REVISION SHEET FP2 (OCR) CALCULUS. x x 0.5. x x 1.5. π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + = + arcsin x = +

Transcription

1 the Further Mathematics etwork wwwfmetworkorguk V 7 REVISION SHEET FP (OCR) CALCULUS The mai ideas are: Calculus usig iverse trig fuctios & hperbolic trig fuctios ad their iverses Maclauri series Differetiatig the Iverse Trig Fuctios Before the eam ou should kow: That ou ca differetiate the trig fuctios, the hperbolic trig fuctios ad their iverses That ou ca appl the stadard rules for differetiatio (product rule, quotiet rule ad chai rule) to fuctios which ivolve the above That ou ca itegrate trig fuctios ad hperbolic trig fuctios That ou ca itegrate, arcsi(), arccos(), arcta(), arccot(), arsih(), arcosh() etc usig itegratio b parts Your trig idetities ad hperbolic fuctio idetities ad how to use them i itegratio problems Particularl get familiar with useful substitutios to make for certai problems 5 arccos( ) d d arcta( ) d d arcsi( ) d d It is importat to be aware of what the rage is for each of these, amel: arcsi, arccos, arcta Stadard Calculus of Iverse Trig ad Hperbolic Trig Fuctios arcsi( ) d d arccos( ) d d arcta( ) d d arsih( ) d d + + ar cosh( ) d d arcta + c + a a a a arcsi + c a a ar cosh + c a + a ar sih + c a Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

2 the Further Mathematics etwork wwwfmetworkorguk V 7 Calculus usig these fuctios The eamples below are ver tpical ad show most of the commo tricks Note details of all substitutios have bee omitted, make sure ou uderstad how to do them i this case ad also i the case of a defiite itegral + d arsih d c ( + ) + d d d arcsi arcsi c c d d arcosh c ( ) d arcosh( ) d Some useful itegratio tricks Splittig up a itegratio: eg (to see this use the chai rule, set z d d + d ad the d d dz ) d dz d B ispectio: eg Sice l( + ) gives + whe differetiated, we have d l( + ) + c or + sice ( + ) gives ( + ) whe differetiated, we have d + + c + Usig clever substitutios: eg the substitutio u sih( ) will help ou with + d Maclauri Series The Maclauri epasio for a fuctio f() as far as the term i looks as follows ( ) f ( ) f () + f () + f () + f () + + f ()!!! The Maclauri series is obtaied b icludig ifiitel ma terms (ie ot termiatig the sum as above) It ma ol be valid for certai values of Eamples iclude: 5 7 si + + which is valid for all, 8! 5! 7! which is valid ol whe <, ote that this secod eample is the same as the biomial epasio of ( ) Note You ca fid the Maclauri series of, eg f(), b takig the series for f() ad replacig the s with Reductio Formulae You should be able to derive ad use reductio formulae for the evaluatio of defiite itegrals i simple cases ie to calculate ed Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

3 the Further Mathematics etwork wwwfmetworkorguk V 7 REVISION SHEET FP (OCR) HYPERBOLIC TRIG FUNCTIONS The mai ideas are: Defiitios of the hperbolic trig fuctios ad their iverses Workig with the hperbolic trig fuctios Idetities ivolvig hperbolic trig fuctios The Hperbolic Trig Fuctios These are defied as: e e sih( ), cosh( ), sih( ) e e tah( ) cosh( ) l l e e For eample, sih(l) 99 Before the eam ou should kow: e e The defiitios sih( ), cosh( ), sih( ) e e tah( ) cosh( ) That ou ca prove that arccosh( ) l( + ),arsih( ) l( + + ) + artah( ) l Your trig idetities ad hperbolic fuctio idetities, eperiece will tell ou whe it is best to work i the epoetial form whe dealig with equatios Ad be able to prove hperbolic idetities from the e e defiitios sih( ),cosh( ), it s worth practicig idices for this The Iverse Hperbolic Trig Fuctios Just as the hperbolic trig fuctios are defied i terms of e, their iverses ca be epressed i term of logs I + fact arcosh( ) l( + ), arsih( ) l( + + ), artah( ) l You should be able to prove (ad use) all of these Here is the proof that arcosh( ) l( ) + Let ar cosh( ), the cosh( ) Rearragig this gives e + e Multiplig this b e gives e e + This is a quadratic i e ad usig the formula for the roots of a quadratic gives ± e ± Takig logs gives ar cosh( ) l( ± ) Do ou kow wh the epressio with the mius sig is rejected here? These epressios ca be used to give eact values of the iverse hperbolic trig fuctios i term of logs For eample, arcosh l + l + l() 9 Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

