Module Summary Sheets. C2, Concepts for Advanced Mathematics (Version B reference to new book)

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1 MEI Mthemtics i Educti d Idustry MEI Structured Mthemtics Mdule Summry Sheets C, Ccepts fr Advced Mthemtics (Versi B referece t ew ) Tpic : Alger. Lgrithms. Sequeces d Series Tpic : Trigmetry.Trigmetricl Rtis. Trigles Tpic : Clculus. Differetiti. Itegrti Tpic 4: Curve Setchig Purchsers hve the licece t me multiple cpies fr use withi sigle estlishmet Septemer, 4 MEI, O Huse, 9 Epsm Cetre, White Hrse Busiess Pr, Trwridge, Wiltshire. BA4 XG. Cmpy N Egld d Wles Registered with the Chrity Cmmissi, umer 589 Tel: F:

2 Summry C Tpic : Alger Lgrithms Chpter Pges 9- Eercise A Q. (i), 6, 7(ii) m If = the m= lg The fllwig lws re derived frm the lws f idices. m+. If y= the = lg y, s y= m+ = lg y lg + lg y= lg y Similrly: I umericl wr. lg lg y= lg y lg usully refers t se. I lgeric. lg = lg wr the lws f lgs 4. lg =,lg = re true fr y se, d the se is therefre usully uspeci- 5.lg lg = fied. E.g. lg + lg 4 = lg lg 6 lg = lg lg 4 = lg = lg lg = lg E.g. Evlute lg 8 lg 8 = lg = lg = E.g. Slve = Te lgs: lg = lg = lg lg = =.585 lg Chpter Pge 4 Eercise A Q. (i), (vi) Chpter Pges 6-9 Eercise B Q. Epetil fuctis If y = lg the = y. The epetil fucti is the iverse f the lg fucti d s their grphs re reflectis i the lie y =. Reducti t lier frm The stright lie y = m + c hs grdiet m d itercept c. A stright lie grph c e idetified d these tw vlues fud. A curve ct e idetified s esily. Sme curves c e trsfrmed t stright lies y the use f lges. A set f dt thught t fit e f tw specific curves c therefre e tested. If set f dt is thught t ey the rule y=, the tig lgs gives lgy = lg + lg (Y = M + c) i.e. plt lgy gist, givig lie with grdiet lg d itercept lg. If set f dt is thught t ey the rule y = the tig lgs gives lgy = lg + lg (Y = MX + C) i.e. plt lgy gist lg, givig grdiet d itercept lg. C; Ccepts fr Advced Mthemtics Versi B: pge Cmpetece sttemets,,, 4, 5, 6, 7 E.g. Fid the smllest iteger vlue f fr which. = lg. = lg. = lg = = 6. = 7 lg. (N.B. this ws de usig lgs t se - y se wuld d if yu culd fid the vlue f the lgs.) E.g. Fid vlues f d s tht the dt elw fit the grph y =. 4 5 y Te lgs s tht the frmul ecmes lgy = lg + lg lg lgy These pits lie stright lie with itercept.48. i.e. lg =.48 givig =.48 ~ grdiet =.5 = y= =.7 E.g. Fid vlues f d s tht the dt elw fit the grph y =. 4 y Te lgs s tht the frmul ecmes lgy = lg + lg 4 lgy These pits lie stright lie with itercept.7. i.e. lg =.7 givig =.7 ~5.4.7 grdiet =.55 = lg 4.55 = ~. y = 5(.)

