Mathematics Extension 1 Based on 1983 Syllabus

Transcription

1 HS Mathematics Etesio Mathematics Etesio Based o 98 Syllabus This summay has bee witte to be as shot as possible, to cove eveythig if you aleady have a good idea of the couse cotets, oly eed a quic evisio ad does ot wat to pit too may pages The depth of coveage of a paticula topic does ot ecessaily coespod to its sigificace Poofs have bee give whe they ca be easily deived, i case you foget the fomulae Impotat esults have bee boed fo coveiece The majoity of this wo is based o the ambidge Mathematics tetboo Sectios ae umbeed accodig to the syllabus This documet was last updated o Septembe Basic Aithmetic ad Algeba Iequality with a vaiable i the deomiato: This ca be doe best by multiplyig the iequality by a quadatic facto (eg ) so that you do t eed to woy about chagig the iequality sig Setch the ew fuctio ad obtai the solutios off the gaph hec the solutio bac to the oigial iequality, i case it maes the deomiato zeo (paticulaly whe it s o ) Sample questio: Fid the solutios to 7 Multiplyig both sides by Daw 7, y, ad y fo o But so solutios ae o Note o logaithmic ad epoetial iequality: The iequality symbol is uchaged whe movig betwee epoetial ad logaithmic statemets oly if the base is geate tha Limit: Sample questio: lim lim (divide top ad bottom by ) Plae Geomety icle Geomety The followig poits, theoems ad thei coveses (if ay) should be oted: chod of equal legths (ad hece equal acs) subted equal agles at the cete equal chods ae equidistat fom the cete if ay two ae tue, the othe is tue: a lie is pepedicula to a chod, that lie passes the cete of cicle, that lie bisects the chod a cicle is detemied by thee distict poits a agle i a semicicle (subtedig the diamete) is a ight agle 6 the agle at the cete subtedig a ac (o chod) is twice ay agle at the cicumfeece subtedig the same ac 7 two o moe agles at the cicumfeece stadig o the same o equal acs ae equal 8 opposite agles of a cyclic quadilateal ae supplemetay (also, eteio agle is equal to the opposite iteio agle) 9 a taget is pepedicula to the adius at its poit of cotact two tagets fom a eteal poit have equal legths whe two cicles touch, the two cetes ad the poit of cotact ae colliea the agle betwee a taget to a cicle ad a chod at the poit of cotact is equal to ay agle i the alteate segmet if two chods AB ad PQ itesect at M, AM MB = PM MQ give two secats (which mea chods eteded to outside) AB ad PQ meet at M outside the cicle, AM MB = PM MQ (If AB taget, poits A = B ad AB = PM MQ) Page of 8

2 HS Mathematics Etesio Tests fo cyclic quadilateal: use eithe popety #7 o #8 Diect ad idiect commo tagets: diect if both cicles ae o the same side of the taget Real Fuctios of a Real Vaiable ad Thei Geometical Repesetatio Fidig asymptotes: Divisio by zeo causes vetical asymptote To fid hoizotal ad oblique asymptotes, fid the value of y as appoaches ifiity If this limit is a umbe, the cuve has y as a hoizotal asymptote If this limit appoaches ifiity, it pobably has a 7 oblique asymptote Sample questio: Fid the asymptote(s) of y Fist we eed to divide o simplify it: As, y (sice ) So oblique asymptote is y Thee is a vetical asymptote at sice hee the 7 deomiato i y is zeo Tigoometic Ratios Review ad Some Pelimiay Results ompoud-agle fomulae si si cos cos si ta ta ta cos cos cos si si ta ta These lead to double-agle fomulae si si cos, ta ta ta cos cos si cos si [use Pythagoea idetity] This also eables us to epess si ad cos i tems of cos (eg fo itegatio): si cos ad cos cos The t-fomulae, t ta : t t t ta, si ad cos t t t [Hit: The epessio fo ta simply comes fom the double-agle fomula You ca daw a si t tiagle to fid the si ad cos, o you ca thi of ta ad associate t (the cos t top pat) with si ad t (the bottom pat) with cos The put t as deomiato fo both si ad cos epessios] Geeal Solutios (thee ae seveal ways to wite these) The geeal solutio of si si is OR, iteges The geeal solutio of cos cos is, iteges The geeal solutio of ta ta is, iteges Istead of memoisig, you ca also use the AST diagam to help you Fo eample, fo ta ta it ca be o (same sig i quadats ad ) so combiig it s Fo si si it s i quadats ad, ad so o Moe elegat fom fo si si : Thee-dimesioal Tigoomety: some geeal otes Tiagles ca be iclied o positioed stagely Idetify all tiagles ad see if you ca solve fo a legth o a agle If you ca t, assig a poumeal if you thi the legth o the agle is elevat You may be able to solve equatios (obtaied fom tiagles) simultaeously Loo fo ight agles, because you ca apply tig atios ad Pythagoas theoem to the tiagles Fo othe tiagles you have sie ule ad cosie ule Page of 8

