Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Fuctios Asolute Vlue Fuctio f ( ) f( ), if f( ) 0 f( ), if f( ) < 0 Iverse Fuctio If y f( ), the Rememer: Oject the vlue of Imge the vlue of y or f() f() mp oto itself mes f() f ( y) 0 Qudrtic Equtios Geerl Form + + c 0 where,, d c re costts d 0. *Note tht the highest power of ukow of qudrtic equtio is. Formig Qudrtic Equtio From its Roots: If α d β re the roots of qudrtic equtio α + β αβ The Qudrtic Equtio ( α + β) + αβ 0 or ( SoR) + ( PoR) 0 SoR Sum of Roots PoR Product of Roots c Qudrtic Formul ± 4c Whe the equtio c ot e fctorized. Nture of Roots 4c > 0 two rel d differet roots 4c 0 two rel d equl roots 4c < 0 o rel roots 4c 0 the roots re rel http://www.oe-school.et/otes.html
03 Qudrtic Fuctios Geerl Form Completig the squre: f ( ) + + c where,, d c re costts d 0. *Note tht the highest power of ukow of qudrtic fuctio is. f ( ) ( + p) + q (i) the vlue of, p (ii) mi./m. vlue q (iii) mi./m. poit ( p, q) (iv) equtio of is of symmetry, p > 0 miimum (smilig fce) < 0 mimum (sd fce) Qudrtic Iequlities Altertive method: f ( ) + + c (i) the vlue of, (ii) mi./m. vlue f ( ) (iii) equtio of is of symmetry, Nture of Roots > 0 d f( ) > 0 > 0 d f( ) < 0 4 c> 0 itersects two differet poits t -is 4 c 0 touch oe poit t -is 4 c< 0 does ot meet -is < or > < < 04 Simulteous Equtios To fid the itersectio poit solves simulteous equtio. Rememer: sustitute lier equtio ito o- lier equtio. http://www.oe-school.et/otes.html
05 Idices d Logrithm Fudmetl if Idices Zero Ide, Negtive Ide, Frctiol Ide Fudmetl of Logrithm log y y log log log 0 0 ( ) m m Lws of Idices m m+ m m ( ) m m ( ) ( ) Lw of Logrithm log m log m + log m log log m log log m log m Chgig the Bse log log log log c c log http://www.oe-school.et/otes.html 3
06 Coordite Geometry Distce d Grdiet Distce Betwee Poit A d C ( ) + ( ) y y Grdiet of lie AC, m Or y it ercept Grdiet of lie, m it ercept Prllel Lies Perpediculr Lies Whe lies re prllel, m m. Whe lies re perpediculr to ech other, m m m grdiet of lie m grdiet of lie Midpoit A poit dividig segmet of lie Midpoit, M +, y + y A poit dividig segmet of lie + m y+ my P, m+ m+ http://www.oe-school.et/otes.html 4
Are of trigle: Are of Trigle A y + y + y y+ y + y ( 3 3 ) ( 3 3) Form of Equtio of Stright Lie Geerl form Grdiet form Itercept form + y + c 0 y m+ c m grdiet c y-itercept y + -itercept y-itercept m Equtio of Stright Lie Grdiet (m) d poit (, y ) give y y m( ) poits, (, y ) d (, y ) give y y y y -itercept d y-itercept give y + Equtio of perpediculr isector gets midpoit d grdiet of perpediculr lie. Iformtio i rhomus: A D C B (i) sme legth AB BC CD AD (ii) prllel lies mab mcd or mad mbc (iii) digols (perpediculr) m m (iv) shre sme midpoit midpoit AC midpoit BD (v) y poit solve the simulteous equtios AC BD http://www.oe-school.et/otes.html 5
Rememer: y-itercept 0 cut y-is 0 -itercept y 0 cut -is y 0 **poit lies o the lie stisfy the equtio sustitute the vlue of d of y of the poit ito the equtio. Equtio of Locus ( use the formul of distce) The equtio of the locus of movig poit P (, y) which is lwys t costt distce (r) from fied poit A, ) is ( y PA r ( ) + ( y y r ) The equtio of the locus of movig poit P (, y) which is lwys t costt distce from two fied poits A (, y) d B (, y) with rtio m : is PA m PB ( ) + ( y y) m ( ) + ( y y ) The equtio of the locus of movig poit P (, y) which is lwys equidistt from two fied poits A d B is the perpediculr isector of the stright lie AB. PA PB ( ) + ( y y ) ( ) + ( y y ) More Formule d Equtio List: SPM Form 4 Physics - Formule List SPM Form 5 Physics - Formule List SPM Form 4 Chemistry - List of Chemicl Rectios SPM Form 5 Chemistry - List of Chemicl Rectios All t http://www.oe-school.et/otes.html 6
Mesure of Cetrl Tedecy 07 Sttistics Me Ugrouped Dt Σ N Without Clss Itervl Σ f Σ f Grouped Dt With Clss Itervl Σ f Σ f Medi me Σ sum of vlue of the dt N totl umer of the dt m T N + Whe N is odd umer. me Σ sum of f frequecy vlue of the dt m T N + Whe N is odd umer. me f frequecy clss mrk (lower limit+upper limit) m L+ N F C f m T m + T N N + Whe N is eve umer. TN + TN + m Whe N is eve umer. m medi L Lower oudry of medi clss N Numer of dt F Totl frequecy efore medi clss f m Totl frequecy i medi clss c Size clss (Upper oudry lower oudry) Mesure of Dispersio Ugrouped Dt Without Clss Itervl Grouped Dt With Clss Itervl vrice σ N σ f f σ f f σ vrice σ vrice σ vrice Stdrd Devitio σ Σ ( ) N Σ σ N σ Σ ( ) N Σ σ N σ ( ) Σ f Σ f Σ f σ Σ f http://www.oe-school.et/otes.html 7