3 Summry C Tpic : Alger Sequeces d Series Chpter 7 Pges 6-6 Eercise 7A Q. (i),(iv),(i), (iv) Chpter 7 Pges 6-6 Pges 69-7 Eercise 7B Q. 8 Chpter 7 Pges 6-64 Pges 76-8 Eercise 7C Q., 4 Chpter 7 Pge 8 Eercise 7C Q. 8, 9 Sequeces d Series A sequece is set f umers i give rder. Ech umer is clled term. The th term is deted. The terms i the sequece my e defie lw f cecti. A series is sequece f umers dded tgether. E.g. A sequece f umers:,, 5, 7,. =, =, etc A series f umers S = If there re fiite umer f terms the the series my e summed t give vlue This is writte: = S = = = A series my hve ifiite umer f terms. Arithmetic Sequeces d Series A sequece f terms where the differece etwee csecutive terms is cstt. i.e. + = d, clled the cmm differece. If = d the cmm differece = d the = + ( ) d The sum t terms, S = ( + ( ) d) If l is the lst term the S = ( + l) Gemetric Sequeces d Series A sequece f terms where the rti f csecutive terms is cstt. i.e. + = r, clled the cmm rti. If = d the cmm rti = r the = r ( r ) ( r ) The sum t terms, S = = ( r ) ( r) Ifiite G.P. If r < the the terms cverge s r s The sum t ifiity, S = ( r) E.g. fid the rule fr the fllwig sequeces: (i),, 5, 7,... (ii),, 4, 8,. (iii), 5,, 7, (i) = + with = r = + ( ) = (ii) = with = E.g. If = ( + ) fid 5 = r = (iii) = + ( ) E.g. give the A.P fid (i) the 7th term d (ii) the sum t terms. E.g. give the G.P Fid (i) the 5th term d (ii) the sum t 7 terms. E.g. give the G.P fid the sum t ifiity. = 6, r = The sum t ifiity, S = r 6 = = ( ) 5 = = = 85 7 = + 6d = + 6 = 5 The sum t terms is S where S = ( + 9 ) = =, r= = r = = 54 ( ) The sum t 7 terms, S7 = = 86 ( ) 7 Chpter 7 Pge 64 Peridicity d Oscilltis A sequece which repets itself t regulr itervls is clled Peridic. A sequece i which terms lie ve d elw middle term is sid t scillte. C; Ccepts fr Advced Mthemtics Versi B: pge Cmpetece sttemets s, s, s, s4, s5, s6, s7, s8, s9, s, s, s E.g. sequece i which +5 = fr ll itegers hs perid 5. A scilltig sequece my e peridic, ut it des t hve t e. E.g. G.P. with = 8 d r = 8, 7, 9,,,,...

4 Summry C Tpic : Trigmetry - Chpter Pges 7-76 Eercise A Q. 6 Rtis fr y gles The vlue f the rtis fr y gles shuld e w y y siθ =, csθ = : tθ = Specil gles There re specil vlues fr the rtis f the gles,, 45, 6, 9 etc. These eed t e lert i surd frm s well s i decimls. E.g. si45 = cs45 =, t45 = si = cs6 =, si6 = cs = t =, t6 = Chpter Pges Grphs The grphs f y = si, y = cs = t shuld e w. E.g. frm the grphs it c e see tht si5 = si si = si cs = cs cs(-) = cs t = t6 t = t y = si y = cs y = t Chpter Pges 8-8 Eercise A Q. Sluti f equtis The use f clcultrs t slve trigmetricl equtis will ly give e swer (the "Pricipl gle"). Whe specific rge is give there my e mre th e rt r rt tht is t the Pricipl gle, s the result frm the clcultr eeds t e used with the ifrmti ve t give the full sluti. E.g. Fid vlues f i the rge 6 fr which si =. clcultrs will give = 7.5 which is t i the rge. Referece t grphs will cfirm tw rts, = = 97.5 = = 4.5 Chpter Pge 76 Eercise A Q. Pythgrs I the trigle + = c + = c c cs + si = c 4 E.g. si =,cs = si + cs = = = Chpter Pges 7-7 Eercise D Q. Applictis - Drw creful digrms!! Agles f icliti d depressi θ θ Agle f icliti Agle f depressi E.g. Fid the gle f icliti f the tp f the twer: 56m 56 tθ= m θ= 6. 6 C; Ccepts fr Advced Mthemtics Versi B: pge 4 Cmpetece sttemets t, t, t, t4, t5, t6, t7

