Rahul Chacko. IB Mathematics HL Revision Step One

Transcription

1 IB Mthemtics HL Revisio Step Oe Rhul Chcko Chpte. Aithmetic sequeces d seies; sum of fiite ithmetic seies; geometic sequeces d seies; sum of fiite d ifiite geometic seies. Sigm ottio. Aithmetic Sequeces Defiitio: A ithmetic sequece is sequece i which ech tem diffes fom the pevious oe y the sme fied ume: {u } is ithmetic if d oly if u u d Ifomtio Booklet u u d Poof/Deivtio: u u d u d u d u d u u u u u d Deivtios: u u d u u d u u d Ifomtio Booklet S u d u u Poof:. S = u + u + u u = u + (u + d) + (u + d) + (u + 3d) + + (u + ( )d) = u + (u d) + (u d) + (u + 3d) + + (u ( )d) S = (u + u ) S u u Deivtios

2 S u u S u u S u u Geometic Sequeces Defiitio: A geometic sequece is sequece i which ech tem c e otied fom the pevious oe y multiplyig y the sme o-zeo costt. u {u } is geometic if d oly if, whee is costt. u Ifomtio Booklet u u Poof: u u u u u u u u u u Deivtios: u u u u u (o-clculto ppe) u u u Compoud Iteest: (clculto ppe) 00% i% u u, whee u iitil ivestmet,, i iteest te pe 00% compoudig peiod, = ume of peiods d mout fte peiods. u

3 Ifomtio Booklet u u S, Poof: S = u + u + u u - + u = u + u + u + u u + u S = (u + u + u + u 3 + u u ) + u S = (S u ) + u S S = u u S ( ) = u ( ) u S Deivtios S u u S u S Sum to ifiity u S, Poof: u S,, 0, u S 0, u S, Sigm Nottio f ) (

4 is the ume of tems, f() is the geel tem d = the fist vlue i the sequece. epesets the sum of these the tems i this pogessio. Chpte. Epoets d ithms. Lws of epoets; lws of ithms. Chge of se. y Defiitio of Epoets: mes y, i.e. the umeto i epoet is the powe to which ume is ised d the deomito is the oot to which it is loweed. Lws of Epoets y y 0 y y Defiitio of Logithms: y mes y. Notes: l e, whee e is the uique el ume such tht the fuctio e hs the sme vlue s the slope of the tget lie, fo ll vlues of. y y y y y y y y THEREFORE Othe Sigifict Equtios e l Poof:

5 e e e l l l Othe Sigifict Equtios c c Poof: c c c c c c c c Lws of Logithms Poof: Lws of Logithms Poof:

6 Chpte.3 Coutig piciples, icludig pemuttios d comitios. The iomil theoem: epsio of,. The Poduct Piciple If thee e m diffeet wys of pefomig opetio d fo ech of these thee e diffeet wys of pefomig secod idepedet opetio, the thee e m diffeet wys of pefomig the two opetios i successio. The ume of diffeet wys of pefomig opetio is equl to the sum of the diffeet mutully eclusive possiilities. Fctoil Nottio! is the poduct of the fist positive iteges fo.! ( )! ( )( )! ( )( )( 3)! etc! ( )! 0! 0! Pemuttios (i lie) A pemuttio of goup of symols is y gemet of those symols i defiite ode. Epltio: Assume you hve diffeet symols d theefoe plces to fill i you gemet. Fo the fist plce, thee e diffeet possiilities. Fo the secod plce, o mtte wht ws put i the fist plce, thee e possile symols to plce, fo the th plce thee e possile plces util the poit whee =, t which poit we hve stuted ll the plces. Accodig to the poduct piciple, theefoe, we hve ( ) ( ) ( 3)... diffeet gemets, o!

7 If symols e fied i plce, evet to the poduct piciple. Sice thee is oly oe possiility fo whicheve plce the symol(s) is fied t, the ume of possiilities is! equl to whee is the plce t which tht symol is fied. Pemuttios (i cicle) The est wy to thik of pemuttios i cicle is s pemuttios i lie whee you hve to divide the oml totl ume of possiilities fo pemuttios i lie y the ume of diffeet ideticl positios the symols c hve whee they hve simply shifted to the ight y oe plce. Logiclly, thee e diffeet positios whee this is! the cse thus the ume of possiilities is equl to ( )!. Comitios A comitio is selectio of ojects without egd to ode o gemet. C C C is the ume of comitios o distict symols tke t time. Sice the comitio does ot tke ito ccout the ode, we hve to divide the pemuttio of the totl ume of symols ville y the ume of edudt possiilities. Sice we e choosig pticul ume of symols, these symols hve ume of edudcies equl to the pemuttio of the symols (sice ode does t mtte). Howeve, we lso eed to divide the pemuttio! y the pemuttio of the symols tht e ot selected, tht is to sy.!!!!!! Biomil Epsio Tkig ech tem i the epsio of, to e symol, it c e see tht the coefficiet fo ech symol, which is diffeet vlue of, ( eig the epoet d eig the powe to which e is ised i this pticul tem) is detemied y the ume of diffeet possile gemets cotiig symols d symols. Thus, the coefficiet fo y symol is equl to. Sice the epsio of, is effectively the sum of ll the symols d thei coefficiets, we e left with the esult tht, d the vlue 0 of ech tem is T. The costt tem is the tem cotiig o viles, ofte. See H&H p.5 emple 8.

8 Whe fidig the coefficiet of, lwys coside the set of ll tems cotiig (see H&H p.5 emple 9). Chpte.4 Poof y mthemticl iductio. Fomig cojectues to e poved y mthemticl iductio. Niety pecet of the poits fo mthemticl iductio questios c e otied simply y usig the coect fom, so it is vey impott to memoise the two foms of mthemticl iductio d ly the poof out ccodigly. The Piciple of Mthemticl Iductio Suppose P is popositio which is defied fo evey itege,. Now if P is tue, d P k is tue wheeve P k is tue, the P is tue fo ll. Sums of Seies Fist step: Pove tht P is tue if povig fo ll (icludig 0). +, o tht P0 is tue if povig fo Secod step: Assume tht Pk is tue, d stte the cosequeces of this ssumptio. Thid step: Usig the ssumptio, mipulte you equtio (the geel equtio fo k tems + the (k+) th tem) to give you the geel equtio whee the vile k hs ee eplced y k+ wheeve it ppes. Fil step, stte: Thus P k is tue wheeve P k is tue. Sice P is tue, P is tue fo ll [foemetioed set of umes]. Note: Alwys look fo commo fctos. Divisiility Pove tht f is divisile y s fo, Fist step: Pove tht P is tue whee is you lowe limit fo which P k is tue. Secod Step: Assume tht P k is tue: f k sa whee A is itege. Thid Step: Septe out k, vlue of k solved fom f k. fom f k d sustitute the tem out fo the Fouth step: Epess k f s poduct of s. Fil step, stte: Thus f k is divisile y s if k tue wheeve P k is tue d P is tue, P is tue. f is divisile y s. Hece, P is k

9 Chpte.5 Comple ume i ; the tems el pt, imgiy pt, cojugte, modulus d gumet. Ctesi fom z i. Modulus-gumet fom i z cos isi e cis. The comple ple, o Agd digm. Ctesi fom I Ctesi fom, ll comple umes z e witte i the fom + i,,. cotis o imgiy compoet d s such is kow s the el pt whees is poduct of the imgiy uit i, d thus is kow s the imgiy pt. It is impott to ote tht el umes e meely comple umes with 0 d imgiy umes e meely comple umes with 0, thus the ctegoy el umes is suset of comple umes. Equlity of Comple Numes Two comple umes e equl whe thei el pts e equl d thei imgiy pts e equl, i.e. if i c di, the c d d. Poof: Assume d. i c di,,, c, d i di c d i c c i d Theefoe the sttemet d must e flse sice i is imgiy d theefoe d c. c is el, d Cojugtes The cojugte of comple ume z i, is i z *. I othe wods, comple cojugtes e comple umes z whee the sig of the imgiy pt is iveted. Popeties: * * z z * * * z z z z d z z z z z z z * z z d, z * z * * z * * z, z, 3 z z * d zz * e el. * * z 0

