Edexcel Core 1 Help Guide

Transcription

1 Edexcel Core Help Gide Steve Bldes 0

2 Aim The im of this booklet is to llow ppils opportity to brek dow topics i Core ito lgorithmic process. It cold be viewed s checklist of wht to do i certi exm qestios. It cold be rged tht mthemticl rigor is lckig d tht the gide is prescriptive list of rles or istrctios. My gol is to llow stdets who my strggle with cocepts opportity to ccess the exm pper rther th ecessrily bildig i depth derstdig of the mthemticl cocepts i ech topic. More rigoros ttorils c be fod o the listed sites for those lookig to bild i depth kowledge of the corse cotet. Cotet Idices Srds Qdrtic Eqtios d Expressios Simlteos Eqtios Ieqlities Grph Sketchig Coordite Geometry Seqeces d Series Differetitio Itegrtio Formle Give i Exm Mesrtio Srfce re of sphere = r Are of crved srfce of coe = rl (where l is the slt height) Arithmetic series ( ) d S ( ( ) d ) S ( l) Steve Bldes 0

3 Idices The rles of idices The idex is simply the power bse is rised to. I the exmple 8, is the bse, is the idex, expoet or power d 8 is the meric vle obtied if we rise to the rd power. Most qestios will sk yo to simplify expressio or fid mericl vles d se oe of the rles below. m m m m 0 m m m m m m 6 Most exm qestios will be or mrks. If they re worth mrks show workigs e.g. Fid Aother typicl qestio might sk s to Fid the vle of x give x x. I kow is power of (or vice vers) so I c rewrite the eqtio sch tht I hve the sme bse (): x x ( ) Usig the rles of idices: x ( x) If the bses () re the sme we c simply eqte the powers d solve for x. x ( x ) x x x To check my swer I c sbstitte x bck i to the eqtio see if it mkes the eqtio tre. We kow 6 6 d therefore we hve the correct swer. Do check yor swer! We my hve to expd brckets d simplify. A exm qestio cold be: Expd d simplify x x givig yor swers i descedig powers of x : Showig fll workigs d sig the rles of idices: x x x x x x x Filly, pttig my swer i descedig powers of x, the swer is: x x x If we hve 6 we do t divide by, we tke the th root. This is commo error. Aother commo error is m with egtive powers. Remember. This will ot give egtive vle. m 8 Oe importt rle to remember is. Yo c se this whe workig with srds d idices.. This will be very importt whe it comes to differetitio d itegrtio i the sectio o clcls. Steve Bldes 0

4 Srds Srds re irrtiol mbers d re sid to be exct vles. Do t be tempted to try d write trcted or roded deciml swer. Leve yor swer s srd. Clcltios i this form will be more ccrte & esy to perform. Geerlly yo will either hve to simplify srds, crry ot bsic clcltios with the opertios or rtiolise the deomitor of frctio with srd i. Some Bsic Srd Lws c b d b cd Jst try these with meric vles. Yo will see my brek dow ito iteger vles Simplifyig Srds Brekig srds dow (simplifyig) is jst cse of prime fctorisig e.g. 8 Addig d Sbtrctig Srds Yo c oly dd d sbtrct like srds BUT some will simplify to llow yo to do tht e.g. or 7 re exmples of srd ddig/sbtrctig withot prior simplifictio. Sometimes yo will hve to simplify first e.g. 0 8 ( ) ( ) 7 Expdig Brckets Yo my be expected to expd sigle or doble brckets. Most of these qestios reqire fll workigs e.g. Write ( )( 7) i the form b ( )( 7) ( )( ) ()() ()( ) ( )() ( )( ) , b Some exm qestios will ot expect tht level of workig. Do check first! Rtiolisig the Deomitor Mthemticis hte hvig irrtiol deomitor. A srd is irrtiol mber. Yo will come cross two types of frctios where yo will hve to rtiolise the deomitor: Whe the deomitor is sigle srd vle sch s or 7 or. With this type simply mltiply the mertor d the deomitor by the srd vle i the deomitor d simplify. For the first oe mltiply mertor d deomitor by, the secod 7 d the third. Simplify yor sig the srd rles. A exmple cold be (7) 9 Whe the deomitor hs two vles d dditiol or sbtrctio sig e.g. or or. 7 With this type we simply mltiply the mertor d the deomitor is the sme vles i the deomitor bt with the opposite sig. This will crete differece of sqres d llow yo to simplify to give rtiol, deomitor. Here is exmple i the form b. ( ) ( ) ( ) ( ) 6 ( )( ) There my be some prcticl pplictios of legth width = re rerrgig, rtiolisig & simplifyig srds sch s res or legths. Here is x typicl qestio: Fid the vle of x give the re of the rectgle below is sq its x 6 8 x Steve Bldes 0

