the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic logrithms C C C C Lws of logrithms C C C C Solvig = b C C C C The grph of = k C C C C Before the em ou should kow: How to mipulte surds. The lws of idices (multiplig, dividig d power of power). How to use zero, frctiol d egtive idices. The defiitio of logrithms. The lws of logrithms (multiplig, dividig d powers). How to solve =b usig logs. Wht epoetil growth/dec is d the sigificce of k d i the grph = k Surds A surd is epressio cotiig irrtiol root, e.g. +. Surds re used whe ect swer is required (d ofte feture i o-clcultor ems such s C). Emples Simplif the followig: ) ) + ) ( + )(7 + 8 ) ) 7 ) + Solutios ) = 9 = 9 = ) + = 8 ) ( + )(7 + 8 ) = + + + = + 7 7 7 ) = = 8 + 6 8 ) = = = + + 6 Idices Idices is the plurl of ide d is other me for powers. I the epressio m, where m mes (m times), is the bse d m is the ide. Emples Solve the equtios: ) = ) ( ) = 6 ) - = 6 ) ⅓ = Lws of idices m m = + m m = m m ( ) = Zero, egtive d frctiol idices = m m m = = m Solutios ) = = = ) ( ) = 6 6 = 6 = ) - = = 6 6 = = ±¼ ) ⅓ = = ( ) = 8 Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 Logrithms Defiitio log = = NB The log ke o most clcultors refers to log Lws of logrithms (to bse) log = log + log log log log = log = log A es w to remember this is log = log log = (becuse = ). = log Further results log = log = log Emples ) Show tht log6 log9 = log ) Epress log i terms of log Solutios ) log 6 log 9 = log 6 log 9 ) 6 = log 9 = log log = log log = log Solvig = b The grph of = k Tke logs of both sides: lo g = logb is the growth-fctor. Whe ou move uit to Brig to the frot: log = logb the right the -vlue is logb multiplied b. Divide through b log : = log k is the k -itercept. = k for ( > ) is show Emple Solve the equtio = 6 d illustrte this solutio o grph of = Solutio = 6 =. = log = log. 6 log = log. log. = =.79 (dp) log.79 Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS CORE MATHEMATICS The mi ides re AQA INTEGRATION OCR Before the em ou should kow: Tht itegrtio c be thought of s the reverse process to differetitio. The differece betwee idefiite d defiite itegrl. Tht costt of itegrtio is required for idefiite Idefiite Itegrls C C C C itegrl. Rtiol Powers of C C C C How to itegrte iteger powers of d rtiol powers of. Defiite Itegrls C C C C How to iterpret defiite itegrl i terms of re. Itegrtio s the reverse How to use the trpezium rule C C C C of differetitio h{ ( + ) + ( + +... + ) Fidig Ares C C C C } Trpezium Rule C C C C to pproimte re. Ed MEI Itegrtio s the reverse process to Differetitio Itegrtio is the reverse process to differetitio. Whe ou itegrte give fuctio ou re required to fid fuctio which would differetite to tht fuctio. Idefiite Itegrls A idefiite itegrl looks like f( ) d (s distict from defiite itegrl which looks like f( ) d b, see below). To fid this ou eed to fid fuctio whose derivtive is f(). If the derivtive of F() is f() the so is the derivtive of F() + c for costt c. The iclusio of + c is ecessr d this is sometimes referred to s costt of itegrtio. Rtiol Powers of d Sums of Powers of. The bsic rule for itegrtig power of is to icrese the power b d divide b the ew power. For emple + d = + c (the + c is ecessr s this is idefiite itegrl). Sums of powers of re delt with s + emple. Emple ) Fid d. Emple ) Fid d. Solutio d = + c. 6 Solutio d = + c = + c. Emple ) Fid d. Emple ) Fid + d. Solutio d = d. Solutio + d. 9 = + c = + c = + d 8 = + + 6 + c Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