10 Pol Fom Comple umes c, howeve, e epessed o Agd digm. A Agd digm is like Ctesi digm whee the y-is vlues epeset the coefficiet of the imgiy pt of pticul comple ume d the -is vlues epeset the vlue of the el pt of tht ume. Thus, comple umes c e epeseted o poits o this digm, whee, s stted ove, its positio eltive to the -is detemies the vlue of its el pt d its positio eltive to the y-is detemies the coefficiet of its imgiy pt. Epessig the comple ume o Agd digm llows us to epess it i tems of its positio eltive to the oigi: with gumet, o gle θ (i dis) eltive to the -is i the positive diectio (goig ti-clockwise) d modulus, o legth of the stight lie dw etwee the poit d the oigi. We c thus fid the vlue of the el pt d imgiy pt of the comple ume i pol fom: As is demostted ove, the el pt c e sid to e equl to cosθ d the coefficiet of the imgiy pt c e sid to e equl to siθ. Thus, the comple ume z c e epessed s z cos i si cos isi cis. This is kow s the pol, o modulus-gumet fom. Note: The modulus of comple ume is the mgitude of its uit vecto o Agd digm. As such, it c lso e witte s z d is equl to (fom the comple ume s epessed i Ctesi fom). Note: The gumet of y comple ume z c e epessed s g z the th θ. Notes (ll these e esily pove so poofs shll ot e mde): z * z z zz * z z z z z z d, z z z z z z3... z z z z3... z d 0 z z, + Notes (Poofs my potetilly e povided t lte dte ): cis cis cis cis cis cis cis k cis, k

11 Covetig etwee Ctesi d Pol Fom z i cos i si cos, si ct (fom Agd digm) (fom Agd digm) Chpte.6 Sums, poducts d quotiets of comple umes. Opetios with comple umes e ideticl to those fo dicls. i c di c d i i c di c d i ic di c di ci di c d d c i i c di c di ci di c di c di c di c d i c d c d c d i c d Chpte.7 De Moive s theoem. Powes d oots of comple ume. De Moive s theoem sttes tht fo y comple ume z: z cis Notes: i cis g z g z z cis - z z z cis ct Poof y mthemticl iductio fo De Moive s Theoem: Requied to pove tht: z cis z cis P : If, the z z P k k Assume P k is tue z cis z cis k k z cis z cis P k+ : z z z k cis k z cis k k cis k cis k cis cis is tue. Thus P k+ is tue wheeve P k is tue d P is tue P is tue.

12 Roots of Comple Numes The th oots of comple ume c e the solutios of z = c. Two methods of solvig: fctoistio d the th oot method. th oot method: z z c z z cis k cis k k cis, k 0,,..., - Note: the th oots of uity e the solutios of z = Chpte.8 Cojugte oots of polyomil equtios with el coefficiets. Rel Polyomils A el polyomil is polyomil with oly el coefficiets: Polyomils Degee Nme, 0 Lie c, 0 Qudtic 3 c d, 0 3 Cuic 3 c d, 0 4 Qutic k k k3... k k, k 0,, c, d, e, k (Geel) o k is the ledig coefficiet d the tem ot cotiig the vile is the costt tem (s stted ove). Polyomil Multiplictio Evey tem i the fist polyomil must e multiplied y evey tem i the othe. Algoithm: Sythetic multiplictio detch coefficiets d multiply s i odiy multiplictio of lge umes 3 Emple: c d e f c d e f f f cf df e e ce de 0 e f e f ce cf de df 4 e 3 e ce c d f e f f cf de df 0

13 Polyomil Divisio Divisio Algoithm: H&H p.70 LEARN (it s hell to type out o wod) If P is divided y util costt emide R is otied: P R Q( ) whee is the diviso, P () is the polyomil, Q () is the quotiet d R is the emide. Deivtios: P Q R P R Q R P Q P R Q Roots d Zeos A zeo of polyomil is vlue of the vile which mkes the polyomil equl to zeo (-is itecept). The oots of polyomil equtio e vlues of the vile which stisfy the equtio i questio. The oots of 0 P e the zeos of P. I polyomils of the fom: P Q the oots of the equtio occu t 0 Whe fidig the oots of polyomils, it is impott to fctoise poduct of lie fctos d qudtic fctos. Lie fctos of qudtics: d the solutios of Q. Q util it is c 0 ( )( ) 4c 4c 4c 4c 4c c 4c * If z, c z z 0, z z d z * e cojugte oots of c 0 Fo polyomils of eve degee, evey oot hs cojugte oot, fo polyomils of odd degee, evey oot ecept oe hs cojugte oot. Evey polyomil of degee hs oots, ut whee thee is fcto 0, the polyomil hs epeted oots.

14 Chpte. Cocept of fuctio f : f( ) : domi, ge; imge (vlue). Composite fuctios f g; idetity fuctio. Ivese fuctio f. Algeic Test: If eltio is give s equtio, d the sustitutio of y vlue fo esults i oe d oly oe vlue of y, we hve fuctio. (Note tht the lgeic test c e used s defiitio fo wht fuctio is). Geometic Test: If t y poit the -is, thee e two y-is vlues, the gph is ot fuctio. I composite fuctios, the ight-most fuctio is fuctio of d those tht e t e meely fuctios of the fuctios the ight of it. Tht is to sy tht if you defie fuctio f s eltioship give s equtio whee the sustitutio of y vlue fo esults i oe d oly oe vlue of y, the defiitio of the fuctio g f : g f is eltioship give s equtio whee the sustitutio fo y vlue fo f esults i oe d oly oe vlue of y. The domi of eltio is the set of pemissile vlues tht my hve. The ge of eltio is the set of pemissile vlues tht y my hve. Note: The stted domi d ge of the eltio i questio must e pplied efoe it is detemied whethe o ot this eltio costitutes fuctio. Itevl Nottio :, :, y y y y :, :, y y y y : y: y y :, :, y y y y [ mes tht is the lowe limit d tht c equl ] mes tht is the uppe limit d tht c t equl ] mes tht is the uppe limit d tht c equl [ mes tht is the uppe limit d tht c t equl. ], [ whee thee is o limit o oe side, must e ecluded i the ottio.