5 Qdrtic Eqtios d Expressio Solvig Qdrtic Eqtios Qdrtic eqtios d expressio will hve highest power of the vrible s sqred term e.g. y x x 0 or p p 9 0 where ech power is iteger vle. A expressio is collectio of terms, eqtio will hve eqls (=) sig i d we cold look to se rge of techiqes to solve. Whe solvig qdrtic eqtio cosider two possible soltios e.g. x will hve two rel soltios, x. A commo error is to simply give the swer of. Check the vlidity of yor soltios s the lgebr my work bt oe soltio my ot be pplicble. A exmple wold be egtive legth i re of rectgle qestio. Sometimes qdrtic eqtio will hve o rel soltios. Whe tryig to solve qdrtic eqtio go throgh checklist of methods yo c se to solve eqtio. Type x. Simply sqre root both sides. p p p Type x bx c 0, tht DO fctor Set RHS side to 0, solve. 6x x 6x x 0 (x )( x ) 0 x or x Remember, ot ll eqtios will iitilly 0. Simply rerrge the eqtio ito the form x bx c 0 d look to solve. 6x x 6x x 0 (x )(x ) 0 x or x Type x bx 0 Set the RHS = 0, fctor d solve x x 0 x(x ) 0 x 0 or x Type x bx c 0 tht DON T fctor. Yo c se either the forml or yo c complete the sqre. Whe qdrtic eqtio is i the form x bx c 0 the soltios to the eqtio will be x exmple: x x 9 0 b b c x () ( ) ()( 9) 97 x Check the vlidity of both soltios whe swerig the qestio. The qestio might, for exmple, stte x which rles oe soltio ot. Steve Bldes Type x bx c 0, tht DO fctor. Set the RHS side to 0, solve. x x 8 0 ( x )( x ) 0 x or x Type x bx c 0 tht DON T fctor. Completig the sqre is optio. The coefficiet of the term i x mst be. x x 9 0 x 9 x 0 9 x x 6 97 x 6 97 x Completig the Sqre Completig the sqre c be sed to solve qdrtic eqtios, fid the mximm or miimm poit d llow esy grph trsformtios if yo eed to sketch qdrtic fctio. I the exm yo might be sked to write qdrtic expressio the form ( x b) c. This is i completed sqre form. Completig the sqre whe the qdrtic expressio is i the form x bx c, If yo wt lgorithm: tke hlf the coefficiet of the term i x ito the brcket with x, sqre the brcket, sbtrct the sqred vle wy x x ( x ) ( x ) 9 This is positive qdrtic expressio. If this ws eqtio where y x x the mximm wold hve the coordites (,9) d the grph trsltio of y x of its i the egtive x directio d 9 its i

6 the positive y directio. Completig the sqre whe the qdrtic expressio is i the form x bx c, I order the complete the sqre the coefficiet of the term i x mst be. If it s ot yo will eed to fctor ot the vle of. If yo re simply solvig eqtio divide throgh the eqtio by. Here is typicl exm style qestio: Express x x i the form( x b) c. x x x x x 6 6 x 6 9 x 6 If yo grphed this the prbol wold ope pwrds (s it s positive vle of ) d the miimm poit will hve coordites, 9 6. If I ws solvig the eqtio x x 0,I cold write x x 0 & solve If yo hd to grph y x the it wold be mximm t (, ) s the vle of is egtive. The Discrimit The Discrimit determies the tre of the roots of qdrtic eqtio. Remember mth error whe sig the qdrtic eqtio o clcltor? Tht ws becse b c 0. The scerios re listed below: Distict Rel Roots x bx c 0 No Rel Roots x bx c 0 Repeted or Eql Roots x bx c 0 b c 0 b c 0 b c 0 A exmple cold be fid the set of vles of k for which the qdrtic eqtio x kx k 0 hs two distict rel roots: x kx k 0, b k, c k b k c 0 for rel roots ()( k) 0 k( k 8) 0 k 0, k 8 Yo c se sketch to help o the lst prt (covered i the ext sectio o qdrtic ieqlities) A qick tip! A tget toches crve. Some more sbtle qestios ivolvig tgets my reqire the se of the Discrimit d i prticlr b c 0 for repeted root. Sketchig qick digrm shold mke sese of this! Steve Bldes 0

