Advanced Higher Formula List

Transcription

1 Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0 C y Geeal Tem i Biomial Theoem C y

2 Patial Factios No-Repeated Liea Facto: (a b) S a b Repeated Liea Facto: (a b) S a b T ( a b) Ieducible Quadatic Facto: (a b c) S T a b c Diffeetial Calculus th Deivative d f d d d d d d f d d d d d times Poduct Rule D (f g ) (Df ) g f (Dg) (f g ) f g f g Quotiet Rule D f g ( Df ) g f ( Dg) g (f / g ) g (f g f g )

3 Recipocal Tigoometic Fuctios sec cos cosec si cot cos si Tigoometic Idetities ta sec cot cosec Deivatives of Recipocal Tigoometic Fuctios ad ta D (sec ) sec ta D (cosec ) cosec cot D (cot ) cosec D (ta ) sec Deivative of Natual Logaithm D (l ) Deivative of Epoetial D (e ) e D (ep ) ep

4 Applicatios of Diffeetiatio Velocity v (t) ds dt s (t) Acceleatio a (t) dv dt d s dt ṡ (t) Itegal Calculus Itegals of e, -, ad sec e d e C d l C sec d ta C Geeic Foms of Itegatio by Substitutio ( Df ) f d f C Df d l f C f Aea Betwee a Fuctio ad the y-ais A d f (y) dy c 4

5 Aea Betwee Cuves about the y-ais d A (f (y) g (y ) dy c Volume of Solid of Revolutio about -ais V π b y d a Volume of Solid of Revolutio about y-ais V π d dy c Rectiliea Motio v (t) s (t) a (t) dt v (t) dt Fuctios ad Gaphs Modulus Fuctio ( 0) ( < 0) Modulus of a Fuctio f f ( f 0) f ( f < 0) 5

6 Domai ad Rage of Ivese Tigoometic Fuctios dom (si ) [, ], a (si ) π π, dom (cos ) [, ], a (cos ) [0, π] dom (ta ) R, a (ta ) π π, Eve ad Odd Fuctios Eve : f () f ( ) ( dom f ) Odd : f () f () ( dom f ) Asymptotes p () q () f () g () q () Vetical Asymptote(s) : solve q () 0 fo f costat Hoizotal Asymptote : y costat f () m c Oblique Asymptote : y m c Gaussia Elimiatio Elemetay Row Opeatios Itechage o moe ows : Ri R j Multiply a ow by a o-zeo eal umbe : Ri k R j Replace a ow by addig it to a multiple of aothe ow : R R k R i i j 6

7 No Solutios a b c j 0 e f k (l 0) l Uique Solutio a b c j 0 e f k (i 0) 0 0 i l Ifiitely May Solutios a b c j 0 e f k Uit Poof ad Elemetay Numbe Theoy Eve ad Odd Numbes k k o k Futhe Diffeetiatio Deivative of Ivese Fuctio D (f ) ( Df ) f d dy dy d 7

8 Deivatives of Ivese Tigoometic Fuctios D (si ) D (cos ) D (ta ) st Deivative of Paametic Fuctios dy d dy dt d dt y d Deivative of Paametic Fuctios d y d y y d y dt Applicatios of Diffeetiatio Displacemet Vecto ad Distace s (t) ( (t), y (t) ) (t) i y (t) j s (t) y Velocity Vecto, Speed ad Diectio of Motio (Velocity) v (t) ds dt ( (t),y (t ) (t) i y (t) j v (t) y ta θ y 8

9 Acceleatio Vecto, Magitude ad Diectio of Acceleatio a (t) dv dt (ẋ (t),ẏ (t ) ẋ (t) i ẏ (t) j a (t) y ta η y Futhe Itegatio Stadad Itegals d a d a si a C d a d a ta a a C Itegatio by Pats u ( D v ) u v ( D u ) v b u ( D v ) d a b b u v ( D u ) v d a a Sepaable Diffeetial Equatio dy d f () g (y) Comple Numbes Defiitio of i i 9