15 Ivese fuctios Whee fuctio f sustitutes vlues fo diffeet vlue, the ivese fuctio f is the fuctio tht sustitutes f vlues fo vlues. Gphiclly, the ivese of fuctio is tht fuctio eflected i the lie y. As diect esult of this, y fuctio i which moe th oe vlue is sustituted to the sme vlue of y (clled my-to-oe fuctio) hs o ivese. This is epeseted gphiclly y the hoizotl test, whee if you c dw lie pllel to the -is tht cosses the fuctio moe th oce, the fuctio hs o ivese (if the covese is tue, the the fuctio is oe-to-oe fuctio). Othewise stted, oe-to-oe fuctios hve ivese whees my-to-oe fuctios do ot. The domi must e tke ito ccout whe ctegoisig fuctio s oe-to-oe the th my-to-oe, fo emple f si ppes to e my-to-oe fuctio, ut if the domi is 0, the the fuctio is oe-to-oe fuctio d hs ivese. Chpte. The gph of fuctio; its equtio y f. Fuctio gphig skills: use of GDC to gph viety of fuctios, ivestigtio of key fetues of gphs, solutios of equtios gphiclly. Whe fuctios e gphed, the fuctio f is lwys epeseted o the y is. The symptote is o y vlue fo which thee is defied vlue of the fuctio, geelly ppeig whee the deomito i the fuctio hs goe to 0. The oots o equtios (o zeos of fuctios) c e foud gphiclly y otig the poits t which the fuctio cosses the is. Fo the solutio to two equtios epeseted s fuctios s f d g espectively whee f g, the solutio(s) c e foud t the itecept(s) of the two fuctios. Chpte.3 Tsfomtios of gphs: tsltios; stetches; eflectios i the es. The gph of y f s the eflectio i the lie y of the gph of y f. The gph of y f fom y f. The gphs of the solute vlue fuctios, y f. y f d Tsfomtios: y f tsltes the gph uits i the positive y diectio. y f tsltes the gph uits i the positive diectio. y pf ( ) stetches the gph pllel to the y-is with fcto p. y f stetches the gph pllel to the -is with fcto q. q

16 y f eflects the gph i the -is. y f eflects the gph i the y-is. If give f d equied to gph f, elise whee the gph of f ( ) gets steepe, the gph of gets fltte, d tht the gete the mgitude of f f y poit, the mgitude of teds towds 0, the slope gets steepe util it eches the symptote t f( ) 0. t f is lesse. Howeve, whee f ( ) is less th d y f( ) The domi whee f( ) 0 is eflected i the -is, ll else is uchged. y f The domi 0 is tke d the gph is eflected i the y-is. Popeties of, R is the distce fom 0 to o the ume lie 0 y y y y y, Z y y y is the diffeece etwee d o the el ume lie y y Chpte.4 The ecipocl fuctio, 0 : its self-ivese tue. Sice the fuctio tht mps oto is kow s the ivese of, 0 d is, the ecipocl fuctio is effectively its ow ivese. Chpte.5 The qudtic fuctio. The fom h k c. The fom. : its gph. Ais of symmety p q All qudtic fuctios c e witte i the fom c. Howeve, some c e witte i the fom p q, 0 while othes c lso e witte i the

17 h k. If k 0, the qudtic c oly e witte i the ltte fom fom. (shot of usig comple oots) howeve if k 0, the qudtic c e epessed i oth foms. The fist fom is eeficil fo fidig the oots of the qudtic, sice they e equl to p d q. The secod, howeve, is much moe useful fo fidig the is of symmety (h) of the qudtic d detemiig whee the miimum (o mimum fo 0 ) poit (k) of the qudtic is. Thks to the popeties of the secod equtio, puttig the geel equtio fo c i the fom h k llows us to fid the geel equtio fo the positio of the is of symmety d the positio of the mi/m poit o the gph. Completig the sque: c c c 4 c ch k 4 h, ck 4 h Chpte.6 The solutio of discimit c 4. c 0, 0. The qudtic fomul. Use of the Completig the sque lso llows us to fid the geel equtio fo the oots of equtio.

18 c 0 c c 4 c0 4 4c 4 4c c c 4 4 c 4 c This equtio is kow s the qudtic fomul. Wht is impott to ote i this fomul is the discimit detemies whethe o ot the qudtic hs el solutio. 4c. This Whee 0, the qudtic hs two epeted oots, whee 0, the qudtic hs two diffeet oots d whee 0the qudtic hs o el oots. Chpte.7 The fuctio:, 0. The ivese fuctio, 0. Gphs of y d y. Solutios of usig ithms. The fuctio hs cuvtue tht teds towds zeo s teds towds ifiity. The cuvtue lso teds towds zeo s teds towds egtive ifiity. The gdiet, howeve, costtly iceses fom 0 t to t. Its ivese fuctio is 0 meely eflectio of this i the -is. We c fid the solutios of usig ithms: Fo hde equtios, the solutios c e foud the followig wy:

19 Chpte.8 The epoetil fuctio 0. e. The ithmic fuctio l, It is kow fom topic. tht the equtio fo compoud iteest is: 00% i% u u, whee u iitil ivestmet,, i iteest te pe 00% compoudig peiod, = ume of peiods d mout fte peiods. It is cle tht equtio whee u u i, d tht if we tet the iitil ivestmet to e the 0 th tem, we get u u i 0 0. If is (isted) the pecetge te pe ye, t the ume of yes d N the ume of Nt iteest pymets pe ye, the u u u 0 0 N N Let N t t, the u u u 0 0 l But e whee d e y el ume, thus: t u u 0 N t lim e t u u e fo lge vlues of. 0 Gowth d decy woks i simil wy, puttig the sequece i the fom u u0 f. Chpte.9 Iequlities i oe vile, usig thei gphicl epesettio. Solutio of g ( ) f( ), whee f, g e lie o qudtic. Iequlities c e solved usig sig digm, whee it is key to ememe tht the ow epesets ll the vlues fo which f ( ) 0, egdless of whethe we e tyig to fid the vlues fo which the ovell fuctio is gete th o less th.

20 Iequlity lws: c c, c 0 c c d, c 0 c c d 0.. c c. c c Sig Digm Notes: The hoizotl lie of sig digm coespods to the -is. The citicl vlues e vlues of whe the fuctio is zeo o udefied. A positive sig (+) coespods to the fct tht the gph is ove the -is. A egtive sig () coespods to the fct tht the gph is elow the -is. Whe fcto hs odd powe thee is chge of sig out tht citicl vlue. Whe fcto hs eve powe thee is o sig chge out tht citicl vlue. Fo qudtic fcto c whee 4c 0; c 0, R, 0, c 0, R, 0. Thee is o citicl vlue i eithe cse. Solvig iequlities: Pocedue Mke the RHS = 0 y tsfeig ll tems to LHS. Fully fctoise the LHS. Dw sig digm of the LHS. Solve. Note: Do ot, ude y cicumstces, coss-multiply. This emoves ceti tems fom the equtios d pevets oe fom fidig ll the solutios. Chpte.0 Polyomil fuctios. The fcto d emide theoems, with pplictio to the solutio of polyomil equtios d iequlities. It is impott to ote tht the highe the ode of the polyomil fuctio, the steepe the lie fo d the less steep the lie fo. Odd odes hve egtive vlues whee < 0 ut eve ode lwys hvig positive vlues uless tslted dow. Repeted oots mege o gph to ppe s oly oe oot. Repeted oots sigify poit of iflectio. The fcto theoem Accodig to polyomil divisio, k R P( ) Q( ) Thus, whee = k P( k) Q( k) 0 R P( k) R

21 Theefoe whe polyomil P () is divided y k util costt emide R is otied the R P(k) The emide theoem If k is zeo of P() thus: P( ) P( k) 0 R 0 P( ) ( ) k k is fcto of P() k P k I geel: is zeo of ( ) is fcto of ( ). Usig the emide theoem to fid the emide llows us to fid how f the P( ) miimum/mimum poit of the gph of R 0 is shifted wy fom the -is. Q( ) Usig the fcto theoem llows us to quickly fid the oots (if the fcto is kow) o the fctos (if the oot is kow) of the polyomil equtio d thus solve polyomil equtios d iequlities quickly. Chpte 3. The cicle: di mesue of gles; legth of c; e of secto. Oe di is defied s the gle tht suteds c of legth equl to the dius. Thus, c totl C C c totl Thus di is c This mes tht: 360, so of the gle of full cicle, thus thee e dis pe full tu. Thus, the legth of c i dis is: l Ad the e of secto is: A P c 80 c d. 80 Chpte 3. Defiitio of cos d si i tems of the uit cicle. Defiitio of si t s. Defiitio of sec, csc d cot. Pythgoe idetities: cos cos si ; t sec ; cot csc.