7 Simlteos Eqtios Lier Simlteos Eqtios Simlteos eqtios re simply or more eqtios tht shre commo soltios. I C these will be two eqtios geerlly i x d y. Yo will eed to be ble del with differet scerios i the modle. () Where there re two lier eqtios () Where oe eqtio is lier d oe is ot d () Where both re o lier. So wht does lier me? The powers of x d y will both be oe d there will be o terms i x d y tht re mltiplied or divided by oe other. Two Lier Eqtios Yo hve two choices () Elimitio by ddig or sbtrctig the two eqtios to elimite oe vrible () Rerrgig d sbstittio to elimite oe vrible Here is stright forwrd exmple from GCSE mths: Solve the simlteos eqtios: x y x y Method Method I eed either the terms i x or y to be the sme. Lbellig the eqtios Lbellig the eqtios () x y () x y () x y () x y I m goig to write eqtio () to mke y the sbject I m goig to mltiply eqtio () by sch tht () y x () x y 8 I c ow sbstitte this ito eqtio () to elimite () x y y from the eqtio sch tht Addig the two eqtios will elimite y ledig to: () x (x ) 7x, which gives x. If I simplify this I hve lier eqtio i x sch This vle of x c ow be sbstitted bck ito either tht () 7x. Solvig for x we c see x. eqtio () or () to solve for y. I m ow goig to se this vle i eqtio () to I m goig to se eqtio () solve for y. () () y () y () y Agi from this we c see y. Check yor swers for both x d y stisfy both eqtios () d (). y Check yor swers for both x d y stisfy both eqtios () d (). This ws t the oly wy I cold hve doe the qestio. I cold hve mltiplied eqtio () by d eqtio () by to mke the terms i x the sme d the sbtrcted eqtio () from (). This prticlr eqtio leds itself well to this method s y cold be writte i terms of x with o frctios o the RHS. Some eqtios will hve frctios i! A grphicl represettio is show below. These eqtios wold, i the mi, be solve lgebriclly i C Oe Lier Eqtio, Oe No Lier Typicl exmples will iclde circle d lie, prbol d lie or lie d other type of eqtio where y is implicitly defied s fctio of x (slly where there is prodct of terms i x d y ) sch s d x y 6. xy y y x Steve Bldes 0

8 Let s look t circle d lie. Fid the coordites of poits of itersectio of the eqtiosy x d x y. This is reltively stright forwrd exmple, others c become qite messy. The first eqtio is the eqtio stright lie d the secod eqtio is circle, cetred t the origi d hs rdis of its. Lbellig the eqtios: () x y () y x I m goig to divide both sides of eqtio () by d the sbstitte this ito eqtio () to elimite y () y x Sbstittig ito () eqtio () will give me qdrtic eqtio i x. () x x 9 6 () x x () x 6 () x 6 () x 6 Solvig for x we c see x. At this stge we simply sbstitte or vles of x bck ito the lier eqtio () to solve for y. () y x Whe x, y d whe x, y. This gives s the two poits (,) d (, ). Yo c of corse mke x the sbject of eqtio () t the strt d solve the qdrtic eqtio i y THEN fid the vles of x from the lier eqtio. A grphicl represettio is show below: Two No - Lier Eqtios Yo might eed to solve simlteos eqtios where we hve two qdrtic fctios. These re reltively stright forwrd d we will work thogh typicl qestio: Solve the simlteos eqtios: y x x y x x We hve expressios for y i terms of x for both eqtios so we c simply either set the eqtios eql or sbtrct oe eqtio from the other to elimite y. If y x x d y x x it follows tht x x x x Settig the RHS to 0, x x 6 0 We c fctor d solve for x ( x )( x ) 0 givig the soltios x d x. At this stge some stdets fiish the qestio withot solvig for y. We eed to fid y too! Usig y x x we c solve to fid the two vles of y : Whe x, y 9 which gives y Whe x, y 0 which gives y 9 The soltios re x d x, y d y 9. Steve Bldes 0

9 Ieqlities Lier Ieqlities Ieqlities tell s bot the reltive size of two vles. The vrible x for exmple (which cold represet displcemet) might be more th m i give sittio. Mthemticlly we cold write x d we wold red this s x is greter th. This gives s the rge of vles stisfyig the ieqlity. I C yo will be expected to solve lier d qdrtic ieqlities d sometimes stte the set of vles tht stisfies both lier d qdrtic ieqlity. Lier Ieqlities re solved i similr mer to lier eqtios BUT If yo re mltiplyig or dividig the ieqlity by egtive mber yo mst chge the ieqlity sig rod. A exmple of lier ieqlity my be Fid the set of vles of x tht for which x x x x x x x Qdrtic Ieqlities Qdrtic ieqlities re delt with i similr mer to qdrtic eqtios (d will slly fctor whe writte o the form y x bx c ) d solved ofte with id of sketch. Some exmples re show below. Remember! The x xis is the lie y 0 x x 6 0 x x 6 0 If we were sked to fid the set of vles tht stisfy BOTH ( x )( x ) 0 ( x )( x ) 0 ieqlities below we cold dd the The criticl vles re d - The criticl vles re d - lier ieqlity to or sketch. x x 6 0 x x x x, x We re iterest i the set of vles where there is both lie d shdig. I this cse it wold be x I hve simply sed the iformtio from previos qestios to grph the ieqlities bove Be crefl with strict d iclsive ieqlities d their respective ottio, /, /. My mrks re lost o exm ppers with icorrect ottio or ppils ot cosiderig the sttemet they hve writte dow. x c be red x is greter th A ope dot wold be sed o mber lie. q c be red is greter th q or q is less th. The vles tht stisfy this ieqlity re ll those strictly less th. A ope dot wold be sed o mber lie. p c be red p is eql to or greter th. The set of vles tht stisfy this ieqlity re ll vles or more. A closed dot wold be sed o mber lie. b y c be red b is eql to or less th y or y is eql to or greter th b. A closed dot wold be sed o mber lie. Be wred! - Ieqlities c pper o qestio ivolvig the Discrimit! Steve Bldes 0