4 the Further Mathematics etwork wwwfmetworkorguk V 7 Graphs of the Hperbolic Trig Fuctios cosh( ) d sih( ) d 8 tah( ) d d sech ( ) - - sih( ) d cosh( ) d - Graphs of the Iverse Hperbolic Trig Fuctios - - You must also kow the graphs of the iverse hperbolic trig fuctios, arsih, arcosh ad artah As for a fuctio these are obtaied b reflectig the respective graphs of sih, cosh ad tah i The eamples of arsih ad arcosh are show here Notice that arcosh() is ol defied for greater tha or equal to Idetities Ivolvig Hperbolic Trig Fuctios Idetities ivolvig hperbolic trig fuctios iclude: cosh u sih u, cosh( u) cosh u+ sih u, si( u+ v) sih( u)cosh( v) + cosh( u)sih( v ) The ol differece betwee a hperbolic trig idetit ad the correspodig stadard trig idetit is that the sig is reversed whe a product of two sies is replaced b a product of two sihs This is called Osbor s Rule You ca prove a hperbolic trig idetit usig their defiitios ad should be able to do this for the eam Equatios Ivolvig Hperbolic Trig Fuctios Eample Solve the equatio cosh + 5sih givig our aswer i terms of atural logarithms Solutio e e cosh + 5sih + 5 8e + 8e 9e e + 9e e or e e 9 l or l 9 ( )( ) Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

5 The mai ideas are: What Polar Coordiates are Coversio betwee Cartesia ad Polar Coordiates Curves defied usig Polar Coordiates Calculatig areas for curves defied usig Polar Coordiates θ 5 θ 5 θ θ 5 θ 7 θ θ θ the Further Mathematics etwork wwwfmetworkorguk V 7 5, REVISION SHEET FP (OCR) POLAR COORDINATES θ 5 θ 8, 5,- θ θ θ θ θ θ θ 5 θ θ θ,- θ θ 5 θ 7 θ Before the eam ou should kow: How to chage betwee polar coordiates (r, θ) ad Cartesia coordiates (, ) use r cosθ, r siθ, r + ad taθ You ll eed to be ver familiar with the graphs of si, cos ad ta ad be able to give eact values of the trig fuctios for multiples of ad How to sketch a curve give b a polar equatio β The area of a sector is give b rd θ How Polar Coordiates Work You will be familiar with usig Cartesia Coordiates (, ) to specif the positio of a poit i the plae Polar coordiates use the idea of describig the positio of a poit P b givig its distace r from the origi ad the agle θ betwee OP ad the positive -ais The agle θ is positive i the aticlockwise sese from the iitial lie If it is ecessar to specif the polar coordiates of a poit uiquel the ou use those for which r > ad < θ It is sometimes coveiet to let r take egative values with the atural iterpretatio that ( r, θ) is the same as (r, θ + ) α It is eas to chage betwee polar coordiates (r, θ) ad Cartesia coordiates (, ) sice r cosθ, r siθ, r + ad taθ You eed to be careful to choose the right quadrat whe fidig θ, sice the equatio taθ alwas gives two values of θ, differig b Alwas draw a sketch to check which oe of these is correct Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

6 the Further Mathematics etwork wwwfmetworkorguk V 7 The Polar Equatio of a Curve The poits (r, θ) for which the values of r ad θ are liked b a fuctio f form a curve whose polar equatio is r f(θ) A good wa to draw a sketch of a curve is to calculate r for a variet of values of θ Eample Sketch the curve which has polar equatio r a(+ cos θ ) for θ, where a is a positive costat Solutio Begi b calculatig the value or r for various values of θ This is show i the table The curve ca ow be sketched θ r a a a ( + ) a a It s a good eercise to tr to spot the poits give i the table above i polar coordiates o the curve show here (,) For eample the poit a ( + ) is here The Area of a Sector The area of the sector show i the diagram is Eample β rd α A curve has polar equatio r a(+ cos θ ) for θ, θ where a is a positive costat Fid the area of the regio eclosed b the curve Solutio The area is clearl twice the area of the sector give b θ Therefore the area is Note Eve though r ca be egative for certai values of θ, r is alwas positive, so there is o problem of egative areas as there is with curves below the -ais i cartesia coordiates Be careful however whe cosiderig loops cotaied iside loops ( ) ( ) rdθ a + cosθ dθ a + cosθ + cos θ dθ ( cos cos ) a + θ + θ dθ a θ siθ ( + ) a + + si θ Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