22 The Sie Fuctio The sie fuctio of mes tht fo ight gled tigle with gle (s show i the digm), the tio of the legth of the opposite side to tht of the hypoteuse is equl to the sie fuctio of. Pythgos s theoem sttes tht c whee is the opposite side, is the djcet side d c is the hypoteuse, thus we c stte tht: si c cos c si t c cos c The uit cicle hs dius. This mes tht the hypoteuse hs dius. We c thus use Pythgos s Theoem to stte tht d tht theefoe d. Thus, i tems of the uit cicle, cos c si c Ad t

23 This tells us the vice of ech fuctio i tems of the uit legth of the opposite d i tems of the uit legth of the djcet. I the plottig the sie cuve, we plot o the y-is d o the -is. Thus we oti: si cos cos cos si si cos Ad t t si si t si si si si t si si si cos t sec cos Also si t si si cot t si si si cot csc si si Chpte 3.3 Compoud gle idetities. Doule gle idetities. Compoud gle idetities Equtio ooklet: si( A B) si Acos B cos Asi B

24 Poof Coside P(cos A, si A ) d Q(cos B, si B ) s y two poits o the uit cicle. Agle POQ is A B Usig the distce fomul: A B A B PQ cos cos si si PQ cos A cos Acos B cos B si A si Asi B si B cos Asi Acos Bsi B cos Acos Bsi Asi B cosacosbsiasib cosacosbsiasib But ccodig to the cosie ule, PQ cos AB cosab cos A B cos Acos B si Asi B If we stte tht cos coscos sicos, we c let B d A. A B A B A B B si B B cosb A B A B A B cos coscos sicos c lso e witte s: cos cos cos si si cos cos cos si si si cos cos cos cos si si si A B cos AB cos AB cos Acos Bsi Asi B si Acos Bcos Asi B It is lso cle tht si AB si Acos B cos Asi B t t A t B AB tatb

25 Doule gle idetities si A si A A si Acos Acos Asi A si Acos A si Asi A A cos Acos Asi Asi Acos A si A cos A cos A si A cos A cos A cos A cos cos si si si si A A A A A A t A t A t A t A t t t Tig idetities tle: A A A Agle Sie Cosie Tget Chpte 3.4 The cicul fuctios si, cos d t ; thei domis d ges; thei peiodic tue; thei gphs. Composite fuctios of the fom f c d. The ivese fuctios csi, ccos, si ct ; thei domis d ges; thei gphs. The peiod of the utslted sie d cosie fuctios is d thei ge is., epesets stetch fom the -is of fcto, f si c d I the fom epesets stetch fom the y-is of fcto, c epesets tsltio c uits to the left d epesets tsltio d uits upwd. The ecipocl fuctios c e deived though commo sese. The ivese fuctio c oly e deived y estictig the domi to fo ll the tigoometic fuctios, the eflectig i the lie y

26 Chpte 3.5 Solutios of tigoometic equtios i fiite itevl. Use of tigoometic idetities d fctoistio to tsfom equtios. Fo the fist oe: Use the ivese fuctios, the dd o sutct (o ) ccodigly. Fo the secod oe: Use tigoometic idetities to fid ltetive wys of witig ceti tsfomtios. I shot, put tigoometic fuctios ito the fom f si c d usig fctos d idetities to e le to descie the tsfomtio. Chpte 3.6 Solutio of tigles. The cosie ule: c cosc. The sie c ule:. Ae of tigle s si si A si B si C C. Ay tigle with vetices A, B, C d sides,, c c e mde ito two ight-gled tigles s show. h h si A, si B si A h si B si A si B si A si B si A si B h h si A, si C c csi A h si C csi A si C c si A si C c si C The cosie ule c e poved s follows usig the tigle elow. CD AC AD ccos A

27 BC BD CD si cos c A c A c si A ccos Ac cos A si cos cos cos c A A c A c c A cos c c A Chpte 4. Defiitio of mti: the tems elemet, ow, colum d ode. A mti is ectgul y of umes ged i ows d colums. A ow is hoizotl set of umes. A colum is veticl set of umes. The ode of mti deotes the ume of ows d ume of colums i the mti d is equl to the ume of elemets i the mti: m whee m is the ume of ows d is the ume of colums. Chpte 4. Alge of mtices: equlity; dditio; sutctio; multiplictio y scl. Multiplictio of mtices. Idetity d zeo mtices. Two mtices e equl if they e of the sme ode. It is oly possile to dd o sutct two mtices if they e of the sme ode. Ech elemet of pticul ow d colum is dded o sutcted y the coespodig elemet i the othe mti. Multiplictio y scl ivolves the mee multiplictio of evey tem i the mti y tht scl. Two mtices c oly e multiplied togethe if the fist (mti multiplictio is ot commuttive) mti hs the sme ume of colums s the secod hs ows. The esultig mti hs the sme ume of ows of the fist mti d the sme ume of colums s the secod. p c q p q c. The idetity mti I is the mti such tht AI = A whee A is y sque mti. Fo mtices, I d fo 33mtices, I To pove this, Let A c d AI = A (y defiitio). Let Thus: I p q s

28 p q c d s c d p q s cp d cq ds c d p q s cp d c cq ds d p p p p p p p 0 p p p p 0 p p p p p 0 0 p s 0 q 0 p q 0 s 0 A zeo mti is mti i which ll elemets e equl to 0 such tht AO = O d A + O = A It is impott to ote tht multiplictio y oth the idetity d the zeo mti is commuttive.

29 Hee e some ules: If A d B e mtices tht c e multiplied the AB is lso mti. Mti multiplictio is o-commuttive. If O is zeo mti the AO = OA = O fo ll A. A(B + C) = AB + AC If I is the idetity mti the AI = IA = A A fo c e detemied povided tht A is sque d is itege. Chpte 4.3 Detemit of sque mti. Clcultio of d 3 3 detemits. Ivese of mti: coditios fo its eistece. The ivese of mti A - is such s will stisfy the equtio AA - = A - A = I. p q 0 c d s 0 p q s 0 cp d cq ds 0 p q s 0 cp d 0 cq ds p cp d 0 p d p 0 c d c c c c d d c c d d p c d c d d c d d p d c d c s d c c q d c c d c

30 d p q d c d c d s c d c c d c d c This suggests tht thee is o ivese fo y mti whee d c 0. A mti without ivese is kow s sigul mti, d d c is kow s the detemit ecuse it detemies whethe the mti will e sigul o ivetile. The detemit of mti A is witte s A d s det A. Rules: detab = detadetb. The detemit of 3 3 mti: c Whee A = c, 3 3 c 3 c c A c. c c Chpte 4.4 Solutios of systems of lie equtios ( mimum of thee equtios i thee ukows). Coditios fo the eistece of uique solutio, o solutio d ifiity of solutios. AX B X - A B Aove is how to solve simulteous equtios. Row Opetios (simulteous equtios): The equtios c e itechged without ffectig the solutios A equtio c e eplced y o-zeo multiple of itself Ay equtio c e eplced y multiple of itself multiple of othe equtio. Augmeted Mti Fom y c pqy c e witte s c ` p q We c ow mipulte the equtios usig ow opetios to get mti i the fom c Fom which we my get uique solutio. 0 p q c If mti is otied, the equtio hs o solutio. d

