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 blow th ais in th ais. f () f () f () f () f () Equations of th form f ( ) k ; whr k is a positiv (obviousl) constant Solv f ( ) k, f ( ) k for solution st. Eampl, b b ac a 8 7 Inqualitis of th form f ( ) k or f ( ) k f () will most likl b linar.
Pag Eampl Fin th rang of valus of for which (i) (ii) 5 (i) or (ii) 5 5 5 6 Subtract b Equations f ( ) g( ) an alli inqualitis. A graphical approach is ssntial hr. Eampl (i) Solv th quation (ii) Fin th rang of valus of for which First raw a sktch of an on th sam as th graph of. For point A A (ii) NB Clarl from th iagram, solving has no rlvanc!
Functions Pag A function of ma b rgar as a rul for mapping input valus (of ) to output valus (of ). Thr is a rstriction that for ach input valu thr will b onl on output valu. Input valus ar call th omain of th function. Output valus ar call th rang of th function. A function might b a on to on mapping, or a man to on mapping. Eampl f : for Rang Domain, Illustrats on to on Eampl f ( ) for {i.. taks all ral valus} Illustrats man to on Rang - (-, -) Domain, Composition of functions Th composit function fg is unrstoo to man g first an thn f. Eampl If f ( ) an ( ) g (unspcifi omains) fg ( ) f ( ) gf ( ) g( )
Pag Th invrs of a function Th invrs of f is writtn f an is fin b f (i) f ( ) ff ( ). An invrs function onl ists for a on to on mapping (ii) To obtain f ( ) first chang variabls in f () an rarrang for. Th st of numbrs which is th omain of f bcom th rang of Th st of numbrs which is th rang of f bcom th omain of (iii) On th graph of f (), using qual scals on th as. f f (a) rflct in to obtain th graph of f ( ) (b) a vrtical asmptot k on f () bcoms a horizontal asmptot k on f ( ) (c) a horizontal asmptot h on f () bcoms a vrtical asmptot h on f ( ) (iv) Th quations f ( ), f ( ) an f ( ) f ( ) will all hav th sam solutions. This follows sinc an intrsctions of f () an f ( ) occur on. Eampl f ( ) for (a) sktch th graph of f () an stat th rang of f (b) sktch on th sam as th graph of f ( ) an stat th omain an rang of f (c) fin f ( ) () solv th quation f ( ) f ( )
Pag 5 (a) rang of f: f () (b) Domain of f : f ( ) rang of f : (c) For f f ( ) () Bst to solv Eponntial an Log graphs = ln is a horizontal asmptot NB ln is th invrs function of, an vic vrsa. is a vrtical asmptot
Pag 6 Natural Logarithms (bas onl!) Dfinition ln N N. ln is usful {liminating N from th abov} ln N N {liminating from th abov} Transformations of th graph of f () a f a involvs a translation f ( ) a involvs a translation a af () involvs a strtch with scal factor a in irction f (a) involvs a strtch with scal factor a in irction f () involvs a rflction in. f ( ) involvs a rflction in. Eampl Show how th graph of can b obtain from th graph of translat b using suitabl transformations,, translat [NB This isn t th onl combination; tr anothr!]
Eampl () Pag 7 f is fin for as shown. On sparat iagrams sktch th graphs of (i) f ( ) (ii) f ( ) (i) Strtch - - Translat - - - - (ii) Rflct in Strtch Translat - - - - - - NB Tr th abov, stp b stp, on graph papr.