3 HS Mathematics Etesio Mae sue you udestad the cocept of agle betwee two lies, agle betwee a lie ad a plae, ad agle betwee two plaes Remembe it s a -dimesioal questio Revise o tue beaig ad compass beaig 6 Liea Fuctios ad Lies Agle betwee Two Lies: m m ta OR ta if oe lie is vetical m m m Note: agle betwee two cuves is the agle betwee thei tagets Deivatio: daw two lies maig agles ad with +ve -ais ta ta Ratio Divisio Fomula l ly y Whe PQ is divided by M (, y) i some atio : l, ad y l l Eteal Divisio M divides PQ eteally i a atio of : meas that M is at oe of the eds of the lie ad that PM : MQ (ot PQ) = : Without witig the wod eteal, the same situatio ca be epeseted as M divides PQ i a atio of - :, o equivaletly : - Put this egative value i the atio divisio fomula (does t matte which oe is egative) If you foget the fomula: ty to daw a pictue ad decide which poit is o the left, top, etc Suppose that M is i the middle, P ad Q at the edpoits You ca mae two ight-agled tiagles, oe has hypoteuse PQ ad the othe has hypoteuse PM o QM Sice they e simila, thei sides ae popotioal ad usig this you ca fid uow coodiates 7 Seies ad Applicatios Poof by Mathematical Iductio Povig LHS > RHS: pove LHS RHS > (eg by usig the fact that ) Povig A is divisible by B: pove A = B, fo some itege Below ae some eamples of acceptable aswe fomats Sample questio #: Pove that fo all iteges, [Pove fo the iitial value, i this case, = ] Whe =, RHS LHS So the statemet is tue fo = Let be a itege fo which the statemet is tue That is, suppose (**) Now, let s pove the statemet fo = + That is, we pove LHS = = =, by the iductio hypothesis (**) RHS Wheeve it s tue fo =, it s also tue fo = + But it s tue fo = So the statemet is tue fo =,,, Sample questio #: Show that l! fo all iteges 6 Let the statemet S() be defied by l!, 6,7,8, lealy, S(6) is tue sice l 6! l! Suppose S() is tue oside S(+), equivalet to LHS = l! l l! l! l l! if S() is tue ad l fo 6 So, LHS > Page of 8

4 HS Mathematics Etesio Hece, S() tue implies S(+) tue But S(6) is tue Hece S() is tue fo 6,7,8, Sample questio #: Show that fo ay positive itege, 6 is divisible by which is divisible by Whe =, LHS = If the statemet is tue fo =, 6 M fo some itege M Fo = +, the umbe is 6 M (if it s tue fo = ) 6M M 6M N, N itegal Hece, if it s tue fo = it s also tue fo = + But it s tue fo = Hece it s tue fo all positive iteges Fial Note: Basically you eed to use the esult fom the iductio hypothesis fo = 9 The Quadatic Polyomial ad the Paabola Paametic epesetatio It is ofte useful to peset a fuctio o elatio i aothe way, such as by elatig ad y to aothe vaiable (called the paamete) Mathematicias ca choose a paametic fom so that the paamete has some geometical meaig athe tha some useless cap Obtaiig atesia equatio fom paametic equatios: I the followig eamples, the paamete does ot ecessaily have a ice meaig because they ae just adom eamples Sample questio #: Give cos --() ad y si --(), fid the atesia equatio Appoach #: substitutio of the paamete Fom (), cos Substitutig ito (), y si cos Daw a tiagle with adjacet = ad hypoteuse = the fid the opposite side usig Pythagoas theoem Sie = opposite / hypoteuse ad thee you go Appoach #: additio We ca use the fact that si cos () : cos cos ; () : y si y si Addig the two, y Sample questio #: Fid the atesia equatio fom t --() ad y t --() t t Appoach: substitutio of a epessio Squaig (), t t t t Puttig ito (), y The atesia equatio may have estictios which follow fom the paametic equatios Sample questio #: ompae the cuves of t, y t to t, y t They both have y as thei atesia equatio Howeve, the latte oe has estictios of ad y afte looig at Paametic diffeetiatio Use the chai ule Suppose the paamete is t Paabola ay t ad t dy d Stadad paametic fom: at ad y at the (taget to the) paabola at that poit at P, y a y y y, t dy dt dt d dt, d d is flipped upside dow dt, t has a meaig of beig the gadiet of Taget: hod of cotact of tagets fom a outside poit: fom P, y ay y oditios fo focal chod Fo all the fou stadad paabolas metioed above, PQ is a focal chod (that is, passes the focus) oly if pq (with p beig the paamete of poit P) osequetly, the tagets fom P ad Q meet at ight agles o the diecti Page of 8

5 HS Mathematics Etesio Sample questio #: Fid the equatio of the omal to the paabola ay ap, ap Because P is sot of movable you may cofuse epessios uique to a poit with geeal a epessios Fo eample, diffeetiatig y you may the thi that m N ad the a a a a omal is y ap ap This is wog m N at P is ot, ad ap Sample questio #: The poits P ap, ap ad Q aq, aq ae o the paabola ay ad p q M is the midpoit of PQ Fid the locus of M M ap aq a p q a vetical lie y M a at P ap aq ap q a p q pq a apq a ap p This y M has estictio By diffeetiatio, we have a miimum TP whe p, ym a Hece, the locus of M is the lie a, y a Itegatio Itegatio by substitutio (o a chage of vaiable ) This method wos quite similaly with the evese chai ule, but is a etesio of it Basically, i the evese chai ule we do t wite the vaiable u i the mai pat of ou woig, but i this method we do You will be told the equied substitutio [let u o ] uless it is vey easy to see (fo eample, you have a fuctio et to its deivative lie whe you use the evese chai ule) The followig eample is ot i the easy categoy Type #: let u be some fuctio of Type #: let be some fuctio of u Sample questio #: Fid d usig the substitutio u Solutio: d u u du u u u du lie(*) du side woig : lie() let u du lie() the d lie() which meas du d u u lie() also, because u, lie() u Sample questio #: Fid d usig the substitutio u side woig : Solutio: d u u u u u u u du du lie() let d lie() the u du lie() which meas d u du lie() u fom u, lie() we have u Page of 8