31 c If mti c is otied the thee e ifiitely my solutios. Reduced ow echelo fom llows us to fid uique solutios to simulteous equtios: c d c d c d 0 e f g 3 3 c3 d h i If h 0 we ive t uique solutio. If h 0 d i 0, thee is o solutio d the system is icosistet. If h 0 d i 0, thee e ifiitely my solutios of the fom p kt, y q lt d z t, t A udespecified system (ot eough equtios) is the sme cse s the lst oe ove. The system my hve o solutios, s my e see y lookig fo icosistecies. If simulteous equtio i ugmeted mti fom is ivetile, it hs uique solutio. If it is sigul, it hs eithe o solutios o ifiitely my. Covetig the ugmeted mti to educed ow echelo fom llows us to detemie which. Chpte 5. Vectos s displcemets i the ple d i thee dimesios. v Compoets of vecto; colum epesettio v v viv jv3k. Algeic d v 3 geometic ppoches to the followig topics: the sum d diffeece of two vectos; the zeo vecto, the vecto v ; multiplictio y scl, kv ; mgitude of vecto, v ; uit vectos i, j, k; positio vectos OA. A vecto v v v viv jv3k epesets tsltio of v 3 v uits the -is, v uits the y-is d v 3 uits the z-is. This is ecuse i, j, d k e uit 0 0 vectos (vectos with mgitude of ) whee i 0, j d k 0 d thus 0 0 epeset tsltios i ech of the thee es. The distce etwee y two poits i thee (o two) dimesios is give (y defiitio) y the mgitude of the vecto tht mps oe of the poits oto the othe. This is give i the equtio v v v v. 3 Fo two geel poits A(, y, z ) d B(, y, z) i thee dimesiol spce, AB ( ) ( y y ) ( z z )

32 0 The zeo vecto is vecto 0 such tht If v mps poit A oto poit B, the v mps poit B oto poit A. Summy of vecto ithmetic if,, ( ) c( c) 00 ( ) 0 Also, k k k k if OA d OB, the AB, BA ( OA is positio vecto, mppig the oigi oto poit A). Defiitios Two vectos e equl if they hve the sme mgitude d diectio, ut do ot hve to e o the sme lie: p p d q s q s Poits e collie if they lie o the sme lie: A, B d C e collie if AB kbc fo some scl k. is pllel to k fo some scl k. v is the legth of the uit vecto i the diectio of v. v

33 Chpte 5. The scl poduct of two vectos, vw v w cos ; vwvw vw v3w3. Algeic popeties of the scl poduct. Pepedicul vectos. The gle etwee two vectos. The scl poduct of two vectos, lso kow s the dot poduct o ie poduct of two vectos, v w gives us scl swe. The scl poduct is defied y the secod equtio, vwvw vw v3w3, ut it c e poved tht the fist equtio is lso tue, vw v w cos usig the method descied o p.38, H&H. A cosequece of vw v w cos is tht fo pllel vectos, whee 0, the equtio gives vw v w cos 0 v w d fo pepedicul vectos, whee, the equtio gives vw v w cos 0. Thus, it is possile to fid the solutio fo missig vw vw vw v v v w w w d y settig viles y settig vw vw vw Algeic popeties of the scl poduct c c cd cd cd Chpte 5.3 Vecto equtio of lie. The gle etwee two lies. I 3-D: 0 l y y0 m is the vecto equtio of lie whee R yzis,, y poit o the z z 0 l lie d A0, y0, z 0 is y poit o the lie. m is the diectio vecto of the lie 0 (see collie poits). Thus, y0 mps oe poit i thee-dimesiol spce fom the z 0 l oigi d m tells us the geel positio of ll collie poits.

34 The pmetic equtios of the lie, desciig the lie s two-dimesiol lie o the., y d z ples espectively, e used whe witig the lies i pmetic fom: l, y y m, z z, Ctesi fom, settig ech equtio equl to d thus equl to oe othe, gives us the followig: 0 y y0 zz0. l m The gle etwee the two lies i thee dimesiol spce c e foud usig the scl poduct of thei diectio vectos: ccos ll mm l m l m (vlues tke fom the equtios of the lies). Chpte 5.4 Coicidet, pllel d skew lies, distiguishig etwee these cses. Poits of itesectio. Two lies e coicidet if the Ctesi equtios of oe e scl multiple of the othe. Two lies e pllel if the gle etwee the two lies foud usig the scl poduct of thei diectio vectos is 0 gul uits, ut the Ctesi equtios of oe is ot scl multiple of the othes. Two lies e itesectig if the gle etwee the two lies foud usig the scl poduct of thei diectio vectos is θ gul uits d they itesect t poit foud y epesetig thei equtios s mtices d solvig them (see chpte 4.4). Two lies e skew if the gle etwee the two lies foud usig the scl poduct of thei diectio vectos is θ gul uits d epesetig thei equtios s mtices gives o solutio (see chpte 4.4). Aothe wy of fidig poits of itesectio is s follows. Tke two lies L d L whee: L : l, y y m, z z L : l, y m, z 3 0 y0 3z0 If:, the lies itesect t coodites foud y sustitutig ll mm the otied vlue of ito the pmetic equtios.

35 Chpte 5.5 The vecto poduct of two vectos, v w. The detemit epesettio. Geometic itepettio of v w. The vecto, o coss poduct of two vectos, v w is fuctio of the two vectos which gives vecto pepedicul to the two vectos. Thus, the vecto poduct of two vectos v d w v w whee v v d w w is give y: v 3 w 3 vw 3 3 i j k v v 3 v3 v v v vw v3wvw 3 v v v3 i j k. w w3 w3 w w w vw vw w w w3 i j k v v v3 is kow s 3 3 detemit. w w w 3 A oml is lie pepedicul to ple. Thus, give two vectos (o thee poits) o ple, oml to the ple c e foud. Sice it s diectio vecto, y scl multiple of oml vecto i its simplest fom is usle. Vecto poduct lge vwv, w vv0 vwwv 3 c 3 d is clled the scl tiple poduct. c c c3 cc cd c d c d vw vw 3vw 3 vw 3 vw 3 vw vw v wsi, vw. Popeties If u vw, vw v wsiu v w vw0 v w. If tigle hs defiig vectos v d w the its e is v w.

36 Thus, if pllem hs defiig vectos v d w the its e is v w. Chpte 5.6 Vecto equtio of ple c. Use of oml vecto to oti the fom. Ctesi equtio of ple y cz d. Sice ple i thee dimesiol spce c e descied usig miimum of two lies o the ple, d the vecto equtio of oe lie i spce is give y whee is vecto mppig the oigi oto oe poit o the lie d is the diectio vecto of the lie, the vecto equtio of the ple c e foud y ddig the diectio vecto of othe lie to the equtio, i.e. c whee d c e two o-pllel vectos tht e pllel to the ple. If A is poit o ple d R is othe poit o the ple, the AR OR OB whee is the positio vecto of R (which is geel poit (, yz, ) o the ple) d is the positio vecto of A. Thus, the oml to the ple will lso e pepedicul to tht lie, so: 0 0 This is diffeet wy of epessig the vecto equtio of lie. This mes tht if oml vecto psses though poit ( 0, y0, z 0) the: c y cz d, whee d is some costt. This gives us the Ctesi equtio of the lie: 0 y0 cz0 y cz d whee c is oml vecto of the ple. Chpte 5.7 Itesectios of: lie with ple; two ples; thee ples. Agle etwee: lie d ple; two ples. Whe give the Ctesi equtio of ple, the itesectio of lie with the ple c e foud y usig the pmetic fom fo epessig the lie i tems, y, z d d sustitutig ech ito the Ctesi equtio of the ple, thus solvig fo (s show i H&H, p. 446 emple 0). Tke two ples P d P whee: P : l d, y y m e, z z f P : l d, y m e, z f 3