10 Sketchig Crves Yo will be expected to sketch the grphs of differet fctios. Qdrtic grphs (prbols), cbic grphs d reciprocl grphs. A sketch is ot plot from tble of vles. The exmier is lookig for bsic derstdig of the shpe, key fetres, y symptotes d poits of itersectio. Do ot try d write ot tble of vles. Qdrtic Grphs Qdrtic eqtios c be writte i the form y x bx c d their grphs re symmetric prbols. Positive (whe 0 ) qdrtic grphs with ope pwrds d hve miimm. Negtive (whe 0 ) will ope dowwrds d hve mximm. The grphs will cross the y xis whe x 0 d the x xis whe y 0. These soltios, or roots, c be fod sig the techiqes bove & mx/mi from completig the sqre. Positive Qdrtic Grph Negtive Qdrtic Grph The grph of The completed sqre form y x hs its vertex (which is miimm) t the origi d its xis of symmetry is the lie 0 y ( x b) c c help sketch the mi fetres of qdrtic grph e.g. x. y ( x ) will hve miimm poit t (, ), ope pwrds (s it s positive) d the xis of symmetry will be the lie x. The y itercept will be (0, ) s whe x 0, y. The sketch shold be smooth, ot collectio of stright lies. Ay soltios i srd form shold be left s exct vles. Cbic Grphs Cbic eqtios c be writte i the form y x bx cx d. I C they eqtios will either be fctored e.g. y ( x )( x )( x ) or will hve commo fctor of x e.g. y x x 6x sch tht the eqtio c be fctored to give y x( x )( x ). If the eqtio ws sch tht y 0 it will hve the soltios x 0, x or x which will ssist i sketch. Positive cbic grphs (whe 0 ) will strt i the th qdrt d leve i the st. Negtive cbic grphs (whe 0 ) will strt i the d qdrt d leve i the th. A exmple of egtive cbic grph cold be y ( x )( x )( x). We c see if we expded the brckets the term i x wold be egtive. If the eqtio ws sch tht y 0 the roots, or soltios, to the eqtio wold be x, x or x. These poits wold be plotted o the x xis. Whe x 0 y ()()(), which gives the (0, 0) s the poit of itersectio o the y xis. Positive Cbic Grph Negtive Cbic Grph Steve Bldes 0

11 Some cbic grphs hve repeted roots. The grph y ( x )( x ) will pss throgh the x xis t d toch the x xis t. If the eqtio hd bee sch tht t. Below is the grph of y ( x ) ( x ) y x x ( ) ( ) the it wold toch t d pss throgh Reciprocl Grphs The bsic reciprocl fctio hs the eqtio y, x 0 s divisio by 0 is defied. The grph will hve x two symptotes, the x xis (or y 0 ) d the y xis (or x 0 ). I C yo cold be sked to pply grph trsformtios sch tht reciprocl fctio wold be writte i the form y c, x b. These will be ( x b) covered i the sectio o trsformtios below. The stdrd y, x 0 grph is sketched below. As x gets x lrge y gets smll both i the positive d egtive directio. As x gets smll y gets lrge both i the positive d egtive directio sch tht y ( y teds to positive d egtive ifiity) Grph Trsformtios There re types of grph trsformtios yo my be sked to perform. Trsltios, Reflectios d Stretches. Most re firly ititive bt wy to remember them is If it s o the otside of the brcket it chges the y coordite, if it s o the iside of the brcket it chges the x coordite. Usig meric vles re ofte good wy of cofirmig this. I the exm yo will be give mrks for the correct shpe of the grph d the coordites of give poits fter ech trsformtio hs bee pplied. The sketches do t hve to be perfect d yo my eve fid describig wht yo hve doe my help ot if yor sketch is s bd s mie! For exmple Scle fctor stretch i the x directio or Scle fctor stretch of ½ i the egtive y directio Steve Bldes 0

