Calculus Revision A2 Level

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1 alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ of (iii) (iv) sin (v) cos Lt u u d u (i) d 8 d

2 Thr ar thos of ou who will do ths without introcing u. d diffrntiat with rspct to brackt drivativ of brackt Th raindr will b don in this wa, but th substitution will also b givn for thos who would prfr to us it. (ii) {ssntial to insrt brackts!} d (iii) d u (iv) sin sin d u cos cos u (v) cos cos d cos sin cos sin u cos Proct rul For uv d v d u d Eapl Find th drivativs of (i) (ii) (iii) cos (iv) 5 {A foral us of th forula isn t ncssar, unlss ou insist!}

3 uv d u d v d (lav st diffrntiat nd ) + (lav nd diffrntiat st ) (i) d. u, d v d (ii) d u, d v d (iii) cos d sin sin cos cos u d, v cos d sin (iv) 5 d. 6 5 u d, v d 5 5 Quotint rul For u v d v u d v d

4 Eapl Diffrntiat with rspct to (i) (ii) (iii) {Again, us a foral approach if ou wish, as in (i) blow} (i) v d u d v d (ii) {without th u and v} d (iii) d d u, d v

5 Tabl of intgrals n sin cos d c n n cos sin n frop An iportant tchniqu will b usd with standard intgrals fro th givn tabl. If f ( ) d F( ) thn f ( a b) d F( a b) a Eapl d d 8 Eapl d Eapl d c Eapl d Eapl sin d cos sin d cos cos Eapl cos d sin

6 Eapl d sin cos Eapl Find th intgral of 6 d Eapl Evaluat d. d d 6 Log intgrals () Eapl d Eapl d sinc d sinc d f d f f ) ( ) ( ) (

7 Eapl d d d cos sin Eapl cot d d sin Log intgrals () You will notic that log intgrals in th data booklt includ olus signs. Ths ar not gnrall ncssar, but.. Eapl d Eapl d Rtrospctivl hr. Introc hr, and d {it has bn plaind wh this procr is accptabl} Iplicit Diffrntiation So far w hav t curvs with artsian quation in th for f () i.. is prssd plicitl in trs of. So curvs can t convnintl b prssd plicitl in this wa whn th rlationship btwn and is containd iplicitl in an quation..g. a circl 5 d Using as an oprator d d () f ( ) f ( ) d

8 d d f ( ) f ( ) chain rul d d Eapl, d () Appling to procts d d d d d Eapl.. d () Tangnts and norals d d d d Eapl Find th quation of th tangnt to at th point (, ) d d d,, d d At (, ) d d d, d 6 d d d Equation d d Intgration tchniqus Intgration as th rvrs of diffrntiation d Eapl Find 5. Hnc valuat d d d 5 5 d d 5

9 d d Sipson s rul for approiat intgration f Th dfinit intgral boundd b b a f d is givn b th ara f, a, b and a b Divid th ara into an vn nubr of strips n, ach of width h. Thr will b n ordinats f...,,, n a b Th valu of th intgral is givn approiatl b b a f d h.. n n.. n whr b a h n Eapl Evaluat 5 Strip width = = 9 = = =.665 = =.6555 = 5 =.8798 = 7 =.6 5 = 9 5 = = 6 = = 7 = = 5 8 = 5 9 d using ight strips

10 5 65. d Paratrics. Equations of th for f () or f (, ) ar calld artsian quations. Eapl, 5. Equations of th for f ( t), g( t) whr t is a third variabl ar calld paratric quations; t is th paratr Th dfin a curv which has points with coordinats of th for f ( t), g( t). As t varis th curv is dfind. Paratric diffrntiation Whr t is a paratr d dt d dt Eapl Givn that t and t obtain an prssion for (i) d d (ii) in trs of t. d (i) d dt t dt d t t (ii) d d d d d d dt d dt d t t t Tangnts and noral to a curv at a spcific point Find th gradint through diffrntiation, and us Eapl Find th quations of th tangnt and noral to th graph of at th point whr

11 Whn,. d d ( ) and Tangnt Noral Grad = Intgration tchniqus Intgration b substitution th substitution will b givn Eapl Find (i) sin d b substituting u cos (ii) d (i) sin d sin sin (ii) cos u u u d b substituting u sin d sin d u A cos cos A In a dfinit intgral th liits ar changd according to th substitution d u u u u u u cos d d,, u u

12 Intgration b parts This thod is usd for so procts such as for apl: sin, cos,,, It can also b usd to intgrat, for apl:, sin, tan d Forula u d uv Eapl sin d sin d v d d cos cos cos cos d cos sin.d u d, sin d v cos d In words, for th valuation of first scond intgral First tis of scond inus Intgral drivativ Intgral of Tis alra found of first Eapl d Whthr ou us th forula or words th ordr hr has to b changd d d d A 6 A

13 Eapl d tan Hr, introc as th scond d d tan. tan A d d tan tan tan (Us sa thod for d, d sin, tc) Do th! For a dfinit intgral Eapl. d d d Using partial fractions Eapl Evaluat (i) d (ii) d (i) d d

14 (ii) A A 6 A 6 A 6 A d 6 d d d d d Intgration of sin and cos tc Hr w nd th rarrangd doubl angl forula for cos i.. cos, cos sin cos Eapl d cos d cos sin d sin sin Using th tabls of standard drivativs and intgrals in th forula booklt Rlvant to or ar drivativs of invrs trig functions, and of (Pag 5) Th rsults can b rvrsd. sc, cot and cosc

15 Eapl d sin Eapl cosc d cot Eapl d tan On pag 6 th intgrals of tan, cot, cosc and sc ar rlvant to or Volus of rvolution Whn th shadd rgion is rotatd through c about th volu of rvolution will b givn b a b V a b d b a.about V b a Eapl Find th volu of rvolution of th graph of cos fro to about through c, V cos d cos d sin sin sin

16 Diffrntial quations () Equations of th for d f () or g ( ) intgrat at onc d d Eapl d * Eapl d d sin cos A d sin A B * * Ths ar th gnral solutions of th diffrntial quations whr A, B, ar arbitrar constants. Particular solutions to diffrntial quations can b found if boundar conditions ar givn d Eapl Solv th quation t dt d dt t t t A t, A A t t givn that whn {gnral solution} t, () Equations which rc to f ( ) d g( ) ar calld variabls sparabl. Intgrat both sids but includ just on arbitrar constant. () Eapl d d A c A () An iportant application is to rat of growth and rat of dca. It is iportant to rcall that a rat of chang (RO) (with rspct to ti unlss othrwis spcifid) is a drivativ with rspct to ti). RO positiv thr is growth (incras) RO ngativ thr is dca (dcras)

17 Eapl Th rat of dca of a crtain radioactiv lnt at an ti is proportional to th ass of th lnt raining at that instant. Aftr das, on third of a givn ass has disintgratd. How uch is lft aftr a furthr das? Lt b th ass raining at ti t. Th initial ass is (whn t ) Dca iplis loss of ass, and rat of dca is givn b a drivativ d dt Sparat variabls k whr k d kdt d kdt kt A whn t A kt kt kt kt kt whn t k k k t Whn t 9

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv