Mathematics 1 Outcome 1a. Pascall s Triangle and the Binomial Theorem (8 pers) Cumulative total = 8 periods. Lesson, Outline, Approach etc.

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1 prouce for by Tom Strag Pascall s Triagle a the Biomial Theorem (8 pers) Mathematics 1 Outcome 1a Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 Itrouctio to Pascal s Triagle via routes alog a set of roas leaig to (a + b) 1,, 3,... Defie! (-1)(-).3..1 (calculator) ( + 1) 5, ( 4 1) 3, ( + 5) 6, ( + 1 ) 4, ( ), ( 3 + 1). etc + Pg3, E1, Qu Defie C r as choosig r from ; C r r! ; r!( r)! Show quick way of fiig : Go over 10! !! 3 1 r 1 r + r ; a r + 1 r Pge 7, E B Qu 1a,, 4a,b 5a, 6a, 7a,b () 4 3 Look at 1 0, 4 4 1, 4 6, 4 4 3, a compare with Pascal s triagle. Itrouce to Biomial Theorem 1 1 > ( a + b) a + a b + a b +... b Epa (1 + 3 ) 5 up to term i 3 r r 5 Defie Geeral term Tr + 1 a b a its importace r i fiig particular terms like the term i 7 i (3+) or the term iepeet of i + ( ) 6 Show how to fi term i 3 i ep of ( )(1 + ) 7 Use Raibows Pge 9, E 3A Pge 5, E, Qu 1 Qu 1a- a(i), (iii), & b 6 ( 1 + ) up to 3, ( + 3 ) up to 3 7, ( ) up to Pge 9, E 3A Qu 3a-, 4f Pge 11, E 3B Qu 1a, c, 4a,b, 5a, 6 7 Obtai (0 97) 6 correct to 3 ecimal places Pge 13, E 4 Qu 1 Pge 5, E Qu -4(b) 8 Checkup a Rou-up Cumulative total 8 perios This is page umber 1

2 prouce for by Tom Strag Partial Fractios (4/5 perios) Mathematics 1 Outcome 1b Lesso, Outlie, Approach etc. Nelso MIA - AH M1 P( ) 1 Defie Ratioal Fuctio as f() Q ( ) where P() a Q() are polyomials a show that Pge 18 E Page 7 E 1 Qu 1, 5, 1, 18, 19 Qu 1a, b, c, > i reverse + 1 ( + 1)( ) ( + 1)( ) + (Partial Fractios) + 1 Deal with repeate fractios i eomiator : A B C + + ( + )( 1) + 1 ( 1) etc. Pge 19 E 3 Pge 8 E Qu 1, 3, 5, 10 14, 18 3 Deal with irreucible quaratic factor i eomiator: A B + C + etc. ( + 1)( + + ) Pge 0 E 4 Pge 9 E 3 Qu 1, 5, 7, 9, 11* * ees polyl rem theorem 4 Go over ee to ivie out if egree of umerator is greater tha or equal to that of eomiator : etc ([A/B] Pge E 6 Qu 1 a, b, e, j, l 5 Review Cumulative total 13 perios This is page umber