12 Trsltios i the x directio. f( x ) The grph is trslted (trslted simply mes moved ) its i the egtive x directio or, if yo like, move left by its sch tht the vector is 0 A exmple cold be f( x) x d yo my be sked to sketch f( x ). The grph hs moved its to the right. Whilst this my seem coterititive, sig meric vles my help yo see why. Reflectios i the y xis. f( x) The sig of the x coordite chges d the reslt is reflectio i the y xis. If yo try this with f( x) x yo my be little disppoited! Stretches i the x directio. f( x ) This is scle fctor stretch i the x directio. A esier wy to thik bot this is to divide the x coordites by. If yo re give f( x) for exmple, the x coordites re divided by. As reslt the grph is sqshed towrds the y xis. f x wold see the x divided by ½, or mltiplied by, sch tht the grph looks more stretched ot. Usig s exmple Trsltios i the y directio. f( x) The grph is trslted its i the positive y directio sch tht the vector is 0. A exmple cold be f( x) x d yo my be sked to sketch f( x) Reflectios i the x xis. f( x) The sig of the y coordite chges d the reslt is reflectio i the x xis. Stretches i the y directio. f( x ) This is scle fctor stretch of i the y directio. Yo c simply mltiply the y coordites by. The exmple below shows d beig pplied to f( x) where f( x) x The grph hs moved its pwrds. Steve Bldes A stretch i the y directio will come BEFORE trsltio i y

13 Coordite Geometry Before we strt Oe tip! If i dobt, sketch it ot. Drwig c relly help with qestios o coordite geometry o mtter how strightforwrd they my seem. The whole topic is simply bot stright lies i the x, y ple. Exm qestios rge from bsic exmples of fidig eqtio of stright lie to more chllegig qestios o the re of shpes d distces betwee the vertices of shpes. The Grdiets of Lie or Lie Segmet y y The grdiet is the chge i y over the chge i x sch tht the grdiet m is give s m. x x The grdiet of the lie pssig throgh the poits A(,) d B(, ), for exmple, is ( ). Ofte errors re mde with sigs. Yo c of corse write sketch will show this is positive. The Eqtio of Stright Lie For the eqtio of stright lie yo eed thigs. () A grdiet d () A poit the lie psses throgh. A typicl qestio might be: Fid eqtio of the stright lie pssig throgh the poits A(,) d B(, ). We kow the grdiet, m, is from the previos sectio. We c ow choose either A(,) or B(, ) s poit the lie psses throgh. I m goig to choose A (, ). If I hd chose B(, ) my fil swer wold be the sme. At this stge I c tke either of the pproches below d simply sbstitte the vles i to fid eqtio. y y m( x x) y mx c Usig A(, ) d m, Usig A(, ) d m, y ( x ) () c y x 8 8 c y x c y x I hve writte the lie i the form y mx c. We my be sked to write the eqtio i the form x by c 0. The exmple bove wold be x y 0. Let s look t typicl exm style qestio o stright lies: Fid eqtio of the stright lie tht psses throgh the poit P(, ) tht is prllel to the lie y 6 x i the formx by c 0 where, b d c re itegers. We eed grdiet d poit the lie psses throgh. We hve the poit P(, ) d we c see the grdiet will be from the lie prllel to it (s the grdiet will be the sme). Usig the first method y y m( x x), where P(, ) d m y ( ) ( x ( )) y ( x ) y x 6 x y 0 We cold of corse write the eqtio i the form y mx c sch tht y x. A Remider! A stright lie crosses the y xis whe x 0 d crosses the x whe y 0. These re importt bsic fcts ofte overlooked or forgotte by stdets d my be importt prts of exm qestios. The poit of itersectio of two lies c be solved by lier simlteos eqtios s show i previos sectio. Not ll soltios will hve iteger vles so beig cofidet with frctios is very importt. Prllel d Perpediclr Lies or more prllel lies hve the sme grdiet. Perpediclr lies re t right gles d the prodct of the grdiets of two perpediclr lies sch tht m m where m d m re the grdiets of the lies. A esier wy to thik bot it cold be to cosider the grdiet of perpediclr lie is the egtive Steve Bldes 0