37 Thus, the lie of itesectio of ple c e foud y settig the equtios of eithe, y o z equl to oe othe d solvig fo i tems of o i tems of d thus sustitutig it ito the two emiig equtios to solve fo the emiig costt. The two vlues foud, sustitutig them ito the pmetic fom equtios gives the coodites of poit of itesectio. This method is etemely d iefficiet, howeve. Keepig the equtios i tems of will yield lie descied i pmetic fom, howeve. The itesectio of thee ples is foud y usig the Ctesi foms of the thee vectos d iputtig them ito ugmeted mti to solve fo, y d z. If o solutios e yielded, thee is o commo poit of itesectio. If uique solutio is yielded, the thee ples meet t poit. If thee is e ifiite ume of solutios, the ples meet t lie with pmetic equtios give y the mti whee z is sustituted y vile t. Altetively, if the thee ples meet t poit, the ivese mti method metioed elie c e used: AX B X - A B Fidig whethe o ot the detemit of A eists c e quick method fo detemiig whethe o ot the thee ples meet t poit. The gle etwee lie with diectio vecto d d ple with oml vecto is foud usig the equtio csi d (see H&H p.449). d If two ples hve oml vectos d, the cute gle etwee the two itesectig ples is give y ccos. The otuse gle 80. Chpte 6. Cocepts of popultio, smple, dom smple d fequecy distiutio of discete d cotiuous dt. A popultio is the set of ll idividuls with give vlue fo vile ssocited with them. A smple is smll goup of idividuls domly selected (i the cse of dom smple) fom the popultio s whole, used s epesettio of the popultio s whole. The fequecy distiutio of dt is the ume of idividuls withi smple o popultio fo ech vlue of the ssocited vile i discete dt, o fo ech ge of vlues fo the ssocited vile i cotiuous dt.

38 Chpte 6. Pesettio of dt: fequecy tles d digms, o d whiske plots. Gouped dt: mid-itevl vlues, itevl width, uppe d lowe itevl oudies, fequecy histogms. Mid itevl vlues e foud y hlvig the diffeece etwee the uppe d lowe itevl oudies. The itevl width is simply the distce etwee the uppe d lowe itevl oudies. Fequecy histogms e dw with itevl width popotiol to width d fequecy s the height. Bo d whiske plots A o-d-whiske plot is visul disply of some of the desciptive sttistics of dt set. It show The miimum vlue (Mi ), the lowe qutile (Q ), the medi (Q ), the uppe qutile (Q 3 ) d the mimum vlue (M ). These qutities e kow s the five-ume summy of dt set. Chpte 6.3 Me, medi, mode; qutiles, pecetiles. Rge; itequtile ge; vice, stdd devitio. i f i i i i Me: f i fi i Medi: m whee f i Mode: Mode whee f f, i i m The popultio me,, is geelly ukow ut the smple me, seves s uised estimte of this me. s A qutile (Q s ) is the vlue of i which hs of the totl fequecy fllig elow this 4 vlue d s of the totl fequecy fllig ove this vlue. This is oly pplicle to 4 cotiuous dt. s A pecetile is like qutile, ut fo. 00 The ge is the diffeece etwee the highest d lowest vlue i the dt set.

39 The itequtile ge is Q 3 Q. The vice is mesue of sttisticl dispesio (to wht etet the dt vlues devite fom the me). The popultio vice of fiite popultio of size is give y: i i The popultio vice is, howeve, geelly ukow d hece the djusted smple vice is used s uised estimte of the popultio vice: s s whee s is the udjusted smple vice d s is the estimte. Chpte 6.4 Cumultive fequecy; cumultive fequecy gphs; use to fid medi, qutiles d pecetiles. Cumultive fequecy is the fequecy of ll vlues less th give vlue. A tle c e dw s show: Pmetes f f 0 l l l l l l l l l l l Dwig cumultive fequecy gph (sed o the uppe limit of ech pmete) eles oe to fid the medi, qutiles d pecetiles y tkig the equied fctio of the totl fequecy (cumultive fequecy of the highest vlue) d fidig the coespodig vlue o the -is. Chpte 6.5 Cocepts of til, outcome, eqully likely outcomes, smple spce (U) d A ( ) evet. The poility of evet A s P( A). The complemety evets A d U ( ) A (ot A); P( A) P( A). The ume of tils is the totl ume of times the epeimet is epeted. The outcomes e the diffeet esults possile fo oe til of the epeimet. Eqully likely outcomes e epected to hve equl fequecies. The smple spce is the set of ll possile outcomes of epeimet.

40 Ad evet is the occuece of oe pticul outcome. A ( ) P( A) whee P( A) is the poility of evet A fom occuig i oe til, U ( ) Ais ( ) the ume of memes of the evet A d U ( ) is the totl ume of possile outcomes. Sice evet must eithe occu o ot occu, the poility of the evet eithe occuig o ot occuig must e. This c e stted s follows. P( A) P( A) Chpte 6.6 Comied evets, the fomul: P( AB) P( A) P( B) P( A B). P( AB) 0 fo mutully eclusive evets. Give two evets, A d B, the poility of t lest oe of the two evets occuig, (c lso e stted s the poility of eithe A o B occuig) o P A B is give y the equtio P( AB) P( A) P( B) P( AB) whee P( A) is the poility evet A occuig, P( B) is the poility of evet B occuig d P( A B) is the poility of oth evets occuig. It is impott to ecll (fom the poduct piciple) tht P( AB) P( A) P( B), whee A d B e idepedet evets, o i geel P( AB) P( B) P( A B). This implies tht P( A B) 0 fo mutully eclusive evets A d B sice PAB ( ) would e 0 y defiitio. Chpte 6.7 Coditiol poility; the defiitio: evets; the defiitio: P( AB) P( A) P( AB). P( A B) P( AB). Idepedet P( B) The two defiitios ove simply equie leig. Howeve the followig c thus e deived. P( A B) P( AB) P( B) P( AB) P( A B)P( B) P( AB) P( A)P( B) A impott theoem is Byes Theoem fo two evets (ot ecessily idepedet). P( B)P( A A) P( B A) P( B)P( B B) P( B)P( A B). This c e ptilly deived (o witte i othe fom) i the followig wy:

41 P( A B) P( AB) P( B) P( A B) P( B)P( AB) P( BA) P( A) P( A) Chpte 6.8 Use of Ve digms, tee digms d tles of outcomes to solve polems. Ve Digms A ( ) The poility is foud usig the piciple P( A). U ( ) It is impott to ote tht ( A) A A B A B d ( U ) ( A) ( B) (o ). Tee digms A moe fleile method fo fidig poilities is kow s tee digm.