5 Summry C Tpic : Trigmetry - Chpter Pge 7 Eercise D Q. Berigs The erig f e plce frm ther is the cmpss directi mesured s gle clcwise frm Nrth t the lie etwee the tw plces. A B E.g. A t B leves hrur A. It trvels fr 4m erig. Hw fr rth must it trvel efre it is due west f A? si7 = 4 = 4 si7 =. 76m B A 7 4m Chpter Pges Eercise B Q. (i), (i) Eercise C Q. (i), (i) Eercise E Q. (iv) Geerl Trigle Csie rule : = + c ccsa c Sie rule : = = C sia sib sic (cre shuld e te i the 'miguus' cse: eg.si =.5 = r 5 ) Are = sic Prlem slvig Drw gd cler digrm, especilly i -D prlems Pic ut d redrw y relevt trigles Use pprprite trigmetricl frmule A c B E.g. Slve the trigle = cs 6 = = C = si6 si C 6 si 6 sic = C = B = = 5 Are= 5 6 si 6 = uits A B Chpter Pges 99- Eercise F Q. (i),(iv),(i) (i),(iv), (i) Circulr Mesure Agles re mesured i degrees d ls i rdis. Rdis re prticulrly useful fr circulr mesuremet. rdi = the gle sutede rc legth f uit i circle f rdius uit. revluti = 6 = rdis rd = = rd = 7.9 E.g. rd E.g. si =, cs =.5 Chpter Pges -4 Eercise G Q. 5 Chpter Pges -4 Eercise H Q. (ii),(iv),(v)5 Legth f rc d re f sectr Arc legth, s = rθ Are f sectr, A = r θ Trsfrmtis y = si + represets trslti uits prllel t the y -is. y = si represets e wy stretch f scle fctr prllel t the y-is. y = si represets e wy stretch f scle fctr prllel t the -is. θ s r E.g. Fid the rc legth d re f the sectr f circle, rdius cm d gle 6. Are = = cm 6 = Arc = = cm C; Ccepts fr Advced Mthemtics Versi B: pge 5 Cmpetece sttemets t8, t9, t, t

6 Summry C Tpic : Clculus; Differetiti Chpter 8 Pges 9-96 Chpter 8 Pges 97- Eercise 8B Q. (i),(i) Chpter Pges 9-4 Eercise A Q. (i), (viii) Chpter 8 Pges 6-7 Eercise 8D Q. 5 Chpter 8 Pges -5 Pges 7- Eercise 8E Q., Eercise 8F Q. Chpter 8 Pges -6 Eercise 8G Q. The grdiet f curve t pit is give y the grdiet f the tget t tht pit. δy The grdiet fucti is defie Lim = δ δ The Grdiet fucti. Give fucti y = f() we c differetite t fid the grdiet fucti r the derivtive. If y = f( ) = the = f '( ) = The fucti must e summti (r differece) f terms efre it c e differetited. If y = f + g( ) the = f ' + g ' Tgets d Nrmls The grdiet f the tget t pit the curve is the grdiet f the curve. The rml t the curve t pit is perpediculr t the tget. Hece the equti f the tget t the pit (,y ) Is y y = m( ) where m is the grdiet f the curve t the pit fud ve. The equti f the rml is y y = m'( ) where mm' =. Sttiry pits At sttiry pits, (mim, miim r pits f iflei) the grdiet =. The ture f sttiry pit c e determie lig t the sig f the grdiet just either side f it. -ve ( ) ( ) ( ) +ve +ve Miimum +ve +ve -ve Mimum Sttiry pit f iflecti d derivtives The grdiet f curve is itself fucti d s hs grdiet. This is deted f "( ) = d -ve -ve E.g. Fid the grdiet fucti f the fllwig: (i) y = (ii) y = (i) y = = (ii) y = +4 + =6 +4 (Nte tht the grdiet fucti f is d f cstt is.) E.g. Fid the grdiet f the tget t the curve y = + t the pit (, 5). y = + = Whe =, = =4 E.g. Fid the sttiry pits the curve y = d determie which is mimum d which is miimum. y= = +9 =whe ( 4 +)= ( )( ) = =,. Whe =, y= ; Whe =, y= 4 = ( )( ; ) Fr the pit (,); Whe <, = -ve.- ve = +ve d whe >, = -ve. + ve = -ve, givig mimum. Fr the pit (,-4); Whe <, = -ve.+ ve = -ve d whe >, = +ve. + ve = +ve, givig miimum. E.g. Fid the equti f the tget d rml t the curve y = 5 + t the pit (, ). = 5. Whe =, = Grdiet f tget =, Grdiet f rml = Tget : y+ = ( ) y+ = Nrml : y+ = ( ) y = 4 Fr the use f d derivtives t idetify the ture f sttiry pits, see Pge 8. C; Ccepts fr Advced Mthemtics Versi B: pge 6 Cmpetece sttemets c, c, c, c4, c5, c6, c7, c8