3 prouce for by Tom Strag Differetial Calculus (8/9 pers) Mathematics 1 Outcome a Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 First Pricipals > f ( ) Lim h 0 f( + h) f( ) Go over fiig f ( ) for f() 4, 3 +, (Not teste formally, but shoul uersta). Go over basic S5 rules for ifferetiatio a show importace of chai rule 3. Itrouce Prouct Rule 4. Itrouce Quotiet Rule 5 Defie secatθ secθ c b 1 cosθ etc. use Graphics calcr to sketch graphs of secθ Look at simple properties > secθ 1 Show (ta ) sec, (cos ec ) cosec cot, h 5 etc c θ b (sec ) sec ta (cot ) cos ec a Page 9 E 1A Page 11, E 1 Qu 1, 4, 5, 7 Page 3 E 3A Page 1 E Qu 1a, a,c, Page 13 E 3 3a, 4a, 6a Page 3 E 4A Page 14 E 4 Qu 1, b, 3 Page 36 E 4B Qu 1b, 3, 4 Page 37 E 5A Page 15 E 5 Qu 1,, 3, 4, 6 Page 38 E 5B Qu 1-3 Page 40 E 7 Page 18 E 6 Qu 1,, 3a,c,e,g, 4a Qu Remi about Epoetial fuctio f() a (a e ) Differetiate from 1st P > f e ( ) e use spreasheet to show Lim e h 0 h 1 1 h Page 43 E 8A f() e, e Qu 1a,c,e, a, 3e e si, e 4a-c, 5b,, 6a, Prove also that (l ) 1 7 Go over (l ), (l ), (l(cos )) l Remi e l( e ) 8. Defie y, 3 y 3, 4 y, 4 f ( ), f ( ) etc. erivative test for ature of statioary poits Page 43, E 8A Page 18, E 6 Qu 1b, b,c, 3a,b,c 4,e 5a,c,e 6b,c,e (If time > E 4B Qu 1) Page 46 E 9A Qu 1,, 3, 4, 6 9. Review Cumulative total perios This is page umber 3

4 prouce for by Tom Strag Applicatios of Differetial Calculus (5/6 pers) Mathematics 1 Outcome b Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1/ Go over (isplacemet) (s i Physics) v t a v t spee (velocity) acceleratio t Page 51 E 1 Page 0 E 7 Qu 1a, b,, f, a, c, e 3, 4, 6, 7, 8, 10, 1 Home eercise 1 o all topics to ate 3/4 Go over local maima/miima Do t go ito i aythig like epth of MIA Book Cover split omai fuctios (piece-wise) Basically fi ma / mi i a close iterval (a, b] etc Stuy (i) local ma/mi, (ii) e poits (ot a above) a (iii) critical poits a erivative test (where appropriate) for ature. (pg56) Page 56 E Page 4 E 8 Qu 1, 3 a,c,e,g,i page 60 E 3 Qu 1 a,c,e, 5 a- 5 Optimisatio - as for Fifth Year work with ew ifferetiable fuctios Chose eamples carefully from Nelso Page 63 E 4A Page 5 E 9 Choose 5 or 6 eamples from E 4A/4B (har) 6 Review Cumulative total 8 perios This is page umber 4

5 prouce for by Tom Strag Itegral Calculus (10/11 perios) Mathematics 1 Outcome 3 Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 Go over iea of ati-erivative a area uer a curve (See page 69 for eplaatio of area) Page 70 E 1A a 1B Page 9 E 1 + special rule for itegratio:- ( a + b) etc. 1 l + c, sec ta + c a + b 1 a b e c a e , a b l( a + b) + c etc + a e e + c, Page 7 E A Page 31 E + some of B if time Page 33 E 3 (some) 3. Itegratio by substitutio Simple oes by ispectio More complicate - substitutio will be give Go over :- 4 ( + 5 ), , 4 si cos page 74 E 3 Page 36 E 4 Qu s - o umbers 4 Itegratio by Substitutio part - substitutios give l Go over, give u l Go over 4, give siu a cosuu 5 Show si 5 4 si si si ( 1 cos ) half of perio go over efiite itegratio a chagig the limits for the ew variables. 6 Special case f ( ) f( ) l f() + c 3 e Go over, 3 1, ta 3 e + 1 etc 7 Areas uer curves a betwee curves (rev of S5 work) 8 Areas betwee curves a y -ais 9 Volumes of solis of revolutio 10 Go over motio :- att () vt () a vtt () t () See eamples o page 88 Page 75 E 4A Page 37 E 5 Qu s - O Numbers Page 38 E 6 Page 76 E 4B Qu 1-5 Page 77 E 5A o o s + 1 or from E 5B page 80 E 6A Qu 1 c,, 10 + few more Some of E 6B if time Page 40 E 7 Page 41 E 8 page 4 E 9 Page 89 Qu 4, 5, 10 Page 44 E 10 Page 88 E 10A Qu 1,, 4, 6, 7, 8 (Rest of E 10A + some of E 10B if time) 11 Review Cumulative total 39 perios This is page umber 5