14 reciprocl of the grdiet of the origil lie. A exmple cold be m, m.yo will geerlly hve to stte the fct m m i exm whe workig with perpediclr lies. A bsic exm qestio cold be: The lie l psses throgh the poits A(,) d B(, ). Fid eqtio for the lie l which is perpediclr to l d psses throgh the poit C(, ). For the eqtio of stright lie we eed grdiet d poit the lie psses throgh. We hve the poit C(, ). We fod the grdiet of the lie pssig throgh A(,) d B(, ) i the previos sectio. I will cll this m where m. The grdiet of the perpediclr will therefore be the egtive reciprocl which gives m. All I eed to do is simply sbstitte these vles ito the eqtio of stright lie. I c se either method otlied previosly. I m goig to choose y y m( x x) d se the vles C(, ) d m sch tht: y ( ) ( x ) mltiplyig both sides of the eqtio by d expdig the brckets o the RHS: y 6 x I m goig to give the fil swer i the form x by c 0 where, b d c re itegers: x y 0 The Midpoit of Lie Segmet This c be doe geometriclly or by sig forml. I Lym s terms, dd the x coordites together d divide by, dd the y coordites together d divide by. More formlly the midpoit M is sch tht x x, y M y. The midpoit of the lie AB where A(,) d B(, ) is (, ) 6,, Expect some o iteger swers i exm d be prepred for exm qestios tht give yo the midpoit d oe poit where yo re expected to fid the other. Sbstittig ito the forml is esy wy to tckle these types of qestios. The Distce Betwee Two Poits or the Legth of Lie Segmet Despite its qite bewilderig forml this is simply Pythgors Theorem. Plottig the poits i the x, y ple shold mke this firly cler. The distce forml is sch tht distce d is d x x y y Steve Bldes Usig A(, ) d B(, ) gi we c fid the legth of the lie segmet AB by sbstittig the vles i. ( ) AB AB AB 0 AB This is left i exct form d simplified. Most qestios will sk for the legth i the form p q or similr. Here p, q Here is qestio to fiish: The lie x y crosses the x xis t A d the y xis t B. Fid the re of the trigle AOB whereo is the origi. Whe x 0, y d whe y 0, x We ow hve right gle trigle with bse of d height of. Usig the re of trigle: A Which gives re of sq it. Here is qick sketch

15 Seqeces d Series A seqece is ordered list tht follows give rle. A series is the smmtio of the terms i seqece. The most chllegig spect of this topic for my stdets is the ottio. There re oly few cocepts tht hve to be derstood d implemeted. If yo strggle with this topic, especilly recrrece reltios, I sggest sig tble to write dow yor vles. Bsic Seqeces Yo my be sked to geerte terms i seqece or solve costt sch s p or q. Here is typicl qestio. A seqece is defied be the rle, 0. Fid the first terms. All we hve to do here is sbstitte vles of ito the seqece, strtig with d write dow the vle. I m goig to pt these i the boxes below to keep o top of my work. 6 6 This gives s the first for terms s,, 6 d. Yo c rge the tble is overkill bt it c led to fewer mistkes. We red s sb. is ofte sed i exms too. Recrrece Reltios The terms i recrrece reltio re geerted bsed o rle likig previos terms i the seqece. A exmple cold be, 0 where we might be sked to fid the first terms of the seqece. Yo cold red this seqece s The ext terms is divided by the lst term the sbtrct Usig the tble below we c geerte terms i the seqece. We kow the first term d tht 0, so we c strt with. We kow 6 6 This gives s the first terms s,, d. This is firly simple cse bt gives yo ide o the strctre of the qestios. Ofte the seqece will be defied i terms of costt d qestio will itrodce the sm of mber of terms beig eql to give vle. Yo will hve to solve ccordigly. Let s look t typicl exmple: () Write dow the first terms i the seqece k,, where k is positive costt. (b) Give We kow k k() k i 8 fid the vle of the costtk. i k k( k ) k k This gives s the first terms s, k d k k which is prt completed. We re ow goig to prt b d simply sm the terms we hve d set them = 8 s we re smmig from i toi. Steve Bldes 0

16 k k k k 8 9 k (We re told there is oe vle of k d tht k is positive discrd k ) k 0, k Arithmetic Seqeces d Series,, 6, 8 is exmple of rithmetic seqece, is exmple of rithmetic series. Arithmetic seqeces d series hve commo differece, slly deoted s d. The differece cold be,, - or ½. The seqece or series will icrese or decrese by fixed mot. If it does t icrese or decrese by fixed mot the it s ot rithmetic. The first term is give s or. Most qestios simply ivolve fidig the th piece of iformtio fter beig give yo. Yo my hve to fid term i the seqece, the mber of terms i seqece or the sm of series for exmple. Fidig term i seqece The th term of rithmetic seqece or series,, is fod by sig ( ) d. Let s look t bsic exmple. Fid the rd term i the seqece, 7, 0,,. With y qestio like this we c simply collect the iformtio reqired d sbstitte ito the forml. = = d = (s we c see the commo differece of the seqece is +) It s lwys good ide to write, d d dow the side of the pge to collect iformtio. This will relly help with word bsed qestios too. Sbstittig i: ( ) 6 0 A more chllegig qestio might be: Give the first terms of rithmetic seqece re x, x d x, fid the vle of the th term. We kow rithmetic seqece hs commo differece therefore. Iitilly we eed to solve for x Solvig for x : x ( x ) x (x ) x x 8 x 6 Tht gives first term of (sbstittig i x ), secod term of 9 d third term of. Applyig this to the qestio: = = d = Sbstittig i: ( ) 9 Fidig the sm of Series We c se oe of formle (which yo my be sked to prove i exm) to fid the sm of series or vles give sm. We cold se S ( ( ) d ) or S ( l) where is the first term d l is the lst term or th term. A stright forwrd exmple my sk s to fid the sm of the first terms of the series Do t be tempted to try d do this mlly. Simply collect the iformtio d sbstitte it ito the forml. Steve Bldes 0

