compoets, directio ratios/cosies 3.. calculate scalar ad vector products i three dimesios 3..3 kow that a b = -b a 3..4 fid a b ad a.b c i compoet form 3..5 kow the equatio of a lie i vector form, parametric ad symmetric form 3..6 kow the equatio of a plae i vector form, parametric ad symmetric form, Cartesia form 3..7 fid the equatios of lies ad plaes give suitable defiig iformatio 3..8 fid the agles betwee two lies, two plaes ad betwee a lie ad a plae 3..9 fid the itersectio of two lies, a lie ad a plae ad two or three plaes 3. Matri Algebra 3.. kow the meaig of the terms: matri, elemet, row, colum, order, idetity matri, iverse, determiat, sigular, osigular, traspose 3.. perform matri operatios: additio, subtractio, multiplicatio by a scalar, multiplicatio, establish equality of matrices 3..3 kow the properties of the operatios: A + B = B + A ; AB BA AB C = A BC ; LSM Nov. 00 i geeral; A( B + C) = AB + AC ; ( A ) ( A + B) = A + B ; ( AB) AB = B A ; det AB = det A det B = A ; = B A ; 3..4 calculate the determiat of ad 3 3 matrices 3..5 kow the relatioship of the determiat to ivertability 3..6 fid the iverse of a matri 3..7 fid the iverse, where it eists, of a 3 3 matri by elemetary row operatios 3..8 kow the role of the iverse matri i solvig liear systems 3..9 use matrices to represet geometrical trasformatios i the (, y) plae 3.3 Further Sequeces ad Series 3.3. kow the term power series 3.3. uderstad ad use the Maclauri series: r f = f r = 0 r! ( r) ( 0) fid the Maclauri series of simple fuctios: α e, si, cos, ta,, l kowig their rage of validity + +, fid the Maclauri epasios for simple composites, e.g. e, si e.g. e, e cos use the Maclauri series epasio to fid power series for simple fuctios to a stated umber of terms use iterative schemes of the form = g ( + ), = 0,,,... to solve equatios where g rearragemet of the origial equatio = is a use graphical techiques to locate a approimate solutio kow the coditio for covergece of the sequece give by = g ( ), = 0,,,... + { } 3.4 Further Ordiary Differetial Equatios 3.4. solve first order liear differetial equatios usig the itegratig factor method 3.4. fid geeral solutios ad solve iitial value problems kow the meaig of the terms: secod order liear differetial equatio with costat coefficiets, homogeeous, o-homogeeous, auiliary equatio, complemetary fuctio ad particular itegral solve secod order homogeeous ordiary differetial equatios with costat coefficiets d y dy + + = 0 a b cy d d fid the geeral solutio i the three cases where the roots of the auiliary equatio: (i) (ii) (iii) are real ad distict. coicide (are equal) are comple cojugates solve iitial value problems solve secod order o - homogeeous ordiary differetial equatios with costat coefficiets d y dy a + b + cy = f usig the auiliary d d equatio ad particular itegral method

6 3.5 Further Number Theory ad Further Methods of Proof 3.5. kow the terms ecessary coditio, sufficiet coditio, if ad oly if, coverse, egatio 3.5. use further methods of mathematical proof: some simple eamples ivolvig the atural umbers kow how to epress the g.c.d. as a liear combiatio of the two itegers 3.5. use the divisio algorithm to write itegers i terms of bases other tha direct methods of proof: sums of certai series ad other straightforward results further proof by cotradictio further proof by mathematical iductio prove the followig result r = ( + )( + ); N 6 r = kow the result 3 r = r = apply the above results ad the oe for r = direct methods results cocerig other sums r to prove by. 3 e.g. r ( r + ) = ( + )( + ) r = 4. e.g. ( 3i + ) i= e.g. r r + r + = + r = e.g. r ( r + ) r = kow the divisio algorithm ad proof use Euclid's algorithm to fid the greatest commo divisor (g.c.d.) of two positive itegers. LSM Nov. 00

Higher Course Plan. Calculus and Relationships Expressions and Functions

Higher Course Plan. Calculus and Relationships Expressions and Functions Higher Course Pla Applicatios Calculus ad Relatioships Expressios ad Fuctios Topic 1: The Straight Lie Fid the gradiet of a lie Colliearity Kow the features of gradiets of: parallel lies perpedicular lies