42 This llows oe to clculte the poilities of the occuece of evets, eve whee tils e o-ideticl (whee P( A A) P( A) ), though the poduct piciple. Tles of outcomes Refe to H&H p Chpte 6.9 Cocept of discete d cotiuous dom viles d thei poility distiutios. Defiitio d use of poility desity fuctios. Epected vlue (me), mode, medi, vice d stdd devitio. A dom vile epesets i ume fom the possile outcomes, which could occu fo some dom epeimet. Fo y dom vile thee is poility distiutio ssocited with it. A discete dom vile ivolves cout. Thus, discete dom vile X hs possile vlues,, 3,... Thus, fidig P( X ) (the poility distiutio of ) ivolves listig P( i ) fo ech vlue of i. I dom distiutio, the epected outcome E( X ) is the me. The stdd devitio, o V( X ), is the sque of the distce of X fom the me: V( X) E( X ) E( X ) {E( X)}. E( X ) i pi fo discete dom viles d E( X ) f( )dfo cotiuous dom distiutio. Fo discete dom vile, the mode d medi c e foud s outlied ove. The me d vice c e epessed s follows: ipi p p i i i i A cotiuous dom vile ivolves mesuemets. A cotiuous dom vile X hs ll possile vlues i some itevl (o the ume lie). Rthe th poility distiutio, cotiuous dom viles hve poility desity fuctios. A cotiuous poility fuctio (pdf), f ( ), is fuctio whee f( ) 0o give itevl, such s [, ] d f ( )d. Fo cotiuous poility desity fuctio, the mode is tht vlue of t the mimum vlue of f ( ) o m [, ]. The medi m, is the solutio fo m of the equtio f( )d. The me d vice c e epessed s follows: f( ) d f( )d f( )d

43 It is impott to ote tht stdd devitio V( X ) The followig ules summise the popeties of E( X ). E( k) k fo some costt k E( kx ) ke( X ) fo some costt k E( AX ( ) BX ( )) E( AX ( )) E( BX ( )) fo fuctios Ad B This mkes it possile to deive othe fom fo the vice. V( X ) E( X ) EX X ( ) E( X ) E( X) E( X ) E( X ) {E( X)} Chpte 6.0 Biomil distiutio, its me d vice. Poisso distiutio, its me d vice. Biomil Distiutio I the cse of tils whee thee e successes d filues, P( X ) C p q whee q = p d = 0,,, 3, 4,,. p is the poility of success d q is the poility of filue. P( X ) is the poility distiutio fuctio. Thee e thee citei tht must e met i ode fo dom poility distiutio to e iomil distiutio.. The poility distiutio is discete.. Thee e two outcomes success d filue. 3. The tils e idepedet the poility of success is costt i ech til. If is dom vile which is iomil with pmetes d p, the the me of is p d the vice of is pq. Clculto: P( ) iompdf (, p, ) d P( ) iomcdf (, p, ) Poisso Distiutio m me The Poisso distiutio is defied s P( X ) whee m is the pmete.! f m. f

44 Thee e thee citei tht must e met i ode fo dom poility distiutio to e iomil distiutio.. The vege ume of occueces (μ) is costt fo evey itevl.. The poility of moe th oe occuece i give itevl is vey smll. 3. The ume of occueces i disjoit itevls e idepedet of ech othe. Chpte 6. Noml distiutio. Popeties of the oml distiutio. Stddiztio of oml viles. If X is omlly distiuted the its poility desity fuctio is give y f ( ) e fo. The gd mjoity of cotiuous distiutios e oml distiutios, whee the poility desity deceses ccodig to how f the vlue is fom the me. This is pticully tue fo viles i tue. Popeties The cuve is symmeticl out the lie lim f ( ) 0 f ( )d M f ( ) is uiquely detemied s the hoizotl distce fom the veticl distce fom the veticl lie to poit of iflectio. I oml distiutio, 68.6% of vlues lie withi oe stdd devitio of the me, 95.4% of vlues lie withi two stdd devitios of the me d 99.74% of vlues lie withi thee stdd devitios of the me. A ( ) Sice P( A), the poility of X lyig withi ceti itevl is equl to the U ( ) pecetge of vlues tht lie withi tht itevl. This is otied fom the dt ooklet d fom GDCs. The stdd oml distiutio, o Z-distiutio, is the pplictio of the tsfomtio Z X to oml X-distiutio, such tht the me is t = 0 d thee is oe stdd devitio pe uit o the -is. Whee the poility desity fuctio fo oml distiutio hs two pmetes d, the Z-distiutio hs oe. This mkes it useful whe compig esults fom two o moe diffeet oml distiutios, sice compig Z-vlues llows oe to tke ito ccout the stdd devitio d me whe compig esults. Fidig poilities with GDC ivolves usig omlcdf(,,, ) fo lowe limit d uppe limit (ude DISTR ). It is impott to ote tht P( Z ) P( Z ). To fid poilities fo omlly distiuted dom vile X, covet X vlues to Z usig the tsfomtio, sketch the stdd oml cuve (shde the equied egio) d fid the stdd oml tle o gphics clculto to fid the poility. omlpdf(,, ) gives the poility fo pticul -vlue.

45 Chpte 7. Ifoml ides of limit d covegece. Defiitio of deivtive s f h f f lim. Deivtive of, si, cos, t, e d h0 h l. Deivtive itepeted s gdiet fuctio d s te of chge. Deivtives of ecipocl cicul fuctios. Deivtives of d. Deivtives of csi, ccos, ct. All fuctios ppoch pticul vlue s the vlue of the vile they e i tems of ppoches give vlue. Howeve, i ceti cses, it is ot possile to diectly detemie the vlue of the fuctio t tht pticul vlue of the vile ecuse the swe ivolves divisio y zeo o the vlue of the vile i questio is, i fct, ifiity. Thus, whee fuctio coveges towds pticul vlue, it c e sid tht the limit of the fuctio t tht vlue of the vile is equl to the vlue the fuctio ppoches. Fo emple, si 0 oly whee 0 d sice the fuctio si is cotiuous, it mes tht fo 0,si. Wht is moe, sice the closest poit of iflectio to the sie fuctio occus t 0, the close is to 0, the close si is to. si Thus, give the fuctio f Howeve, 0 is udefied. Thus, we c sy tht sice 0 si towds 0, lim, o the limit of si s teds to 0 is. 0., the close is to 0, the close f of poit f teds towds s teds The deivtive, f ( ) f ( ), o fuctio is the gdiet of the tget o istteous te of chge of the fuctio t tht poit. This c e foud y usig the e of limits: Tke two poits A d B o the cuve of fuctio f f ( ),. Let A hve coodites d B, h uits wy fom A o the -is, theefoe hve coodites f ( h),( h). Thus, the gdiet of the c AB o the cuve joiig the two lies is f ( h) f( ) f( h) f( ) equl to. It c e see tht the close poit B is to A, h h the close the gdiet of the c is to the gdiet of the tget t poit A. Epessed usig limit ottio, this gives us the equtio: f h f f( ) lim. h0 h Usig this equtio to fid the deivtive of fuctio t poit is kow s fidig the deivtive fom fist piciples. This is doe y mipultig the equtio util h is tke out of the deomito, sice this will geete ectly the sme esult s the pevious equtio ut with the dditiol solutio fo whee h 0, thus tellig us wht the limit of the fuctio s h teds towds 0 is t the poit y settig the h 0 d solvig fo the deivtive. This llows us to geete some geel ules fo the deivtives.

46 The fist of these is the powe ule: Let f ( ) Thus: h... h... h h f( ) lim lim h0 h h0 h h... h... h lim lim... h... h h0 h h0 0 This esult is kow s the powe ule. Also ote: If f ( ), h h h f ( ) lim lim lim h0 h h0 h h0 h It is lso fily evidet tht y costt if f ( ) k, f ( ) The followig e ules tht e impott to ote: f ( ) f ( ) f ( ) si f( ) cos f ( ) cos f( ) si f ( ) t f( ) sec f( ) e f( ) e f( ) l f( ) Whe epessed s f ( ), the deivtive of f ( ) suggests the te of chge fuctio. Whe epessed s d y, the deivtive of ( ) d f y suggests gdiet fuctio. The two e, howeve, completely itechgele, though d y is y f moe useful d esie d to mipulte fom. This is ecuse d y epesets the poit gdiet o the cuve s the tio etwee d ifiitely smll displcemet dy i the y-diectio d ifiitely smll displcemet d i the -diectio, epesetig y give cuve s seies of coected ifiitely smll lie segmets with gdiet equl to the tget of the cuve t tht poit.