Defiite Itegrls the Further Mthemtics etwork www.fmetwork.org.uk V 7 b A defiite itegrl looks somethig like f( ) d. To clculte defiite itegrl, itegrte s orml but isted of ddig the costt of itegrtio ou tke the differece betwee itegrl evluted t = d the itegrl evluted t = b. Emple. Fid Emple. Fid d. Solutio 6 8 d= 6 = = = + d. Solutio d [ ] + = = = Fidig Ares Defiite itegrls c be iterpreted i terms of res. For emple, the vlue of the defiite itegrl ( + + ) d is the re betwee the - is, the lie =, the lie = d the curve = + +. This re is therefore ( + + ) d= + + = 7. = 7 Whe curve goes bove d below the -is the betwee the limits of the itegrtio, the prt below the -is will mke egtive cotributio to the vlue of the defiite itegrl. The digrm o the left shows the curve = +. It crosses the -is t = (d t = ). Evlutig the itegrl ( + ) d gives 9. The itegrl ( + ) d= d the itegrl ( + ) d= Emple. Fid the re betwee the curves i the digrm below 6 = Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.. The re shded i the digrm is + =. Are Betwee Curves To clculte the re betwee two curves, subtrct the lower curve from the upper oe d clculte the defiite itegrl betwee the two vlues of where the re begis d eds. Solutio. The re is give b 6 ( 7 ) d = 7 6 d = 7 + 6 = 8 Trpezium Rule Use this to pproimte the re beeth curve. The ide is to use trpeziums to pproimte the re. 6 6 The re uder the curve, bove the -is, betwee the two verticl lies is pproimted b the totl re of the three trpeziums. This is ( + 6) ( 6+ 6) ( 6+ ) + + = ( + ( 6) + ( 6) + ) = 9.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS CORE MATHEMATICS POLYNOMIALS The mi ides re AQA Ed MEI OCR Arithmetic opertios C C C C Divisio of polomils C C C C Fctor theorem C C C C Remider theorem C C C C Biomil epsios C C C C Before the em ou should kow: How to dd, subtrct d multipl polomils. How to use the fctor theorem. How to use the remider theorem. The curve of polomil of order hs t most ( ) sttior poits. How to fid biomil coefficiets. The biomil epsio of ( + b). A polomil fuctio of hs terms i positive iteger powers of, d m hve costt term. The order of polomil is the highest power of pperig i the polomil. Nme Order Emple Qudrtic f() = + 7 Cubic f() = + + Qurtic f() = + + The curve of polomil of order hs t most ( ) sttior poits Qudrtic: oe sttior poit Cubic: t most two sttior poits Opertios with Polomils Polomil fuctios m be dded, subtrcted d multiplied. Emples: ) ( + ) + ( + 7 ) = + + ) ( + ) ( + 7 ) = 9 + 9 ) ( + )( + 7 ) = ( + 7 ) ( + 7 ) + ( + 7 ) = + 9 6 + 8 A polomil fuctio c be divided b lier fuctio to give quotiet with or without remider. Emple: Give tht ( ) is fctor of ( ) fid ( ) ( ) Solutio: Let ( + b + c)( ) so + (b ) + (c b) c Equtig coefficiets gives =, b =, c = Usig log divisio Qurtic: t most three sttior poits + + + + + Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 Fctor d Remider Theorems Let f() be polomil fuctio i Fctor Theorem f() = so ( ) is fctor of f() = is root of f() f(/b) = so (b ) is fctor of f() Remider Theorem The remider whe f() is divided b ( ) is f() Emples: Show tht = is root of f() = +. Hece