The vrice is mesure of the me for the squre of the devitios from the me. The stdrd devitio refers to the squre root for the vrice. Effects of dt chges o Mesures of Cetrl Tedecy d Mesures of dispersio Mesures of Cetrl Tedecy Mesures of dispersio Termiology Dt re chged uiformly with + k k k k Me, medi, mode + k k k k Rge, Iterqurtile Rge No chges k k Stdrd Devitio No chges k k Vrice No chges k k 08 Circulr Mesures Covert degree to rdi: Covert rdi to degree: rdis 80 π degrees o π ( )rdis 80 80 rdis ( ) degrees π π 80 Rememer: 80 π rd 360 π rd??? 0.7 rd O???. rd http://www.oe-school.et/otes.html 8
Legth d Are r rdius A re s rc legth θ gle l legth of chord Arc Legth: Legth of chord: Are of Sector: Are of Trigle: Are of Segmet: s rθ θ l rsi A r θ A r siθ A r ( θ si θ ) 09 Differetitio Grdiet of tget of lie (curve or stright) Differetitio of Algeric Fuctio Differetitio of Costt y is costt 0 Emple y 0 δ y lim ( ) δ 0 δ Differetitio of Fuctio I y Emple 3 y 3 Differetitio of Fuctio II y Emple y 3 3 0 http://www.oe-school.et/otes.html 9
Differetitio of Fuctio III Chi Rule y Differetitio of Frctiol Fuctio y Rewrite y Emple y y Emple 3 y (3) 6 + y u u d v re fuctios i du du Emple 5 y ( + 3) du y u, therefore 5u du du du 4 5u 4 u + 3, therefore 4 5 4 5( + 3) 4 0 ( + 3) 4 4 Or differetite directly y ( + )..( + ) y ( + 3) 5 5( + 3) 4 0 ( + 3) 4 4 Lw of Differetitio Sum d Differece Rule y u± v u d v re fuctios i du dv ± Emple 3 y + 5 (3) + 5() 6 + 0 http://www.oe-school.et/otes.html 0
Product Rule Quotiet Rule y uv u d v re fuctios i du dv v + u Emple 3 y (+ 3)(3 ) 3 u 3 v 3 + du dv 9 4 du dv v + u + + 3 (3 )() ( 3)(9 4 ) Or differetite directly 3 y (+ 3)(3 ) + + 3 (3 )() ( 3)(9 4 ) u y u d v re fuctios i v du dv v u v Emple y + u v + du dv du dv v u v + ( + ) ( )( ) () + + (+ ) (+ ) 4 Or differetite directly y + ( + )( ) () ( + ) 4 + + (+ ) (+ ) http://www.oe-school.et/otes.html
Grdiets of tgets, Equtio of tget d Norml Grdiet of tget t A(, y ): grdiet of tget Equtio of tget: y y m( ) Grdiet of orml t A(, y ): If A(, y ) is poit o lie y f(), the grdiet of the lie (for stright lie) or the grdiet of the tget of the lie (for curve) is the vlue of whe. m orml m tget grdiet of orml Equtio of orml : y y m( ) Mimum d Miimum Poit Turig poit 0 At mimum poit, 0 d y 0 < At miimum poit, 0 d y 0 > http://www.oe-school.et/otes.html
Rtes of Chge Chi rule da da dr dt dr dt If chges t the rte of 5 cms - 5 dt Decreses/leks/reduces NEGATIVES vlues!!! Smll Chges d Approimtio Smll Chge: δ y δ y δ δ Approimtio: y y + δ y ew origil yorigil + δ δ smll chges i δ y smll chges i y If ecomes smller δ NEGATIVE http://www.oe-school.et/otes.html 3
0 Solutio of Trigle Sie Rule: si A si B c si C Use, whe give sides d o icluded gle gles d side A B A 80 (A+B) Cosie Rule: c + c c cosa + c c cosb + cosc cos A + c c Use, whe give sides d icluded gle 3 sides A c Are of trigle: C A si C C is the icluded gle of sides d. http://www.oe-school.et/otes.html 4
Cse of AMBIGUITY A 80 - θ θ C B B Cse : Whe < si A CB is too short to rech the side opposite to C. If C, the legth AC d legth AB remi uchged, the poit B c lso e t poit B where ABC cute d A B C otuse. If ABC θ, thus AB C 80 θ. Rememer : siθ si (80 θ) Cse : Whe si A CB just touch the side opposite to C Outcome: No solutio Cse 3: Whe > si A ut <. CB cuts the side opposite to C t poits Outcome: solutio Cse 4: Whe > si A d >. CB cuts the side opposite to C t poits Outcome: solutio Outcome: solutio Useful iformtio: c θ I right gled trigle, you my use the followig to solve the prolems. (i) Phythgors Theorem: c + (ii) Trigoometry rtio: si θ, cos θ, tθ c c (iii) Are ½ (se)(height) http://www.oe-school.et/otes.html 5
Ide Numer Price Ide Composite ide I P P 0 00 I ΣWi I ΣW i i I Price ide / Ide umer P 0 Price t the se time P Price t specific time I Composite Ide W Weightge I Price ide I I I AB, BC, AC, 00 http://www.oe-school.et/otes.html 6