Thomas Whitham Sixth Form

Size: px

Start display at page:

Download "Thomas Whitham Sixth Form"

Dustin Sutton
5 years ago
Views:

1 Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui

2 Pag Algbra Moulus 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.

3 Pag Eampl Fin th rang of valus of for which (i) (ii) 5 (i) or (ii) 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. A For point A (ii) NB Clarl from th iagram, solving has no rlvanc!

4 Pag Functions A function is a rul for mapping input valus (of ) onto output valus (of ). Thr is th rstriction that for ach input valu thr will b onl on output valu. Hnc on to man mappings can t b functions. Functions ar ithr on to on or man to on. Input valus form a st call th omain of th function. Intrval notation might b us for th omain. ( ) ) ( ( ) ) ( ) Onc th omain is fin, th output valus follow. Th st of output valus is call th rang of th function an it too might b rfrr to as ( ) an ( )... familiar notation. f for ) Eampl (-,) Rang (,) Domain, Th rang of f, in intrval form ) Eampl f : for sktch th graph of ( ) an stat th rang of f.

5 Pag Th notation us hr is mapping notation raing f is such that is mapp to an th omain is Putting i.. ( ) i.., locats th intrsctions with an th turning point at (, ), a maimum. Whn, (, ) (, -) Th rang of f is or 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( ) Th omain of fg Larn as bing th st of all in th omain of g for which ( ) is in th omain of f Eampl f g for ( ) an for )

6 Pag 5 (i) Fin th rang of g (ii) Fin ( ) an fin th omain of. (iii) Fin th rang of. (i) Th omain of g is. Th rang of g is Or in intrval form ( (ii) If our can t writ th omain own consir ths numbrs from th omain of g.... output ( ) F can accpt ths th omain of fg is ) F can t accpt ths (iii) Th rang of fg is or in intrval form ) Eampl f for ) an (i) Writ own th rang of g. (ii) Writ own th omain of fg. g for ( ) (i) Th rang for g is ( ) (ii) F can onl accpt numbrs which ar Hnc th omain of fg is )

7 Pag 6 Not sur? Thn tr th following tabulation of som valus of from th omain of g, an outputs output ( ) F can t accpt ths Th invrs of a function Th invrs of f is writtn F can accpt ths 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.

8 Pag 7 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 ( ) (a) rang of f: (b) Domain of rang of f : f : (c) For f f ( ) () Bst to solv =

9 Eponntial an Log graphs Pag 8 ln is a horizontal asmptot NB ln is th invrs function of, an vic vrsa. is a vrtical asmptot 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.

10 Pag 9 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 f () is fin for as shown. On sparat iagrams sktch th graphs of (i) f ( ) (ii) f ( ) (i) Strtch - - Translat

11 (ii) Rflct in Strtch Translat Pag NB Tr th abov, stp b stp, on graph papr.

12 Calculus Tabl of rivativs NB a n ln sin cos tan sc cos Pag an cos sin sc n From c * sc cos Chain rul othrwis known as th function of a function or composit rul. u u Eampl (i) (ii) (i) Obtain th rivativ of (iii) ln Lt u u u u u (iv) sin (v) cos

13 u 8 u Pag Thr ar thos of ou who will o ths without introucing 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

14 Pag Prouct rul For uv Eampl Fin th rivativs of (i) (iii) ln (ii) cos (iv) 5 {A formal us of th formula isn t ncssar, unlss ou insist!} uv u v v u v u u v (lav st iffrntiat n ) + (lav n iffrntiat st ) (i). u, u v v (ii) ln ln ln (iii) cos sin sin cos cos u u, v cos v sin

15 (iv) 5. Pag 6 Quotint rul For u v u v u v v 5 u 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) u v v u v u u, v v

16 Pag 5 (ii) ln {without th u an v} ln ln ln ln (iii)

17 n Tabl of intgrals c n n ln sin cos cos sin n Pag 6 ( ) ( ) fromp ( ) ( ) ( ) ( ) ( ) Th alli tabl might b larnt. Othrwis us th first tabl along with th following as in th first ampls. If f ( ) F( ) C thn f ( a b) F( a b) C a Eampl C C 8 C Eampl ln C ln C Eampl c

18 Eampl Eampl Pag 7 sin cos C sin cos C cos cos sin C cos sin C C Eampl 6 Fin th intgral of C 6 C 6 Eampl Evaluat. 6

19 Pag 8 Log intgrals () f ( ) ln f ( ) C f ( ) ln Eampl C ln Eampl C Eampl C ln cos cot ln sin sin Eampl C Log intgrals () You will notic that log intgrals in th ata booklt inclu moulus signs. Ths ar not gnrall ncssar, but.. Eampl ln ln ln ln Eampl ln ln ln Rtrospctivl hr. Introuc hr, an ln ln ln {it has bn plain wh this procur is accptabl}

20 Pag 9 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 Using as an oprator () f ( ) f ( ) f ( ) f ( ) chain rul Eampl, () Appling to proucts Eampl.. () Tangnts an normals Eampl Fin th quation of th tangnt to at th point (, ),,,

21 Pag At (, ) 6 Equation Intgration tchniqus Intgration as th rvrs of iffrntiation Eampl Fin 5. Hnc valuat 5 5 5

22 Pag Numrical Mthos Solving quations using itrativ procurs 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 f (.6) f (.7) 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 with =.5 to writ 6 own from our calculator an corrct to 6 cimal placs, an show that this is th valu of corrct to 6 cimal placs. (i) {NB f (.5). 565 tlls us nothing. Us an intrval which n contains.5} f ( ) 6 f (.).6 f (.6).77 Sign chang.. 6 i.. nar.5

23 Pag (ii) (iii) f f sign chang Simpson s rul for approimat intgration to 6 p 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,,..., n n- n a b

24 Pag Th valu of th intgral is givn approimatl b b a f whr h.. b a h n.. n Eampl Us Simpsons rul with 7 orinats to fin an approimat valu for Strip with = (6 strips!) I NB No rouning off at th tabulation stag. n 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 t t n

25 Pag Eampl Givn that t an t obtain an prssion for (i) (ii) in trms of t. (i) t t t t t (ii) t t t t t Trigonomtr Thr furthr trig ratios sc, cos cosc, sin Graphs Duc from thos of cos, sin an tan Eampl sc cot tan cos -

26 Pag 5 Intitis cos 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 T Drivativs cos,.,. 9 sc cos cos sin Similarl A C sin sin. cos cos cos PV =. cosc cot cosc tan sc cot cosc S T A C PV =.9 Do thm!

27 Pag 6 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 - (iii) f ( ) tan has omain ar horizontal asmptots Rang

28 Pag 7 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 tan Do thm! Disproof b countr ampl Eampl Show b countr ampl that th following intitis ar fals. (i) log log log (ii) cos cos (iii) a b a b

29 Pag 8 (i) Lt an log log. RHS = log log LHS RHS Hnc intit is fals. LHS = (ii) Lt cos cos9 LHS = RHS = LHS cos RHS Hnc intit is fals. (iii) Lt an LHS = 5 RHS = LHS RHS Hnc intit is fals.

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