6 HS Mathematics Etesio O the ight-had colum, I label a few lies You ca choose to wite eithe lie() o lie(), ad you ca sip lie() O the left-had colum of sample questio #, I label lie(*); it is ecommeded that you leave out this lie I both types, we wat to eplace all -lettes by u befoe itegatig [which is with espect to u] Afte we ve foud the itegal (i u) the we covet it bac to the epessio The above eample gives a good illustatio of how eithe type ca ofte be used, although oe may be pefeed to the othe Sample questio #: Fid d u d let Solutio: u d u du d usig the substitutio u side woig : u the du we eplace d by du o the left I type #, i the mai woig [left-had colum], we eplace d by somethig du I type #, we usually eplace somethig d by du But if you ca t see how you would fit du ito the itegad, you ca just eplace d by somethig du I the ase of Defiite Itegals We ca chage the limits of the itegatio to avoid havig to put bac the -tems Let s use sample questio # as ou illustatio Note o the Limits The bigge value of the ew limits ( i this eample) does ot have to be o the top It wos based o coespodece: = coespods to u = etc You should chage the limits ( ad - to ad ) as soo as you have du i the itegal [eve though you still have thee] If you pefe ot to chage the limits (so you e puttig bac the epessios), you must ot wite d u du o u u du Do the idefiite itegal, ad the at the ed wite: the defiite itegal = The Tigoometic Fuctio u 7 d Deivatives of the Si Tigoometic Fuctios d d si cos cos si d d d d sec sec ta cos ec cos ec cot d d d d ta sec cot cos ec d d The deivatives of co- fuctios have egative sigs They [the deivatives of the cofuctios] loo simila to that of its coespodig o-co- fuctios (fo eample, cosec coespods to sec, cot coespods to ta), which ae all o table of stadad itegals Pimitives [o Itegals] of ta ad cot u du u side woig : lie(6) at, u lie(7) at, u Page 6 of 8

7 HS Mathematics Etesio si ta d d logcos [umeato is a deivative of deomiato] cos Pimitives of sec ad cos ec ae beyod the scope of this couse Pimitives of squaes of tig fuctios: The pimitives of sec ad cos ec ae obvious fom the table of stadad itegals Now fo othe fuctios: Wite cos as cos, similaly si cos [fom double agle idetities] Wite ta as sec [Pythagoea idetity], do a simila thig fo cot The Sum of Sie ad osie Fuctios Epessed as Oe Fuctio Ay fuctio f asi bcos ca be witte i ay of these fou foms: y Rsi y Rsi, y Rcos, ad y Rcos,, R, 6, is called the auiliay o subsidiay agle To do that, epad the auiliay agle fom usig the compoud-agle fomula, ad equate the coefficiets of si ad cos Squae ad add the esultig two equatios to fid R ad to detemie the quadat of The fid usig eithe equatio Sample questio: Wite si cos i y Rsi fom Let si cos Rsi si cos Rsi cos cos si Equatig coefficiets, R cos (eq) ad R si (eq) (eq) + (eq) : R cos si R sice cos si Fom (eq) ad (eq), cos is +ve ad si is ve So is i quadat, 97 Solvig asi bcos c : you ca ewite this i the auiliay agle fom ad solve it Alteatively, use the t-fomulae, but be caeful because 8 will eve come up as a solutio fom t whe it is (due to ta beig udefied whe 8 ) Applicatios of alculus to the Physical Wold Related Rates of hage ad the hai Rule of Diffeetiatio This sectio coces situatios whee we have two elated quatities ad thee vaiables, oe of which is time Oe of the two quatities usually chages at a fied ate Questios ofte ivolve geometical objects such as a ectagle o a sphee, so evise fomulas o them dy dy du We use the chai ule: ; ad some volume ad aea fomulae dt du dt Nomally we put egative sig i the ates which ae deceasig Also mae sue that you use the coect uits Sometimes you eed to flip a deivative upside dow Sample questio #: A spheical balloo is pumped so that its adius iceases at a fied ate of m/s Show that the ate of chage of volume is Aswe: Idetify what we eed, dv dv dv which is So, just wite dt dt fist Now let s loo at what we ve got We have dt d so the is d dt Sample questio #: A obseve at A is watchig a plae at P fly ovehead, ad he tilts his head so that he is always looig diectly at the plae The aicaft is flyig at 6 m/h towads A at a altitude of m Let be the agle of elevatio of the plae fom the obseve, ad suppose that the distace fom A to B, diectly below the aicaft, is m (a) With the aid of a diagam, show that ta Hece show that d d si (b) Fid the ate at which the obseve s head is tiltig whe 6 (i degees pe secod) Page 7 of 8