7 the Further Mathematics etwork wwwfmetworkorguk V 7 The mai ideas are: Sketchig Graphs of Ratioal Fuctios Graph Sketchig Ratioal fuctios To sketch the graph of REVISION SHEET FP (OCR) N( ) D( ) : GRAPHS Before the eam ou should kow: There are three mai cases of horizotal asmptotes Oe is a curve which is a liear polomial divided b a liear + polomial, for eample This has a horizotal Fid the itercepts that is where coefficiet of o the top the graph cuts the aes coefficiet of o the bottom Fid a asmptotes the vertical asmptotes occur at values of which make the deomiator zero Eamie the behaviour of the graph ear to the vertical asmptotes; a good wa to do this is to fid out what the value of is b dividig out for values of ver close to the How to sketch graphs of the form vertical asmptote Eamie the behaviour aroud a o-vertical asmptotes, ie as teds to ± coefficiet of o the top asmptote at Here this would coefficiet of o the bottom be The secod is a curve give b a quadratic polomial divided b a quadratic polomial, for eample, + as ±, This has a horizotal asmptote at 5 Here this would be Thirdl, whe the curve is give b a liear polomial divided b a quadratic polomial, it will geerall have the -ais ( ) as a horizotal asmptote That if the degree of the umerator is oe greater tha the degree of the deomiator, the there is a oblique asmptote, which ca be foud f( ) Eample Sketch the curve Sketch Solutio (remember to label all the ke features of our graph whe aswerig eam questios) (+ )( + ) The curve ca be writte as ( + )( + ) If the 5 So the itercept is (, 5) Settig gives, -5 ad - So the itercepts are (-5, ) ad (-, ) - The deomiator is zero whe - ad whe - so these are the vertical asmptotes + + ( + + 8) Also so is a horizotal asmptote - Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

8 the Further Mathematics etwork wwwfmetworkorguk V 7 REVISION SHEET FP (OCR) THE SOLUTION OF EQUATIONS The mai ideas are: Uderstadig geometricall staircase ad cobweb diagrams/sequeces ad how their covergece relates to gradiet cosideratios The Newto Rapsho Method Before the eam ou should kow: How to draw staircase ad cobweb diagrams ad aticipate the behaviour of sequeces from the pictorial represetatios The relatioship betwee the gradiet of F ear to the root ad the rate of covergece of the sequece to that root The workig of the Newto-Raphso Method i geometrical terms Ad be able to use both methods to approimate roots of equatios Basics It is possible to rearrage a equatio ito the form F( ) It ma be that the recurrece relatio + F( ), where is give or chose appropriatel, produces a coverget sequece If this is the case, the limit of the sequece will be a root of F( ) ad therefore it will be a root of the origial equatio Eample Cosider the equatio to decimal places Solutio The iteratio + si Use the iteratio r+ + si r with to fid this root correct r+ + si r with, gives r r si() + si(578) The last three values calculated are all 9 to decimal places I fact the root, to decimal places, is 9 as with f( ) + si, f(955) - ad f(85) +7 Importat ote: You ca get our calculator to perform these calculatios ver quickl usig the ANS feature Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl

9 Staircase ad Cobweb Diagrams the Further Mathematics etwork wwwfmetworkorguk V 7 The steps take to form such a diagram to fid a approimatio to a root of F() are Draw a graph with the lie ad the curve F() ad mark the poit at which the cross whose -coordiate ou are trig to approimate Startig with our iitial estimate of, draw the vertical lie through to meet the curve F() from this poit draw a horizotal lie to meet from this poit draw a vertical lie to meet F() from this poit draw a horizotal lie to meet 5 ad so o Eample: Approimatig a root of cos, the straight lie is ad the curve is cos (Notice how the lie is used to trasfer the value of o the -ais to the -ais so that it ca be iput ito g agai to fid ) Rate of Covergece With eperiece i drawig staircase ad cobweb diagrams ou will see how the covergece of the sequece produces depeds o the gradiet of F() aroud the root of F() I fact if ε is the error i as a approimatio to the root α (to which the sequece is covergig) the: ε F( αε ) if F( α ) ad ε + is approimatel proportioal to ε if F( α ) + Newto Raphso-Method To geerate a sequece of values covergig to a root of f(), ear to, use the followig iterative f( r ) formulae: r+ r This method has secod order covergece f( ) r Below the Newto Raphso method is beig used to approimate a solutio of + So we have r + r r+ r + i this case Usig a startig value of gives r The 577, 5975, , 5975 Notice how quickl the sequece coverges, this is i fact because the method is a eample of a iteratio of the form F( ) with F( α ) + Importat ote: Oce agai, ou ca get our calculator to perform these calculatios ver quickl usig the ANS feature Disclaimer: Ever effort has goe ito esurig the accurac of this documet However, the FM Network ca accept o resposibilit for its cotet matchig each specificatio eactl