stte fctor of f() f() = + = so ( ) is fctor of f() Show tht ( ) is fctor of f ( ) = + 6 ( ) is fctor if = ⅔ is root of f() = f(⅔) = (⅔ ) (⅔ ) +(⅔) 6 = so ( ) is fctor of f() Fid the remider whe f() = + is divided b ( ) f() = + = whe f() is divided b ( ) remider is Solvig equtios / Curve sketchig Let = f() = + b + c + d Grph of = f() itersects -is t (, d) Fctorise f() to solve f() = Use roots of f() = to fid co-ordites of poits of itersectio with -is Sketch grph Biomil Epsios Emple: Sketch the grph of = f() = + 6 Grph psses through (, ) Fctorisig: f() = ( + 6) = ( + )( ) hece f() = =, = or = For turl umber ( + b) = + C - b + C - b + + C - b - + C - b - + b Biomil coefficiets c be foud b usig Pscl s trigle usig tbles usig the formul C r = Emples:! r!( r)! Write the biomil epsio of ( + ) ( + ) = + + 6 + + = 6 + + + 8 + Fid the term coefficiet of i epsio of ( ) 8 Term i = 8 C () = 6 7 () = coefficiet of is - - - - - - Use the first three terms of the biomil epsio of ( + ) 6 to fid pproimtio for.7 6. ( + ) 6 = + 6 + +. Usig =.7 gives ( +.7) 6 + (6.7) + (.7 ) +. =.9.7 6 =.7 Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS CORE MATHEMATICS SEQUENCES AND SERIES The mi ides re AQA Ed MEI OCR Defiitios of sequeces C C C C Sigm ottio for series C C C C Arithmetic progressios C C C C Geometric progressios C C C C Before the em ou should kow: A sequece is ordered set of umbers. A series is the sum of the terms of sequece. The formule for k th term d sum of rithmetic series (progressio). The formule for k th term d sum of geometric series (progressio). How to uderstd d use ottio. The coditios for geometric sequece to coverge. The formul for the sum of ifiite geometric sequece. A sequece is ordered set of umbers u, u, u,, u k,, u where u k is the geerl term. A series is the sum of the terms of sequece: u + u + u + + u There re differet ws to defie sequece: Iductive defiitio: u k+ = f(u k ) with first term u Deductive defiitio: u k = f(k) for k =,,, Emples A periodic sequece is oe such tht for some fied iteger p, u k+p = u k for ll k A oscilltig sequece ltertes either side of middle vlue. The sequece show o the digrm below is,,,,,,,, 6 - - -6 6 8 d is periodic with p =, i.e. u k+ = u k for ll k ) A sequece is defied b u k+ = u k + with first term u =. Write the first five terms of the sequece Solutio:, ( ) + =, ( ) +=, ( ) + =, ( )+= so the first five terms re,,,, ) A sequece is defied b u k = k - for k =,,, Write the first five terms i the sequece Solutio: =, - =, - =8, - =, - = so the first five terms re,, 8,, ) Fid the vlue of for = k k k Solutio: the first four terms re = 6, =, =, = 8 So k = 6 + + + 8 = 9 Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 Arithmetic Progressios A sequece i which there is costt differece (d) betwee successive terms. If the first term is, the the geerl term is u k = + (k )d The lst term, u = l is give b l = + ( )d The sum of the terms, S is give b S = ( + l) = ½ [ + ( )d] Emple A rithmetic progressio hs terms, 7,, d S = Fid the umber of terms d the lst term. Solutio: First term =, costt differece d =, S = ½[ + ( )d] = so ½[6 + ( )] = d + ( ) so + = d ( )( + ) = hece = [igore =.]