8 HS Mathematics Etesio Fo (b), d d d dt d dt d 6m, d si 6 ad dt h d m d ad 8 deg deg dt h 6 s s Natual Gowth ad Decay with a Limitig Value I may situatios, the ate of chage of a quatity is popotioal to how much that quatity diffes fom some fied value Fo eample, the ate at which a hot coffee cools dow is popotioal to how hot it is compaed to the suoudig ai dp t P B ad so P B A e whee It s decay whe the value of A is positive, dt ad vice vesa Questios usually give dp dt ad P ad as you to veify them by diffeetiatig the P epessio Typical questio is as follow The tempeatue of a body is chagig at the ate dt dt T, whee T is the tempeatue at time t miutes ad is a positive costat The tempeatue of the suoudig is o The iitial tempeatue of the body is 6 o ad it falls to o i miutes t (i) Show that T Ae, whee A is a costat, is a coect fuctio t t Diffeetiatig, dt dt A e A e T (ii) Fid how log, coect to the eaest secod, it will tae the tempeatue to fall to 7 o You eed to fid A ad fist, by substitutig give values t At t =, T = 6 ad you put them ito T Ae to fid A At t =, T = ad you put t them ito T Ae (ow A is ow) to fid, emembeig that e ad l ca cacel each othe Havig ow A ad, you ca put T = 7 ito the same equatio to fid t Also evise the log ules because some questios as fo eact values (iii) Eplai why the body will eve each a tempeatue that is half of its iitial tempeatue t Put t ito T Ae ad you will get T (we wat T 8) Motio Velocity as a fuctio of displacemet : Tae the ecipocal to give dt as a fuctio of, d the itegate with espect to to get a equatio i t ad d Acceleatio as a fuctio of displacemet : Use v to get v as a fuctio of If d the questio ass fo a elatio betwee time ad displacemet, tae the squae oots of v ad ty to detemie the sig of v ad pefom simila steps as i velocity as a fuctio of displacemet Detemiig the sig of v: As youself questios such as: is v iitially positive o egative, whee is v zeo, ad what is the value of whe v is zeo Remembe that acceleatio cotols how the object is movig It ca be associated with foce If a object is at est, pushig it (givig foce ad acceleatio) fowads will cause it to move fowads Simple Hamoic Motio (SHM) Somethig is movig i simple hamoic motio with cete if its acceleatio is popotioal to its displacemet fom the cete but i the opposite diectio, that is,, By itegatig twice usig appopiate methods, it ca be show that the displacemet ca be modelled as a t OR acost si Amplitude a, Peiod ( T) Phase t, Iitial Phase hoosig the ice SHM fuctio: ty to avoid ad Use si if the object stats at the cete of its motio, use cos if it stats at the eteme If it stats aywhee else, chage the oigi of time (ie put t whe the object is eithe at the cete o the eteme), but do t Page 8 of 8

9 HS Mathematics Etesio foget to covet the time bac i the fial aswe if it s ased Othewise use asit OR acost, o use bsi t ccos t The gaphs of asi t, ad of acos t : asi t asi t, which is asi t shifted left by We ca itegate it with The acceleatio fuctio of SHM is always d espect to to fid the velocity as a fuctio of Remembe that v ad that sice d the object is oscillatig, the velocity at a paticula poit ca be positive o egative Pojectile Motio A pojectile is somethig that moves ude the ifluece of gavity aloe, which is a foce diected dowwads Displacemet, velocity ad acceleatio ae vectos (have magitude ad diectio) All vectos ca be boe up ito vetical ad hoizotal compoets, which ae idepedet of each othe Fom the tiagle, pat vecto ( magitude ) cos vecto pat y pat y pat y pat vecto ( magitude ) si ta pat I all of ou woigs, we should stat with the acceleatio fuctios ad itegate with espect to time to fid the velocity ad displacemet fuctios, substitutig give values to fid the costat of itegatio ad y g, g Equatio of path of a pojectile : So, we ow have a fuctio of ad of y, both i tems of t (which ae Vt cos ad y Vt gt si, V ad θ deote the magitude ad agle of the iitial velocity) etaily the equatio loos simple ad we ca mae t the subject ad substitute this ito the y equatio (substitute t ), hece we get the y-equatio i tems V cos of (this is what the path of the pojectile loos lie i the ai) Ivese Fuctios ad the Ivese Tigoometic Fuctios Ivese Fuctio Ivese fuctio is made by eflectig the gaph of the oigial fuctio i the lie y Algebaically, this echages the ad y vaiables i the oigial fuctio If the oigial fuctio does t pass the hoizotal lie test, the ivese wo t be a fuctio Howeve the domai of the oigial fuctio ca be esticted to give a ivese fuctio The domai of the oigial fuctio will be the age of the ivese, ad vice vesa The ivese fuctio is witte as f f ad f f The compositio of the fuctio ad its ivese sed evey umbe bac to itself: f The ivese of a (always) iceasig fuctio is also a iceasig fuctio, ad vice vesa Sample hade questio: If f, show that f is a possible y ivese f : y ad f : y y y y y, y y b b ac which is a quadatic i y y a Deivative: If we have y f ad y f, the deivative of the ivese at y is / the deivative of the oigial fuctio at Page 9 of 8