7 Summry C Tpic : Clculus; Itegrti Chpter 9 Pges 4-6 Eercise 9A Q. Eercise 9B Q. (i) Chpter Pges Eercise B Q. (i), (viii) Chpter 9 Pges 9-46 Pges 5-5 Eercise 9B Q. (i), (i), 5 Eercise 9C Q. (ii),(viii) Itegrti is the reverse prcess f differetiti. + d = + c prvidig + c is the cstt f itegrti. f ( ) d = f ( ) ( ) ( ) ( ) ( ) f +g d = f d + g Evlutig c. We must e give crrespdig vlues f. These my e pit the curve y = f() r they my e frm iitil vlues f prlem. We sustitute these vlues i t the equti, d the fid c. Defiite itegrti d If ( F ( ) ) = f ( ) the the defiite itegrl frm t f f( ) is give y ( ) ( ) ( ) ( ) f d = F = F F The re uder grph The re uder grph c e fud s the limit f sum f res f rectgles. Are = Lim yδ = y δ Nte tht the re uder the -is will e egtive. = E.g d = d d d d d 4 = c E.g. Fid the equti f the curve thrugh (, 4) fr which =. y = ( )d y = + c Whe =, y = 4 y = c c = 4 y = + 4 E.g. Evlute ( +) d ( +)d = = + + = = 6 6 E.g. Fid the re ude the curve =, the lies =, = d the is. y - Are = d = 8 8 = ( ) = Chpter 9 Pges 6-64 Numericl itegrti Trpezium Rule The re is estimte trpezi: E.g. Estimte the re uder the curve y= etwee = d = 4. Eercise 9F Q. 6 h y d ~ y + ( y + y y-) + y where h= C; Ccepts fr Advced Mthemtics Versi B: pge 7 Cmpetece sttemets c9, c, c, c, c, c4, c5, c6 [N.B. This re c e fud directly y 4 itegrti - the tpic y.5..5 ppers i C.] 4 h = = 4 [ + (.5 +.) +.5] =.9=.46

8 Summry C Tpic 4: Curve Setchig See C: Tpic 5 Curve Setchig E.g. Setch the curve y = Plttig curve gives idicti f sme precisi. Pits (, y) re clculted (d perhps tle f vlues is cstructed). Setchig curve gives idicti f shpe. I C this icluded the pits where it crsses the es d the ehviur fr lrge. I dditi it is helpful t w the psiti d ture f the sttiry pits sympttes. Whe yu setch curve: Fid where it crsses the es i.e. fid f() d ls the vlue(s) f fr which f() = Chec fr y disctiuities. If f() ctis frcti ivlvig d if there is vlue f tht mes the demitr zer, the there is disctiuity t tht vlue f. E.g. f( ) = hs tw disctiuities, t = d = = gives y = 6 There re disctiuities As, y As y y = 6+ + = + = = ( )( + ) = i.e. =, Sttiry pits:,,,6 6 6 whe ( ) ( ) = 6 ; Whe =, > miimum Whe =, < mimum Emie the ehviur s ± L fr y sttiry pits. Chpter 8 Pges -6 Eercise 8G Q. Eercise 8H Q. 6 d derivtives = gives sttiry pit. > gives miimum. < gives mimum. Yu c visulise this y seeig tht the grdiet fucti is chgig fucti. If the grdiet is egtive just efre the sttiry vlue, zer t the vlue d psitive fter the sttiry vlue the it is icresig fucti givig miimum (See the digrms pge 6) E.g. The equti f curve is give y y = 4 +. (i) Shw tht there is ly e turig pit d fid the crdites. (ii) By tretig the epressi 4 + s qudrtic i, fid where the curve cuts the -is. (iii) Setch the curve. (i) = 4 + = whe ( 4 + ) = This is s ly t = y = = + > whe = s miimum. 4 (ii) + = ( 4)( + 5) = = 4, d, ( ) ( ) (iii) C; Ccepts fr Advced Mthemtics Versi B; pge 8 Cmpetece sttemets C, C

BC Calculus Review Sheet. converges. Use the integral: L 1

BC Calculus Review Sheet. converges. Use the integral: L 1 BC Clculus Review Sheet Whe yu see the wrds.. Fid the re f the uuded regi represeted y the itegrl (smetimes f ( ) clled hriztl imprper itegrl).. Fid the re f differet uuded regi uder f() frm (,], where