17 d Usig this iformtio to fid S S S S (() ( )( )) (0 7) ( ) S 77 The secod forml S ( l) cold be sed if yo fid the lst term ( th term) sig the forml show previosly. Sigm Nottio my lso be sed. A typicl exmple cold be r r. This is rithmetic series with commo differece of. The qestio is skig s to sm the first terms of the series from r to r. We c fid the first term by sbstittig i r which gives d fid the lst term by sbstittig i r which gives. There re terms. Be crefl with the mber of terms s r my ot strt t. l Sbstittig i the vles: S ( ) S 6(7) S If yo hve word bsed problem simply extrct the iformtio from the qestio d decided whether yo re fid term, sm or other piece of iformtio. Simply tke the vles d sbstitte ito the correct forml d solve checkig yor swer is logicl. Steve Bldes 0

18 Differetitio Differetitio is brch of clcls stig rtes of chge. We might sk orselves how does oe qtity chge i respose to other qtity chgig? A ice exmple to look t is displcemet ( s ), velocity ( v) d ccelertio ( ). All re fctios of time ( t ). The rte of chge of displcemet with respect to time is velocity. We cold sy ds v which is prooced dee ess, dee tee. We kow this from dt bsic work i mths d physics. Distce/Time = Speed. The rte of chge of velocity with respect to time is ccelertio. We cold sy dv dt. We re differetitig velocity with respect to time These re bsic exmples lthogh pplictios geerlly will ot be tested i C. The Grdiet of the Tget t Poit o Crve. y y Whe yo fid the grdiet of stright lie yo will se m s we sw i the sectio o coordite x x geometry. Whe yo re fidig the grdiet of the tget to crve t give poit yo will se the grdiet fctio or f '( x ). These re the sme thig sig differet ottio. We sy we re goig to differetite the fctio, i this cse, with respect to x. I C yo will oly be expected to differetite fctios of the form y x d will ot be expected to prove this from first priciples (despite it beig very iterestig!) Let s look t stdrd reslt for differetitio: If y x the x. If yo wt lgorithm, Mltiply dow by the power d drop the power by oe. This gives s the grdiet fctio d we c fid the grdiet of the tget to the crve t y poit by simply sbstittig the x coordite of tht poit ito either or f '( ) x. Some bsic exmples re show i the box below. We sy we re differetitig both sides of the eqtio with respect to x i ech give cse. x d y will ot lwys be the vribles of choice. We might hve to fid ds dt give s t t where we wold be differetitig s with respect (WRT) to t. Yo will eed to be comfortble with bsic frctio work, the rles of idices d se the rles for mltiplyig egtive mbers throghot differetitio d itegrtio. My mrks re lost i exm qestios throgh sloppy frctio work. Be crefl! Fid whe y x. Fid f '( x) give f ( x) x 6x Fid whe y x x y x f '( x) x x Usig the rles of idices to first simplify: ( x ) I hve ot show fll workigs y x here. Yo mst check with yor x 6x techer or exm bord o the level y x x of workigs reqired. 6 0x x At this stge I feel it s importt to discss reslts tht my be ititive bt ofte cse some cofsio. Differetitig term i x will give costt, differetitig costt will give 0. For exmple, if y x the. Vislly this shold be firly cler. The grdiet of the lie y x is. Remember is the grdiet fctio. Altertively yo c sy iitilly the power of x is sch tht y x. Whe yo redce the power by it will be 0 d by the rles of idices x 0. Steve Bldes 0