10 HS Mathematics Etesio Ivese Tigoometic Fuctios y si D : ad R : y ODD y cos D : ad R : y cos cos y ta D : eal ad R : y ODD Mae sue you ca daw them Use calculato to help you (fo eample, to fid si ) Sample questio: Fid the domai ad age of y cos Domai: 6 Rage: cos cos y Note o the Gaph of si The gadiet at the oigi is Nea the eds the cuve esembles vetical lies (ulie omplemetay idetity (alteative otatio): si cos d d d Deivatives: si, cos ad ta d d d Note o Itegatio: si 6 Polyomials cos D Fom: P a a a a ( cadial umbes) OR Some Tems Leadig tem: the tem of the highest ide, with ozeo coefficiet Leadig coefficiet: the coefficiet of the leadig tem Degee: the ide of the leadig tem Moic polyomial: has leadig coefficiet of Zeo polyomial: P, o leadig tem ad hece o degee P a Page of 8 y ) Liea polyomial: its gaph is a staight lie, icludes costat ad zeo polyomials Quadatics, cubics, quatics, quitics, ad so o (quadatic ca be ou o adjective) Roots: the solutios to a polyomial equatio, fo eample o Zeoes of P is zeo (must ot have = sig lie i equatio) P : the values of fo which m I P Q, whee Q is ot divisible by, is called a zeo of multiplicity m A zeo of multiplicity is called a simple zeo, a zeo of multiplicity geate tha is called a multiple zeo (o double zeo, tiple zeo ad so o) Some Results Degee of poduct: degp Q deg P deg Q fo ozeo polyomials Evey polyomial of odd degee has at least oe zeo, by eamiatio of its behaviou fo lage Behaviou of zeoes: eve m, the cuve is taget to the -ais hee ad does t coss it odd m, the cuve is taget to the -ais hee i the fom of hoizotal poit of ifleio (ie cosses the -ais) m =, the cuve cosses the -ais but is ot taget Suppose that P() is a polyomial ad α is a costat, the emaide afte divisio of P() by -α is P(α) (Remaide Theoem) -α is a facto if ad oly if P(α) = (Facto Theoem) Sice a polyomial of degee caot have moe tha zeoes, The gaph of a polyomial is detemied by ay + poits o the cuve

11 HS Mathematics Etesio Two distict polyomials P() ad Q() caot itesect i moe poits tha the maimum of the two degees, sice degp Q the maimum of the two degees A lie is also a polyomial so thee s a limit as to how may times it ca itesect P() How two cuves meet is detemied by the multiplicity of the zeoes of " P Q" as outlied i the behaviou of zeoes sectio above Fo eample, if the multiplicity is the the two cuves ae taget to each othe ad do t itesect at that poit If it s the the cuves cut each othe but thei tagets at that poit coicide Sums ad Poducts of Roots i Relatio to oefficiets We ca fid the sums ad poducts of oots eve though the oots do t eist i the eal umbe system Fo eample, oots of have sum - ad poduct + Let s tae P a b c d Let zeoes be,, ad, the: This should be ead: the sum of evey zeo, the sum of the b a poducts of pais of zeoes, ad the sum of the poducts of c tiple(s) of zeoes Istead of witig the summatio we ca a use sigma otatio d a The mius ad plus sigs eep alteatig, begiig with mius b o a The patte is simila fo ay polyomials Fo eample, i P a b c d e, e a Some Pattes (should ot be memoised) [ lettes divided by lettes leave lette ude, is lettes] [ divided by leaves ude] somethig [ whe epaded will give etc] Factoig Polyomials Use tial ad eo to fid as may itege zeoes of P() as possible [If the coefficiets of P() ae all iteges, the ay itege zeo of P() must be oe of the divisos of the costat tem] Use sum ad poduct of zeoes to fid the othe zeoes Alteatively, use log divisio of P() by the poduct of the ow factos Appoimate Solutio To obtai a solutio to f() =, fist locate the solutio oughly by a table of values o gaph The, use eithe halvig the iteval method o Newto s method Usually they tell you how may applicatios you eed ad how may sigificat figues they wat Below is also some guide whe they oly tell you the umbe of sigificat figues Halvig the iteval: Each successive applicatio will halve the ucetaity Gettig the last digit ight: suppose that ou fial esult afte applicatios is 6 The appoimate solutio is accuate to because 6 ouded becomes ad so does I cotast is oly accuate to so you eed to halve the iteval agai if you wat thee sigificat figues Sample questio: Eplai why a zeo of f eists betwee ad By halvig the iteval method, fid that zeo to two sigificat figues Hece fid the value of to two sigificat figues ANS: a zeo eists because f() chages sig betwee ad (at f() is -ve, at it s +ve) so f() must equal zeo somewhee i betwee Techically you should also metio that f() is cotiuous fo / / /8 = 7 7/6 = 7 / = 6 f() Page of 8