47 Becuse this epesettio of the gdiet of the tget is i the fom of fctio (of ifiitely smll pts), this llows it to e mipulted i such wy s to yield iteestig esults. Fo istce, whee d y gives the gdiet fuctio of cuve whose y vlues e i tems d d of, dy gives the gdiet fuctio of cuve whose -vlues e i tems of dy d y. Tkig the itegl of this fuctio gives you d d dy which is ot vey useful d esult, howeve the ivese of this fuctio, dy dy gives us (wht used to e) i tems of y, i othe wods, the ivese of the fuctio. To summise: The ivese of the itegl of the ecipocl of the deivtive of fuctio is equl to the ivese of the fuctio. The ivese of the ecipocl of the deivtive of fuctio is equl to the deivtive of the ivese of the fuctio. The ove deivtives d the my moe equied e ll i the fomul ooklet. Icesig d decesig fuctios. A icesig fuctio hs positive gdiet fo ll d decesig fuctio hs egtive gdiet fo ll. The itevls duig which fuctio is icesig o decesig is foud y fidig the gdiet fuctio d usig sig digms to detemie whe the gdiet fuctio is positive d whe it is egtive. If cuve hs gdiet fuctio d y d, the oml to the cuve hs fuctio d. dy Chpte 7. Diffeetitio of sum d el multiple of fuctios i 7.. The chi ule fo composite fuctios. Applictio of chi ule to elted tes of chge. The poduct d quotiet ules. The secod deivtive. Aweess of highe deivtives. A impott ule to ememe i diffeetitio is tht if: y f( ) g( )... ch( ) dy f( ) g( )... ch( ) d The chi ule tkes dvtge of the fctiol popeties of the gdiet equtio to simplify the diffeetitio of fuctios such s y f( ) d to llow fo the detemitio of othe elted tes of chge. The chi ule is pehps est descied s it is i the fomul ooklet: dy dy du y g( u), whee u f( ) d du d

48 dy y f( g( )) g( ) f( g( )) d dy y f( u) u f( u), u g( ) d This hs get my pplictios, ot oly to llow fo the diffeetitio of moe complicted fuctios ut lso to llow fo the deivtio of my othe fuctios. The Chi ule lso llows fo the detemitio of elted tes of chge. dv dv ds is oe emple of this, whee the fuctio fo istteous cceletio is dt ds dt usig the fuctios of istteous velocity d the equtio of speed eltive to displcemet. The poduct ule c e descied i similly cle wy: dy dv du y uv, whee u f( ), v g( ) u v d d d As c the quotiet ule: du dv v u u dy y, whee u f( ), v g( ) d d v d v Geelly spekig, it is sfe to sy tht these e meely equtios to e diectly pplied to the questio s eeded. d y The secod deivtive f ( ) o is the deivtive of the deivtive of the fuctio. It d epesets the cuvtue of the fuctio: the te t which the gdiet is chgig i eltio to. This is useful fo esos outlied i the et chpte. Highe deivtives, epessed s d y d ode dow. o ( f ) ( ), e the deivtives of the deivtive oe Chpte 7.3 Locl mimum d miimum poits. Use of the fist d secod deivtive i optimiztio polems. Locl mim d miim occu whee d y 0 d d d y 0 d. If d y 0 fo tht vlue of d d y, the poit is miimum, if 0 fo tht vlue of, the poit is mimum. These d c theefoe e used i optimistio questios whee the fuctio fo give pmete is foud d the diffeetitio pplied, such s i cses delig with pofit, e o volume.

49 Chpte 7.4 Idefiite itegtio s tidiffeetitio. Idefiite itegl of, si, cos, fuctio. e,. The composites of y of these with the lie Idefiite itegtio gives the geel fomul fo the e ude fuctio fom the oigi to. Oe fom of idefiite itegtio is tidiffeetitio. It c e eplied coceptully i the followig wy. Seeig s diffeetitio is the pocess y which you divide the ifiitely smll ise y the ifiitely smll u t ech poit, the evese pocess, defied s tidiffeetitio, must ivolve the opposite pocess, s i fidig the sum the es of the ifiitely smll tpeziums ude ech ifiitely smll lie segmet. Atidiffeetitio is simply the ivese fuctio of diffeetitio. The tidiffeetil of fuctio is the fuctio which, whe diffeetited, gives the oigil fuctio. Whe sked to tidiffeetite, sk youself: Wht fuctio, whe diffeetited, would give me this fuctio? Itegtio ules: kf ( ) d k f ( ) d kdkc d c e de c d l c f ( ) g( ) d f( ) d g( ) d e d e c ( ) d c d l c du f( u) d f( u) du d Chpte 7.5 Ati-diffeetitio with oudy coditio to detemie the costt tem. Defiite Itegls. Ae etwee cuve d the -is o y-is i give itevl, es etwee cuves. Volumes of evolutio. If give the vlue of f ( ) t give vlue of fo the itegl of fuctio, it is possile to plug i the umes to detemie the costt tem c i f ( )d f( ) c. f ( )d f ( ) f ( ) whee is the uppe limit of d is the lowe limit.

50 This gives the e etwee the cuve d the -is fo those limits. To fid the e etwee the cuve d the y-is, it is simplest to tke the e of the ectgle f ( ) d sutct fom tht f ( )d d the e of ectgle f ( ). It is lso possile to fid the e ude the cuve of the ivese fuctio, i.e. d y the th d. y Fo the volume V of evolutio whe e with limits d is otted out the - (fist cse) o y-is (secod cse), it is simplest to stte the equtios: V y d, V d y Similly to the usge of ectgles outlied ove, cylides c e used whe ottig e etwee the cuve d the is is ot the is of evolutio. If ottig ove lie tht is ot o is, it is ecessy to use tsfomtios to tsfom the is of ottio oto eithe the - o y-is i ode to use the equtios show ove. Chpte 7.6 Kiemtic polems ivolvig displcemet, s, velocity, v, d cceletio,. d d d d v s, v s v v dt dt dt ds The e ude velocity-time gph epesets distce. It c e impott to mipulte deivtives so tht they e i the coect fom to fid the equied solutio. This mipultio must ot e fogotte. Chpte 7.7 Gphicl ehviou of fuctios: tgets d omls, ehviou fo lge ; symptotes. The sigificce of the secod deivtive; distictio etwee mimum d miimum poits. Poit of ifleio with zeo d o-zeo gdiets. The deivtive of fuctio gives the fuctio of its tget, the egtive ecipocl of the deivtive gives the fuctio of its oml o the oml t tht poit. The deivtive idictes whee the fuctio is gettig moe positive d whee it is gettig moe egtive. Veticl symptotes occu whee the deivtive hs ifiite vlue d the equtio of the oml hs vlue 0 d hoizotl symptotes occu whee the deivtive of the ivese fuctio is ifiite o whee the teds to zeo s ppoches (d whee the fuctio of the oml ppoches ifiity). The secod deivtive gives the te of chge of the deivtive, i.e. the cuvtue of the fuctio (whethe the fuctio is gettig steepe o less steep). At sttioy poits, if the secod deivtive is positive, the sttioy poit is miimum, if the secod deivtive is egtive, the sttioy poit is mimum d whee the sttioy poit hs secod deivtive 0, the sttioy poit is poit of iflectio, give tht the thid deivtive hs o-zeo vlue. Chpte 7.8 Implicit diffeetitio.