Calculus Tabl of rivativs NB a n ln sin cos tan sc cos Pag 8 an cos sin sc n From P * sc cos Chain rul othrwis known as th function of a function or composit rul. Eampl (i) (ii) (i) Obtain th rivativ of (iii) ln Lt u u u (iv) sin (v) cos
8 Pag 9 Thr ar thos of ou who will o ths without introcing u. iffrntiat with rspct to brackt rivativ of brackt Th rmainr will b on in this wa, but th substitution will also b givn for thos who woul prfr to us it. (ii) {ssntial to insrt brackts!} (iii) ln u (iv) sin sin u cos cos u cos cos (v) cos sin cos sin u cos
Pag Proct rul For uv Eampl v u Fin th rivativs of (i) (iii) v (ii) ln cos (iv) 5 {A formal us of th formula isn t ncssar, unlss ou insist!} uv v u v (lav st iffrntiat n ) + (lav n iffrntiat st ) (i). u, v v (ii) ln ln ln u, v ln v (iii) cos sin sin cos cos u, v cos v sin
(iv) 5. Pag 6 Quotint rul For u v v u v v 5 u, v v 5 5 Eampl Diffrntiat with rspct to (i) ln (ii) (iii) {Again, us a formal approach if ou wish, as in (i) blow} (i) v v u v u, v v
Pag (ii) ln {without th u an v} ln ln ln ln (iii)
Tabl of intgrals Pag n sin cos c n n ln cos sin n fromp An important tchniqu will b us with stanar intgrals from th givn tabl. If f ( ) F( ) C thn f ( a b) F( a b) C a Eampl C C 8 C Eampl ln C Eampl ln C c Eampl Eampl sin cos C sin cos C cos cos sin C C
Pag C sin cos Eampl Fin th intgral of 6 C C 6 6. 6 6 6 Eampl Evaluat. 6 Log intgrals () C f f f ) ( ln ) ( ) ( Eampl C ln Eampl C ln sinc C sinc C
Pag 5 Eampl C ln cos sin Eampl cot lnsin C Log intgrals () You will notic that log intgrals in th ata booklt inclu molus signs. Ths ar not gnrall ncssar, but.. Eampl ln ln ln ln Eampl ln ln ln Rtrospctivl hr. Introc hr, an ln ln ln {it has bn plain wh this procr is accptabl} Implicit Diffrntiation So far w hav mt curvs with cartsian quation in th form f () i.. is prss plicitl in trms of. Som curvs can t convnintl b prss plicitl in this wa whn th rlationship btwn an is contain implicitl in an quation..g. a circl 5
Pag 6 Using as an oprator () f ( ) f ( ) f ( ) f ( ) chain rul Eampl, () Appling to procts Eampl.. () Tangnts an normals Eampl Fin th quation of th tangnt to at th point (, ),, At (, ), 6 Equation
Pag 7 Intgration tchniqus Intgration as th rvrs of iffrntiation Eampl Fin 5. Hnc valuat 5 5 5 Numrical Mthos Solving quations using itrativ procrs Intrval bisction Eampl Show that th quation 6 has a root btwn. an. 8. Fin an intrval of with., which contains using th bisction mtho. f ( ) 6 f (.8).6 Sign chang.. 8 f (.).856
Pag 8 f (.6)..6. 8 f (.7).6.7. 8 Itrativ formula mtho Eampl (i) Show that th quation 6 has a root nar to.5 (ii) Show that 6 is a rarrangmnt of th quation. (iii) Us th itration n to fin th valu of 6 corrct to 5 cimal placs. (i) {NB f (.5). 565 tlls us nothing. Us an intrval which contains.5} f ( ) 6 n f (.).6 f (.6).77 Sign chang.. 6 i.. nar.5 (ii) 6 6 6 6 (iii). 5.97....5 6.9... Last two agr to 5 p. 9
Pag 9 Simpson s rul for approimat intgration f Th finit intgral b a f is givn b th ara boun b f, a, b an a b Divi th ara into an vn numbr of strips n, ach of with h. Thr will b n orinats f...,,, n a b Th valu of th intgral is givn approimatl b b a f h.. n n.. n b a whr h n Eampl Evaluat 5 Strip with = = 9 = = =.665 = =.6555 = 5 =.8798 = 7 =.6 5 = 9 5 =.58899 6 = 6 =.58576 7 = 7 =.7958 8 = 5 8 = 5 9 using ight strips
5 65. Pag 5.65..58..87.58.795 9 Paramtrics. Equations of th form f () or f (, ) ar call cartsian quations. a. Eampl, 5. Equations of th form f ( t), g( t) whr t is a thir variabl ar call paramtric quations; t is th paramtr Th fin a curv which has points with coorinats of th form f ( t), g( t). As t varis th curv is fin. Paramtric iffrntiation Whr t is a paramtr Eampl Givn that t an t obtain an prssion for (i) (ii) t t in trms of t. (i) t t t t t (ii) t t t t t
Trigonomtr Thr furthr trig ratios sc, cos Pag cosc, sin Graphs Dc from thos of cos, sin an tan Eampl sc cot tan cos - cos Intitis cot, sc tan, sin cosc cot Eampl Solv th quation tan sc for answrs corrct to cimal placs. tan sc sc sc sc sc sc sc sc, S A PV =. cos,.,. 9 S A PV =.9 T C T C
Drivativs Pag sc cos cos sin Similarl sin sin. cos cos cos cosc cot cosc tan sc cot cosc Do thm! Invrs trig functions ar fin hr formall as on to on mappings Graphs (i) f ( ) sin has omain - Rang (ii) f ( ) cos has omain Rang -
Pag (iii) f ( ) tan has omain ar horizontal asmptots Rang An valus of which ar f into ths functions will giv output valus corrsponing to thos from our calculator. Th ar call principal valus. Drivativs Lt sin sin cos cos sin
Similarl cos Pag tan Do thm!