12 HS Mathematics Etesio The fist iteval is <<, the secod is << (because at = f() is +ve so the chage i sig is i << ot <<), the thid is <<, the fouth is 7<<, the fifth is 7<<7, the sith is 6<<7 ad ow you have a value accuate to two sigificat figues This is the value of because f Newto s method: If you foget which oe is o top, test with f ' f, O calculatos, ete the value fo ad pess = the ete the fomula by eplacig evey with As Pessig = successively yields,,, The umbe of coect decimal places geeally doubles with each applicatio Gettig the last digit ight: suppose they tell you to give thee sigificat figues Keep epeatig the pocess util you get the same fouth sigificat figue i two successive applicatios Sample questio: A featue aticle i a local ewspape says: Newto s method fails to estimate whee the cuve of y itesects the -ais fo ay iitial appoimatio Assess the validity of this statemet by usig algebaic agumet o othewise Algebaically:,, Theefoe successive applicatio iceases the absolute value ad alteates the sig Hece the appoimate solutio moves away ad it ca be said that Newto s method fails Othewise: the appoimate solutio i Newto s method is the -itecept of the taget to the cuve at The gaph of y is y (lie y but daw sideways) It ca be see that the -itecepts of tagets move away fom the zeo so Newto s method fails 7 Biomial Theoem coefficie t 6 Sample questios: patte The Epasio of o a b ad the Pascal Tiagle The Pascal tiagle loos lie the oe o the left, ad its costuctio will be eplaied late Its usage is illustated hee a b a b a b a b a b b a b a b a b Q: Fid a appoimatio of 8, coect to five decimal places, by biomial epasio 8 ANS: Q: Fid if i the epasio of 6 the tems i ad a have coefficiets i the atio : ANS: fid the two coefficiets by patly epadig 6 [ 6 ], the 6 Q: Fid i 6 if the coefficiets of 6, ad ANS: 6 6 so the solve 6 Page of 8 ae i aithmetic pogessio A Notatio fo the oefficiet Facto (it s my tem oly) i the Pascal Tiagle

13 HS Mathematics Etesio The otatio is: o [ead: choose ] It is defied as the coefficiet of i the epasio of To fid 6 i a calculato, pess 6 the the ad = The epasio of The Epasio of y : I sigma otatio, I sigma otatio, y y OR y The geeal tem is T y, [thee ae tems] Sample questio: Fid the tem i ANS: we ca wite out the full epasio, o alteatively otice that the vaiable of each tem follows the patte 6 i the epasio of 6 6 It s whe 6 ad ad the tem is Also ty to e-do Q, Q ad Q above usig the geeal tem otatio ad see if it s faste Popeties of ad Pascal Tiagle [the fist poits ae the way we costuct the Δ] Each ow stats ad eds with, ie [the additio popety] Evey umbe i the tiagle, apat fom the s, is the sum of the two umbes above it ad eaest to it, ie Each ow is evesible [o symmetical, eg 6 ], ie The sum of each ow is The fou, ie esults eed to be memoised i Sample questio: Fid the coefficiet of facto ad loo fo the tems i both factos whose poduct gives ] Factoial Notatio,!!!, with! [ote: thee s a factoial butto o calculato, but ote that fo lage,! is vey lage that it wo t fit o the scee ad so the calculato oly appoimates]! 98 7! Sample questio: ! 7! Futhe Applicatio of!!! [o calculato, is also oly appoimate fo lage ] Sample questios: Fid the geeal tem i [hit: patly epad each ad hece fid the tem i Fid the tem idepedet of i the epasio of!! Fid the value(s) of if A: fistly,!!! Now, ad sice Geatest oefficiet ad Geatest Tem Page of 8

14 HS Mathematics Etesio I ay biomial epasio, such as,, 7 o whateve else, the absolute values of the coefficiets of successive tems ise ad the fall [ad eve ise agai] The absolute values of the tems [if we wee to put = some umbe] also ise ad fall t To fid the geatest coefficiet, fist we fid whee (fo which s) t t, ie, whee t small t deotes coefficiet Now the geatest coefficiet is of couse t sice t t Use simila appoach to fid the geatest tem (doated by capital T) Nomally you wo t get the same as the geatest coefficiet Sample questio: Fid the geatest coefficiet of the epasio of Also fid the geatest tem if Solutio fo the geatest coefficiet: fistly fid a epessio fo the t coefficiet: t (you ca also use t ) Now t!!! (ote: but you ca t quote it) Now!!! solve which would give Highest is Geatest coefficiet is t so it s t Sample hade questio (with egative coefficiets ad tems): Fid the maimum coefficiet i the epasio of Solutio: the coefficiets alteate betwee positive ad egative Howeve the size o absolute value of each coefficiet is the same as that of, so they also ise up to a pea ad the fall dow Fist fid the coefficiet with t geatest absolute value, usig, ie as i the pevious eample This gives t t, but suely a egative coefficiet ca t be the maimum Fid the two coefficiets aoud it (ie t ad t ) ad see which oe is bigge This is the maimum coefficiet Idetities o Biomial oefficiets Some thigs to coside: Diffeetiatio o itegatio of both LHS ad RHS to obtai ew epessio Substitutio of values lie, - o ito o y i both LHS ad RHS Equatig coefficiets of tems with cetai i LHS ad RHS Nomally they give you the statig poit o epessio such as sometimes they do t Sample questio #: Show that itegatig both sides of Solutio: Itegate the RHS just lie whe you itegate Page of 8, but by ad substitutig, givig you ito the ew equatio You will have aothe costat of itegatio o the LHS, but you ca combie them ito oe: K ; fid K by puttig Fially, put ad you should get a epessio that loos simila to what is ased Multiple both sides by (-)