10 Catesia Fom z iy Comple Cojugate Modulus ad Picipal Agumet z iy z y θ ag z y ta (θ ( π, π]) Geeic Agumet Ag z { ag z π : Z } Pola Fom z (cos θ i si θ ) cis θ Popeties of Cojugate, Modulus ad Agumet z z z y z ± w z ± w zw z w z w z w zw z w, Ag zw Ag z Ag w 0

11 z w z w, Ag z w Ag z Ag w De Moive s Theoem k k z (cos θ i si θ ) z (cos kθ i si kθ ) th Roots of z w with w (cos θ i si θ) z k cos θ π k θ π k si (k 0,,,, ) Roots of Uity Solve z Sequeces ad Seies Sum to Tems of a Sequece S th Tem Give Successive Sums u u S S Sum to Ifiity S lim S th Tem of a Aithmetic Sequece u a ( ) d (a R, d R {0})

12 Sum to Tems of a Aithmetic Sequece S ( a ( ) d ) th Tem of a Geometic Sequece u a (a R {0}, R {0, }) Sum to Tems of a Geometic Sequece S a ( ) Sum to Ifiity of a Geometic Sequece S a Epasio of ( ) - ( ) i 0 i Defiitio of e as a Powe Seies e lim!! b 0 b! Defiitio of e as a Powe Seies e lim Sum of the Numbe times

13 Sum of the Fist Natual Numbes ( ) Sum of the Squaes of the Fist Natual Numbes ( ) ( ) 6 Sum of the Cubes of the Fist Natual Numbes 4 ( ) Uit Matices Detemiat of a Mati A a c b d ad bc Detemiat of a Mati A a b c d e f g h i e a h f i d b g f i d c g e h Ivese of a Mati A ad d bc c b a

14 Mati ad Detemiat Popeties A B B A (A B ) C A (B C ) k (A B ) ka kb (A B ) T T A T B T T ( A ) A T ( ka ) T ka A (BC ) (AB ) C A (B C ) AB AC T ( AB ) T T B A AB A B ka k A (k R, N, A is ) A T A A A A ( ) A A ( ) T ( A ) T ( ka ) A k A B ( ) B A 4

15 Vectos Vecto Poduct a b a b si θ a b ( a b a b ) i ( a b a b ) j ( a b a b ) k Popeties of the Vecto Poduct i j k j k i k i j i i j j k k 0 a a 0 a b - b a a (b c) (a b) (a c) (a b) c (a c) (b c) Scala Tiple Poduct [a, b, c] a (b c) a a a b b b c c c Catesia Equatio of a Plae with omal (a, b, c) T a by cz d Vecto Equatio of a Plae with b, c Paallel to Plae ad a i Plae a t b u c 5

16 Vecto Equatio of a Lie with Diectio (a, b, c) T ad a o Lie Paametic Equatios of a Lie p a t u at, y y bt, z z ct Symmetic Equatios of a Lie a y y b z c z t Futhe Sequeces ad Seies Maclaui Epasio of a Fuctio f () 0 f (0)! ( ) f () f (0) f (0) f (0)! f (0)! Specific Maclaui Epasios e!! 4 4! 0! si! 5 5! 0 ( ) ( )! cos! 4 4! 0 ( ) ( )! 6

17 ta ( ) l ( ) 4 4 ( ) Biomial Seies ( ) k 0 k k Futhe Odiay Diffeetial Equatios st Ode Itegatig Facto Diffeetial Equatios dy d P () y f () solved by multiplyig both sides by the Itegatig Facto e P ( ) d ad itegatig both sides d Ode Odiay Diffeetial Equatios d y a d b dy d c y f () solved by addig Complemetay Fuctio to Paticula Itegal. Futhe Poof ad Numbe Theoy Liea Diophatie Equatios Solutios of a by c ae ks b ad y kt a, whee Z ad c kd with GCD( a, b) d as bt. 7

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