19 Here is exmple of differetitig costt: If y the 0. The lie y is horizotl lie which 0 mes the grdiet is 0. Altertively yo cold see this s y x d whe yo mltiply dow by the power it will = 0. Both of these reslts my seem obvios bt stdets sometimes s why term disppers The exmples show i the box previosly hve prodced grdiet fctio which llows s to fid the grdiet of the tget to the crve t give poit. If we wted to fid the grdiet of the tget t give poit we cold simply sbstitte i the give x coordite. Here is typicl qestio: Fid the grdiet of the tget to the crve y x t the poit A (,). We eed to fid y x ( x ) 6x Sbstittig i x, 6() This gives grdiet of 6. The grdiet of the tget (which is jst stright lie) is 6. The sketch below shows grphicl represettio. The tget will toch the crve t the poit A (,) The Eqtio of Tget A tget is simply stright lie. As we hve see before, we eed two thigs for the eqtio of stright lie. () A grdiet d () A poit the lie psses throgh. We c fid the grdiet of tget sig the grdiet fctio ( or f '( x ) ) d the simply sbstitte the vles ito the eqtio of stright lie. Here is bsic exmple: Fid the eqtio of the tget to the crve f( x) x t the poit P(, y ). We eed to fid the grdiet d the y coordite of the poit P. Let s strt with the y coordite of poit P : f( x) x f() () f() We c ow write the poit P s P (,). We eed the grdiet fctio so eed to differetite the fctio with respect to x. If f( x) x the f '( x) 8x. We sy f dshed of x. Yo my be sked to show fll workigs remember! I m ow goig to fid f '() which will give me the grdiet of the crve t the poit P (,). f '() 8() This gives grdiet of 8. I simply ow sbstitte this ito the eqtio of stright lie sig oe Steve Bldes 0

20 of the two methods show below (this is covered i coordite geometry). P(,) d m 8 P(,) d m 8 y y m( x x) y mx c y 8( x ) 8() c y 8x 7 7 c y 8x 7 The Eqtio of Norml The orml is stright lie perpediclr to the tget. If we fid the vle of the grdiet of the tget t give poit sig or f '( ) x we c se the reslt m m if perpediclr to obti the grdiet of the orml t the sme poit. Oce we hve this grdiet we c simply sbstitte the vles we re give (or hve to fid) ito the eqtio of stright lie to fid eqtio for the orml. Let s look t bsic qestio: Fid the eqtio of the orml to the crve y x t the poit P (,). We hve poit the crve psses throgh so ll we eed is the grdiet. Differetitig will give s the grdiet fctio to fid the grdiet of the tget: y x y x x If we sbstitte x ito we will fid the grdiet of the tget t the poit (,) P : Whe x, (). We hve grdiet of. Usig the reslt m m we c sy the grdiet of the orml is. It s the egtive reciprocl. All we eed to do ow is sbstitte these vles ito the eqtio of stright lie: y y m( x x) where P(,) d m y ( x ) y x x y 7 0 I hve writte the eqtio of the orml writte i the form x by c 0. The qestio will gide yo i terms of the form reqired. If the qestio sttes eqtio yo c decide the form yo leve it i. Some qestios will exted beyod these bsics cocepts d sk, for exmple, yo my be sked Wht re coordites of the other poit o the crve where the grdiet is lso? or Where does the orml itersect the crve gi? A qick sketch d bsic pplictios of either lgebr (mily simlteos eqtios) or the se of will llow yo to fid the give coordites. Steve Bldes 0

21 Itegrtio I C Itegrtio is merely see s mechicl process d s the reverse of differetitio. Applictios of itegrtio re ot cosidered til lter its. I very lgorithmic mer we c simply sy i order to itegrte we Rise by power, divided by the ew power d dd costt of itegrtio. The first prt shold mke sese s it s the reverse of differetitio x d follows from the previos sectio. The forml reslt is x c,. (s divisio by 0 is defied). I hve sed the itegrl sig here d it s ottio yo will eed to be fmilir with. Ofte qestios will sk yo to fid eqtio for y give. Remember, the grdiet fctio, is fod by differetitig both sides of eqtio with respect to x where y f( x). We c simply write y. We sy we re Itegrtig with respect to x Whe we differetite fctio tht icldes costt we ed p with oe less term s differetitig costt gives 0. A exmple might be y x x which differetites to give x. Cosider y x x 6. This wold lso hve the derivtive x s wold y x x k where k is costt. This is why c is iclded whe we itegrte. Itegrtig gives s geerl soltio d fmily of crves s we do t yet kow the vle of the costt we lost whe differetitig the origil fctio. The prticlr soltio, i.e. eqtio tht ivolves meric vle for c c be fod if we hve iitil coditios or hve eogh iformtio i the qestio to fid these iitil coditios (this is jst vle for x d y t give poit o the crve). Let s cosider bsic some exmples below: Fid (x x ) Give the crve C, y x 6 x where y f( x) psses throgh the (x x ) x x x c poit P(, ) d is sch 6 y x x Which simplifies to give: tht x. Fid 6 x x x c eqtio forc. y x x c y x Which simplifies to give: y x x c 0 y x x c y x x c Sbstittig i the vles for x d y : () () c c This gives s eqtio for C which c be writte y x x As with qestios o differetitio, yo my hve to se the rles of idices to simplify yor eqtio or expressio first. Some qestios will lso ivolve both differetitio d itegrtio. Steve Bldes 0