15 HS Mathematics Etesio Sample questio #: (i) Show that Fid i tems of the sum Fo (i), use (ii) 6 Hits to solutios: ad diffeetiate twice, the substitute both sides i Fo (ii), multiply by, the itegate the esultig epessio (use substitutio u to itegate the LHS) Fially substitute ito the itegal Some eamples fom HS eams: ty Q7(b) ad Q7(b) 8 Pemutatios, ombiatios ad Futhe Pobability This sectio stats with methods of coutig selectios (fo eample, the sample space o the evet space) Late we will use them i pobability questios I odeed selectios, outcome abcd ad outcome adbc ae couted as two I uodeed selectios, they ae oly oe sice ode does ot matte Odeed Selectios Suppose that a selectio is made i stages The fist stage ca be chose i ways, the secod i ways, etc Numbe of possible selectios = A efficiet settig-out: use a bo diagam listig each stage evet ad its value Daw the esticted evets fist ompoud odeigs (odeigs ivolvig goups): fist ode the goups, the ode the idividuals withi each goup [a goup may cosists of a sigle idividual] Sample questio: Fou boys ad fou gils fom a queue Thee is oe couple who wat to stad togethe, the othe thee gils wat to stad togethe, but the othe thee boys do t cae whee they stad How may ways ae thee to fom the queue? Solutio: thee ae five goups: (couple), (the gils), (boy), (boy), (boy) ode the goups! ode the couple! Odeed selectios with epetitio: Numbe of possible selectios =! Pemutatios (odeed selectios without epetitio): P! Numbe of ways =!!! = ( P is o you calculato but some questios equie you to ow the fomula) I paticula, the umbe of distict odeigs of a -membe set: P! Pemutatio with idetical elemets (whee all membes ae pemuted) Suppose that a wod of lettes has l alie of oe type, l alie of aothe type, ad so o! The umbe of -lette wods = l! l! l! ode the gils! If thee ae oly two types of lette (fo eample, A ad B) with oe type havig lettes:! umbe of -lette wods = [otice also that ]!! Usig cases Sample questio: How may si-lette wods ca be fomed by usig the lettes of the wod PRESSES? Solutio: We omit i tu each of the fou lettes P, R, E ad S This leaves si lettes which must be aaged i ode Thee ae fou cases: 6! If a S is omitted, umbe of wods = 8 [sice we have Es ad Ss]!! Page of 8

16 HS Mathematics Etesio 6! If a E is omitted, umbe of wods = [we have Ss but oly E ow]! 6! If P is omitted, umbe of wods = 6 If R is omitted, thee ae also 6 wods!! So, addig esults fom the fou cases, total umbe of wods = Uodeed Selectios (also called ombiatios) If a set S has membes,! umbe of -membe subsets =!! the total umbe of subsets of S (ic those with o- membes) = Sample hade questio: Te poits A, B,, ae aaged i ode aoud a cicle How may pais of tiagles with distict vetices ca be daw fom those poits? Solutio: hoose 6 poits out of Tae of those poits ad choose the othe poits i its Δ 6 Note: this questio is ot to be cofused with the aagemets i a cicle below Ideas of Ode ad Ovecoutig Sample questio: I how may ways ca we mae goups of fom 8 people? Aswe would 8 6 be /! We divide by! because the ode i which the goup is selected is ielevat Aagemets i a icle Thee is essetially oly way to place the fist item o goup i the cicle If it ivolves compoud odeigs, you might pefe the bo method fo detemiig the umbe of goup aagemets ad the multiply that umbe by the umbes of aagemets of idividuals withi each goup Ad as you should ow, wite the goup with the most uique estictio i the fist bo Sample questio: Five boys ad five gils ae to sit aoud a table Fid how may ways this ca be doe if fou couples sit togethe, but Walte ad Maude do t Solutio: The most uique estictio is Walte ad Maude do t sit togethe Walte Maude st couple d couple d couple th couple No of goup aagemets = 7 No of ways = 7 Rewitig the Questio Sample questio: A team of seve etballes is to be chose fom a squad of twelve playes A, B,,, L How may ways ca we choose them if A must be icluded ad H ecluded Solutio: The questio is the same as How may ways ca we choose 6 playes fom a squad of people: B,, D, E, F, G, I, J, K, L omplemetay outig ad Pobability Fidig the complemetay umbe of ways ad pobability may be easie to do, paticulaly whe the questio metios the wod ot, at least, at most ad ecludig Usig outig i Pobability Questios As metioed, coutig methods ca be used to cout the umbe of all possible outcomes ad of favouable outcomes Sample hade questio: A five-digit umbe is chose at adom Fid the pobability that (i) the digits ae distict ad i iceasig ode, (ii) the umbe has thee seves Solutio (i): [Note that a umbe ca t begi with zeo] Evey uodeed five-membe subset of the set of ie ozeo digits ca be aaged i eactly oe way ito a five-digit umbe 9 with the digits i iceasig ode Hece, the umbe of favouable outcomes is 6, while the umbe of all outcomes is 9 9 Solutio (ii): Use cases ase #: all umbes ic those which stat with zeo Page 6 of 8