. Hece lst term l = + ( )d = + 9 = 9 Geometric Progressios A sequece i which there is costt rtio (r) betwee successive terms. If the first term is, the the geerl term is u k = r k- The lst term, u = l is give b l = r - ( r ) r ( ) The sum of the terms, S is give b S = or (these re equivlet) r r Provided < r <, S coverges to limit kow s the sum to ifiit: S = r Emple A geometric progressio hs d term 6 d the th term is.96. Fid the sum of the first terms to d the sum to ifiit. Solutio: d term = 6 so r = 6 th term =.96 so r =.96 r.96 Solvig simulteousl: r.6 r.6 r = 6 = = Substitutig for r i r = 6 gives = (.6 ) hece S = = 8.88.6 S = =.6 Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS CORE MATHEMATICS TRIGONOMETRY The mi ides re AQA Ed MEI OCR Before the em ou should kow: The grphs of si, cos d t. How to fid the si, cos d t for o, o, o, 6 o d 9 o. Grphs of trigoometric How to work out the si, cos d t for gle θ, C C C C fuctios where o θ 6 o. Specil gles C C C C How to use trigoometric idetities Idetities C C C C How to solve simple trigoometric equtios. Solvig trigoometric How to use the sie d cosie rules. C C C C equtios How to fid the re of trigle which is t rightgled. Sie d cosie rules C C C C Are of trigle C C C C Tht gles c be mesured i rdis, where π rdis = 8 o. Covertig betwee C C C C How to fid the legth of rc d the re of rdis d degrees sector usig rdis Legth of rc d re C C C C Trsformtios of grphs of trigoometricl of sector i rdis fuctios. Trigoometricl fuctios of specil gles Agle 6 9 Cosie / / / Sie / / / Tget / Grphs of trigoometricl fuctios = cosθ. θ 6 9 8 7 6 -. - = siθ. θ 6 9 8 7 6 -. Trigoometricl Idetities siθ tθ = cosθ si θ + cos θ = - = tθ θ 6 9 8 7 6 - - Emples: Solve the equtios ( o θ 6 o ): ) cosθ = ) siθ + = cosθ = ⅔ so θ = 8. o (from the clcultor. This is clled the pricipl vlue.) or θ = 6 o 8. o =.8 o so siθ = =. Pricipl vlue θ =.6 o so θ = 8 o +.6 o =.6 o or θ = 6 o.6 o =.8 o ) tθ + = so tθ = =. Pricipl vlue θ = 68. ο so θ = 68. ο +8 ο =.8 ο or θ = 68. o + 6 o = 9.8 o Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.
the Further Mthemtics etwork www.fmetwork.org.uk V 7 Solve: si θ = cos θ ( o θ 6 o ) Substitute i si θ = cos θ givig ( - cos θ) = cosθ cos θ = cosθ cos + cosθ = (cosθ )(cosθ + ) = cosθ = ½ or θ = [o solutio] θ = 6 o or θ = o Sie rule d cosie rule B Emple: I trigle ABC A = 8 o, b = cm, d c = 7 cm. Clculte the legth of, B d C A c b C Solutio: fid the missig side usig the cosie rule = + 7 ( 7 cos8 o ) so = 78.8 d = 8.7cm (d.p.) If ABC is trigle with sides, b, c, the Sie rule = b = c si A si B si C Cosie rule = b + c bc cosa The re of trigle is give b: ½ bc sia = ½ c sib = ½ b sic Circulr mesure Agles c be mesured i rdis where π rdis = 8 o For sector of circle with rdius r d gle t cetre θ rdis: Legth of rc = rθ Are of sector = r θ Fid B usig the sie rule (iverted becuse we re fidig gle o si B si 8 = 8.7 si 8 si B = 8.7 si B =.977679 B = 77. o Thus C = 8 77. 8 =.9 o o Emple: A sector of circle hs rdius cm d gle t the cetre. rdis. Clculte the perimeter d re of the sector. Arc legth =. = 7. cm So perimeter = + + 7. = 7. cm Are of sector = ½. = 8.7 cm Grph showig = si d = si Trsformtios of trigoometric grphs Trigoometric grphs c be trslted d stretched I the sme w s other fuctios Grph showig = cos d = cos Disclimer: Ever effort hs goe ito esurig the ccurc of this documet. However, the FM Network c ccept o resposibilit for its cotet mtchig ech specifictio ectl.