6 prouce for by Tom Strag Fuctios a Graphs (6/7 pers)-graphics Calc s Mathematics 1 Outcome 4 Lesso, Outlie, Approach etc. Nelso MIA - AH M1 1 Revise Fuctio work from Higher + go over iverse fuctio a how to fi f 1. Show also relate graphs of iverses Page 97 E 1 orally Page 100 E 3 Qu 1 a,e,g,h e,c,g + 3 Iverse Trig Fuctios y si 1 etc Go over graphs of cos 1, ta 1, e, f() & omais Go over EVEN a ODD fuctios 3 Vertical asymptotes of ratioal fuctios f() g ( ) h ( ) V.A. occurs at a if h(a) 0 Discuss what happes as > a + a a -. Remi to ivie if eg g() eg h() Page 10 E Qu 1 & page 68 Qu 1 Page 99 E 1 Qu, 5, 6, 7 Page 68 Qu 5 Page 108 E 8 Qu 3 Page 109 E 9 Qu 1 Page 50 E 1 4 Horizotal asymptotes if eg g() eg h() a slopig asymptotes if eg g() eg h() Sketch graphs of f() usig V.A, H.A. S.A, T.Pt, Roots a y-itercept 6 Relate Graphs - from f(), sketch f( - a) af() f(a) f() + a, f 1 ( ), f ( ) etc 7 Review Page 110 E 10 Page 5 E Qu 1 a, b, g, f, k, l Page 11 E 11 page 57 E 3 Qu 1 a, c, e, g, i, k, m Page 114 E 1A Page 68 E 3 Select carefully from Page 117 Cumulative total 46 perios This is page umber 6

7 prouce for by Tom Strag Matrices - Systems of Equatios (4/5 pers) Mathematics 1 Outcome 5 Lesso, Outlie, Approach etc. Nelso MIA - AH M1 Some of the Matri work from Maths 3 ca be icorporate i this chapter if you wish to save time later e.g. iverse of a matri, et A a iverse of a 3 3 1/ Defie a matri, elemet, orer, traspose etc Solve a system of equatios ( a 3 3) i matri form by row operatios to reuce :- a b c a b c e f p to 0 j k r g h i q 0 0 m s a hece fi the solutio Gaussia Elimiatio - Augmete form etc 3 Show that some systems have ifiite solutios a some have oe e.g. Page 17 E 4A Page 78 E 1 Qu 1 a page 130 E 5 Qu 1 a - c, + y + z 6 + y + z 6 + y z - + y z y z y z 3 (ifiite set) (o sols - icosistet) 4 Ill Coitioig (occurs whe small chage i value of a coefficiet(s) results i large chage i solutio. Page 137 Qu page 79 E 5 Review For Sessio E of Mathematics 1 Cumulative total 51 perios Assumig 5 perios per week a weeks i Jue, this uit shoul e arou Thursay 31st October. >3 perios revisio (icluig specime NAB3) + 1 perio for test 55 perios TEST arou Friay November Actual Test Date This is page umber 7

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x

Rational Function. To Find the Domain. ( x) ( ) q( x) ( ) ( ) ( ) , 0. where p x and are polynomial functions. The degree of q x Graphig Ratioal Fuctios R Ratioal Fuctio p a + + a+ a 0 q q b + + b + b0 q, 0 where p a are polyomial fuctios p a + + a+ a0 q b + + b + b0 The egree of p The egree of q is is If > the f is a improper ratioal