17 HS Mathematics Etesio ase #: ase #: the umbes that stat with zeo hoose whee the thee 7s lie Numbe of favouable outcomes: 9 9 Pobability = 77/9 = / 77 9 hoose whee the thee 7s lie Biomial Pobability ostuct the questio so that (i) all stages ae idetical AND (ii) each stage has oly two distict possible outcomes (called success ad failue), ot ecessaily equally liely Suppose that the pobabilities of success ad failue i ay stage of a -stage tial ae p ad q espectively The, P ( successes [ad hece - failues i ay ode]) = p q This is the tem i the epasio of p q The epesets the umbe of ways we ca ode the successes ad failues The most liely outcome ad the maimum tem i a biomial epasio Sample questio: A die is thow times (i) what is the most liely umbe of sies that will be thow (ii) what is the pobability of gettig that paticula umbe of sies Solutio: (i) Let success be gettig a si ad failue be ot gettig a si, the p ad q, 6 6 P P sies P p q Fid usig (ii) Fid P P 6 Sample hade questio: Joe ad Fay mae shits fo a livig Joe maes shits a day, of which % ae defective [this % is called epeimetal pobability] Fay maes shits a day, of which % ae defective If they sed out thei shits i adomly mied pacels of shits, what is the pobability that o moe tha two shits i a bo ae defective? Solutio: Let p = the pobability that a shit i the bo is defective p = P (Joe made it, ad it s defective) + P (Fay made it, ad it s defective) = % % q is automatically p = / Let X be the umbe of defective shits P X PX PX PX Summay of Fomulae Ode is impotat: hoose the fist o-7 9 Thigs ae eplaced as each is tae: Thigs ae ot eplaced: P I the special case whee eveythig is selected ( o! ), thee is aothe fomula! whe thee ae idetical elemets*: Whe thee ae oly types of l! l! l!! elemets, l o l! l! Aagemet i a cicle: thee is oly way to choose o place the fist item Ode is ot impotat: Thigs ae eplaced: eed to coside sepaate cases Thigs ae ot eplaced: Whe it is a pobability questio, you ca also egad it as odeed ad use P, but you eed to use P to cout both the favouable outcomes ad all possible outcomes Thi about it logically ad you ll udestad why it wos P hoose the secod o-7 9 hoose the o-7 9 Page 7 of 8

18 HS Mathematics Etesio * I pobability questios, teat idetical elemets as beig diffeet Fo eample, i Q6(a) below, we ca egad each fish as beig uique Numbe of favouable outcomes = 8 P ad umbe of all outcomes = P (o 8 ad ) Pobability usig U pobability: Q6(a) ca also be doe usig Pobability = (the poduct ule i multi-stage epeimets) Revisio Questios Q Thee ae 8 swimmes i a ace, i how may ways ca they fiish if thee ae o dead heats ad the swimme i Lae fiishes afte the swimme i Lae Q Thee cads ae selected at adom fom a stadad pac of cads Fid the pobability that you get (a) the Jac of spades, the of clubs, the 7 of diamods (b) thee aces (c) thee diamods (d) thee cads of the same suit (e) thee pictue cads (f) two ed cads, oe blac cad (g) oe 7, oe 8, oe 9 (h) two 7 s, oe 6 (i) eactly oe diamod (j) at least two diamods Q Repeat Q but ow the cads ae selected oe at a time ad each cad is eplaced befoe the et oe is daw Q I how may ways ca the lettes i PROPORTIONALITY be aaged so that the vowels ad cosoats still occupy the same place? Q The lettes of PROMISE ae aaged adomly Fid the pobability that the vowels ae togethe Q6 A ta cotais tagged fish ad 8 utagged fish O each day, fish ae selected at adom, ad afte otig whethe they ae tagged o utagged, they ae etued to the ta What is the pobability of (a) selectig o tagged fish o a give day (b) selectig at least oe tagged fish o a give day (c) selectig o tagged fish o evey day fo a wee (d) selectig o tagged fish o eactly days i a wee Aswe coect to sigificat figues Q7 Thee ae distict oud tables, each with seats I how may ways may a goup of people be seated? Aswes: Q 6 ; Q /, /, /8, /, /, /, 6/, 6/, 7/7, 6/ ; Q /7, /97, /6, /6, 7/97, /8, 6/97, /97, 7/6, / ; Q 76 ; Q /7 ; Q6, 97, 7, 9 ; Q7 Some Woigs: Q Let L be the swimme i Lae, ad L, L, be swimmes i the othe laes Daw a table outliig the possible odes comes fist comes last L L 7 => If L comes fist, L ca be aywhee i the 7 L L 6 emaiig places L L => If L comes thid, L ca be i othe places L L Hece umbe of ways = 7 6 6! (6! is fo the othe 6 L s) Q Occupy the same places meas that somethig lie PTOPORRIONALITY is allowed ad PORPORTIONALITY is ot allowed The coditio ca be ewitte as vowels stay togethe ad stat the wod Thee ae 6 vowels (with O s ad I s) ad 9 cosoats (with P s, 9! 6! R s ad T s) Numbe of ways =!!!!! Q Numbe of favouable outcomes =!! (! Is to ode the vowels withi the goup, is to ode the vowels goup withi the actual wod) Q7 Numbe of ways = 9876 The fist is to choose which table is table A The fist is to sit the fist peso i table A The 9 is to sit aothe peso i table A The secod is to fist the fist peso i table B Page 8 of 8